Relativity

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HISTORICAL INTRODUCTION

Lord Kelvin

writing in

in

1893,

his

preface to the

English edition of Hert/ s Researches on Electric

many workers and many

"

says

nineteenth

build up the

ether

for

heat,

light,

have

thinkers school

century

one

and

the

German and English volumes containing Hertz

Ten years first

"the

sight

about

the

world

electrical

last

decade of the of the splendid

the

realised.

will

we

1905, be

find

to

proved

revolution

in

Einstein

be

declaring

At

superflous."

scientific

thought brought decade appears to be almost more careful even though a rapid review

in the course of a single

A

subject

however, show

will,

Relativity gradually became a

Towards the

beginning

the luminiferous ether

came

Young and

elastic

medium

solid in

stationary

Fresnel.

ether

as

of

the

wave

In

its

light

nineteenth

theory

of

waves may

shown by

satisfactory explanation of

the Theory of

in

stationary

was the outcome

which the ether,

how

historical necessity.

century,

into prominence as a result of

the brilliant successes of the of

;

s

monument

in

permanent

later, in

ether

too violent.

of the

a

now

cons jmmation

that

the

to

given

papers,

century, will be

to

of plenum,

magnetism

electricity,

Waves,

helped

hands the

the search for a

"undulate."

Young,

astronomical

the

aspect

also

This

afforded

aberration.

a

But

to a host of new its ver\ success; g:ivc rise questions all bearing on the central problem of relative motion of ether

and matter.

11

I

lilXC

OK KKI.m

II I.F.

The

Aragd n prism experiment.

\

m of a

refractive index

prism depends on the incident velocity of light outside the prism and its velocity inside the prism after On Fresnel s fixed ether hypothesis, the refraction. glass

incident

light

waves are situated

in the stationary ether

outside the prism and move with velocity If the prism moves with to the ether.

with

c

a

respect

u

velocity

respect to this fixed ether, then the incident velocity

with

should be

of light with respect to the prism

+ H.

C

Thus

the refractive index of the glass prism should depend on velocity of the prism,

absolute

the

its

velocity

,

with

Arago performed the experiment

respect to the fixed ether. in

i.e.,

1819, but failed to detect the expected change.

Boscovitch

Airy- Boacovitc/i water -telescope experiment.

had

earlier

still

i7b

in

t>,

the

raised

important

very

question of the dependence of aberration on the refractive

index

the

of

depends on

medium

the

the telescope and

filling

the

Aberration

telescope.

difference in the velocity of light outside its

the telescope.

velocity inside

If

the

changes owing to a change in the medium filling the telescope, aberration itself should change, that is, aberration should depend on the nature of the medium.

latter velocity

up a telescope with water but any change in the aberration. Thus we the case of Arago pri^m experiment and

Airy, in 1871 failed

to

filled

detect

get both

in

Airy- Boscovitch

seem

to

be

medium with Fresnel

s

quite

independent of

conrection

this

hypothesis, Fresnel

implies

According

a definite

s

is

a

the

of

velocity

/{=]

his

.

Possibly

\Vorking on

famous ether convec

to Fresnel, the presence of

condensation

the

2

//x

taking place.

Offered

very

moving medium

stationary ether.

coefficient

some form of compensation tion theory.

in

effects

respect to Fresnel

the

experiment,

water-telescope

startling result that optical

of

ether

matter

within

the

IIISTOKK

occupied portion of

with

its

own

This

matter.

by

region e\cr<s

OIU TI ION

I.VII.

\l,

ether

is

moving

of

pice,

"

condensed

"

he

to

supposed

111

matter.

or

awav

carried

should

It

be

observed that only the "excess" portion is carried away, A complete while the rest remains a- -taunant ;is ever. convection of the

u

is

ether

2

l

(p.

/fj.

convection

this

if

of

the .

k

u. is

then

prism),

within the moving

refraction

after

of light

velocity

coefficient

being the refractive index of the

the velocity

full

convection

partial

fraction of the velocity

a

only

showed that

Fresnel l

with

p,

ether p with the

to a

equivalent

optically

total

"

"excess

prism would be altered to just such extent as would make the refractive index of the moving prism quite indepen dent of

its

"absolute"

on

of aberration

easily explained

The non-dependence

u.

velocity "

absolute

"

the

u, is

velocity

with the help of

also very

this Fresnelian convection-

coeflicieut k. It should be

Stoke** viscous ether.

that Fresnel at

all

s

stationary ether

remembered, however,

absolutely fixed and

is

disturbed by the motion of matter through Fresnelian ether cannot be said

respect

fashion,

physical

respectable

and

to

this

it.

behave

led

is

not

In this in

Stokes,

any in

more material type of medium. Stokes assumed that viscous motion ensues near the surface 1845-H),

of

of separation

He

undisturbed.

efl

cct.

But

motion

order

in

to

while

remains

at

wholly

a viscous ether

would

were differentially

in it

explain

the

null

Arago

compelled to assume the convection of Fresnel with an identical numerical value

Stoke:-

liNpothesis for k,

all

matter,

ether

how such

>howed

if

moving the

unions

aberration

irrotational.

and

ether

distant

sulficifiitlv

explain

a

to construct

namely

ua.-

I

./

convection-cocllicient

-.

Thus

\va>

the prestige of the Fresnelian

enhanced,

theoretical inveBti&fctiona o!

stokes.

il

anything,

by

the

IV

KINI

I

U

Soon

Fizratf* experiment.. direct

OK RELATIVITY

l.K

after,

confirmation

experimental

in

a

in

J8ol,

received

it

brilliant

piece

of

work by Fizeau.

beam

If a divided

of

light

re-united

is

through two adjacent cylinders

filled

interference fringes will be produced. of the cylinders

is

ether within the

now made

interference

shift actually observed agreed very

k=l

l

1

l^

.

The Fresnelian

became firmly established

as

the

"

in

one

condensed

"

would be convected and

the

in

passing

ordinary

water

If the

to flow,

flowing water

would produce a shift

after

with water,

well

fringes.

The

with a value of

convection-coefficient

now

consequence of a direct

a

On the other hand, the negative evidences favour of the convection-coefficient had also multiplied.

positive effect. in

Mascart, Hoek,

changes

Maxwell and others sought

in different optical effects

induced by

for definite

the

But

of the earth relative to the stationary ether.

motion all

such

attempts failed to reveal the slightest trace of any optical disturbance due to the "absolute" velocity of the earth thus proving conclusively that all tne different optical effects shared in the general compensation arising out of the Fresuelian convection of the excess ether. It must be earefully noted that

the

Fresnelian

convection-coefficient

implicitly assumes the existence of a fixed ether (Fresnel) or at least a wholly stagnant

medium

at sufficiently distant

regions (Stokes), with reference to which alone a convection

any significance. Thus the convectionimplying some type of a stationary or viscous,

velocity can have coefficient

yet nevertheless satisfactorily all Micftelxoti-

"absolute"

known

M or ley

ether, succeeded in explaining

optical facts

Experiment.

down In

to 1880.

1881,

Michelson

and Morley performed their classical experiments which undermined the whole structure of the old ether theory and thus served

to introduce the

new theory

of

relativity.

1NTIIOIH

III.VTOUU-U.

The fundamental In

simple.

all

situated in of

idea

the

of

effect

a

in

piece

away by

Fresnelian

it.

convection

with

quite

of

velocity

was compared

waves actually situated

and presumably curried

is

experiment

the

experiments

ether

free

thift

underlying

old

V

Tlo.N

(

light

the velocity

of moving matter The compensatory

of

ether

afforded a

satisfactory explanation of all negative results.

In the Michelson-Morley experiment the arrangement is If there is a definite gap n a rigid body,

quite different.

in

j

waves situated

light

in free ether will take a

If the

crossing the gap.

platform

rigid

time

definite

carrying

the

motion with respect to the ether in the direc tion of light propagation, light waves (which are even now situated in free ether) should presumably take a longer

gap

set in

is

time to cross the gap.

We

cannot do better than

experiment a river.

may

Eddington

quote

famous experiment.

tion of this

The

"

that

descrip

principle

be illustrated by considering a

It is easily realized

s

of

in

swim

takes longer to

it

the

swimmer

point 50 yards up-stream and back than to a point 50 If the earth is yards across-stream and back. moving to a

through the ether there is a river of ether Howing through the laboratory, and a wave of light may be compared to a

swimmer

travelling with constant

current.

If,

then,

velocity

we divide a beam

relative to the

of light into

two

parts,

and send one-half swimming up the stream for a certain distance and then (by a mirror) back to the starting jx)int,

and

setul

the other

stream and back, the

back

half an

equal distance across

across-stivam

beam should

arrive

first.

Let the ether be the

o

n

apparatus

direct ion

B

the

()

.

with

and

relative to

llowinir

velocity let

OA

-n.

,

in

OB,

the

be

two arms of the apparatus of equal

VI

I

length

/,

OA

being

and back /

8= (1

Let

up-stream.

for the

<

be

the

double journey along

is

= nz

r

if

/c"

)"*,

=

/

+

--

c

where

placed

The time

velocity of light.

OA

KKLATUITY

KINCIL LK OF

+n

rs

^

=

^l

z

2I

B*

c

a factor greater than unity.

For the transverse journey the light must have a compo nent velocity n up-stream (relative to the ether) in order to avoid being carried below OB and since its total velocity :

is <,

its

component across-stream must be

time for the double journey

OB

is

2 \/(c"

),

the

accordingly

But when the experiment was tried, it was found that both parts of the beam took the same time, as tested by the interference bauds

produced."

After a most careful series of

Morley failed to detect the slightest effect due to earth s motion through ether.

The Michelson-Morley experiment seems there

is

Michelson

observations;

and

trace of

to

no relative motion of ether and matter.

any

show that Fresnel

s

stagnant ether requires a relative velocity of 11. Thus Miehelsou and Morley themselves thought at iirst that their e\]H rinient eoniirmed relative

Stokes

motion can ensue

slipping of ether

on

Stokes

off

at a

:i,t

the

:-

theory this

very rapid

rate

-niTrice

viscous as

of separation. Michelson and

ment

viscous

on account

we

ether, in

of

of separation.

How

which abs.MK

the

r

no of

Hut even

of ether would fall

recede from the surface

Morley repeated

their experi

at different heights from the surface of the earth, but

invariably obtained the sttme negative results, thus failing to confirm Stokes

theory of viscous Mow.

his

rotating

in

Further,

o//.

formed

INTKOlim TION

ISTOIMCAI,

II

How

viscous

A

l.V.

-

divided

Lodge per should

i,

wiiich

experiment

sphere

complete absence of any moving masses of matter.

Til

due to

ether

of

beam

of light, after

repeated reflections within a very narrow gap between two massive hemispheres. \\-;is allowed to re-unite and thus

When

the

two hemispheres

produce interference

bands.

are set rotating,

conceivable that the ether in

is

it

due

would be disturbed

would be immediately detected by detect

But

bands.

interference

the gap

viscous How, and any such flow

to

actual

the slightest disturbance

disturbance of the

a

observation of

failed

to

the ether in the gap,

due to the motion

of

ment thus seems

show a complete absence of any viscous

to

tbe

Lodge

hemispheres.

s

experi

flow of ether.

from

Apart

these

experimental

theoretical objections were

urged

discrepancies,

grave

against a viscous ether.

Stokes himself had shown that his ether must be incom pressible

at the

and

all

motion

in

it

differentially

irrotational,

same time there should be absolutely no slipping

Now

the surface of separation.

be simultaneously

satisfied

for

all

at

these conditions cannot

any conceivable material

medium without certain very special and arbitrary assump tions. Thus Stokes ether failed to satisfy the very motive which had lity of

led

Stokes to formulate

constructing a

modified forms of

"

physical"

namely, the desirabi offered

theory which seemed capable of

>tokes

being reconciled with

it,

medium. Planck

the

Micheleon-Morley experiment,

The very complexitv

but required very special assumpt ions.

and the very arbitrariness of these assumptions prevented Planck s ether from attaining any degree of practical importance

The

in

the further development of the subject.

sole criterion of the value of

must ultimately be

its

capacity

any

for

scientific

offering

a

theory simple,

PRINCIPLE OK

Vlll

unified, coherent

In

a

usefulness

and

NATIVITY

K

fruitful description of observed facts.

a theory

as

proportion

It

theory

becomes complex

which

is

obliged

to

it

loses in

a

requisition

whole array of arbitrary assumptions in order to explain special facts is practically worse than useless, as it serves to disjoin, rather than to unite, the several

The

optical experiments of the

last

groups of of

quarter

teenth century showed the impossibility of

facts.

nine

the

constructing a

simple ether theory, which would be simenable to analytic treatment and would at the same time stimulate further It

progress.

shown

should

no

that

be observed

logically

that,

consistent

could scarcely be

it

ether

theory

was

H- A. Wilson offered a consiswhich was at least quite neutral with

possible; indeed in 1910,

sent ether theory respect to is

all

available

almost wholly

optical

negative,

its

data.

only

But Wilson

does not directly contradict observed facts. direct confirmation nor a direct refutation is it

does not throw any light on the various

A

mena.

s

ether

virtue being that

it

Neither any possible and

optical

pheno

theory like this being practically useless stands

self-condemned-

We

must now consider the problem

of relative motion of

ether and matter from the point of view of electrical theory.

From 1860 vector

as

an

became gradually established

as

brilliant

the identity

"displacement

Maxwell and

his

of

light

current"

electromagnetic result of the

a

hypothesis

of

further -analytical investigations.

Clerk

The

became gradually transformed into the electromagnetic one. Maxwell succeeded in giving a fairly satisfactory account of all ordinary optical phenomena elastic solid ether

;vnd little

the

room was

general

any serious doubts as regards Hertz s re Maxwell s theory.

left for

validity

of

in l^Mi, on >lectric waves, first carried out succeeded in furnishing a Mrong experimental confirmation

searches

INTRODUCTION

MMi U,

of Maxwell

s

thenn.

behaved

waves of ven lar^e wave length.

like light

Maxwellian

orthodox

Tlie

waves

Klef-tric

IX

view located

polarisation in fte electromagnetic a transformation of

iVe-ncl

s

the dielectric

her which was merely

el

The Mag

ether.

stagnant

was looked upon as wholly Secondary m origin, being due to the relative motion of the dielectric netic

polarisation

tul.i- of

On

polarisation.

Thomson

in

view

this

comes out to he

vcetion r-octllcient

instead

1880,

of

Fresnelian con-

shown by

as

1 I

as

//*-

This obviously

optical experiments.

the

i,

J. J.

by

required a

complete failure to account for all those optical experiments which depend for their satisfactory explanation on the assumption implies

of a value for the eonvect.ion coefficient equal to

2

l

1

//*

.

The modifications proposed independently by Her!/ and Heaviside

medium

no better.*

fare

to be the seat of

the

emphasised

They postulated the

all electric

reciprocal

polarisation

relation

actual

and further between

subsisting

and magnetism, thus making the field equations more symmetrical. On this view the whole of the

electricity

polarised ether

and

is

consequentty,

becomes unity

carried

the

in this

away by

convection

theory,

a

as that obtained on the original

Thus neither Maxwell

s

the

moving medium,

co-efficient

naturally

value quite as discrepant

Maxwellian assumption.

original theory

nor

its

subse

quent modifications as developed by llertx. and Heaviside succeeded in obtaining a \alne for Fresnelian co-efficient equal to

1 1

1^-

,

and consequently stood totally condemned

from the optical point of view. Certain direct electromagnetic relative

motion of polarised

sive against

experiments

involving-

dielectrics were no less

conclu

the generalised theory of Hertx and Heaviside. * See Note

1.

X

HKIXC1PLE OF RELATIVITY

According to Hertz a moving dielectric would carry away the whole of

its electric

electromagnetic effect be proportional to to

with

displacement near

the

the total

moving

electric

would

displacement, that

specific inductive capacity of

K, the

Hence the

it.

dielectric

the dielectric.

is

In.

1901, Blondlot working with a stream of moving gas could not detect any such effect. H. A. Wilson repeated the experiment in an improved form in 1903 and working

with

found

ebonite

K

portional to

equal to

1

s

This

gives

negative

results.

Rowland had shown due

a

to

the

that

rotating

a

L876

in

K

For gases effect

satisfactory

condenser

was pro

observed effect

instead of to K.

and hence practically no

in their case.

Blondlot

1

is

nearly

will be observed

explanation

of

that the magnetic force dielectric

(the

remaining

K, the sp. ind. cap. On the other hand, Rontgen found in 1888 the magnetic effect due to a rotating dielectric (the condenser remain stationary) was proportional to

K

ing stationary) to be proportional to

K.

Eichenwald

Finally

in

1,

condenser and dielectric are rotated

together,

was quite independent of K, a consistent with the two previous experiments. observed

land effect proportional to

Rontgen

and not

1903 found that when

to

both

the effect

result

quite

The Row

K, together with the opposite K, makes the Eichenwald

effect proportional to 1

independent of K.

effect

All these experiments together with those of Blondlot

Wilson

and

made

effect

due

K

and not to

to

a

it

K

that

the

electromagnetic

dielectric

was

proportional to

clear

moving

by Hertz s theory. Thus the above group of experiments with moving dielectrics 1,

directly

as required

contradicted the

internal discrepancies

Hertz- Heaviside theory. The in the classic ether theory

inherent

had now become too prominent.

It

wao clear that the

III-TOUK

H her

!

;,<(>

outgrown its usefulness. The become too contradictory and too

be

reduced to an organised whole with

heterogeneous to tin-

from the

tion

finally

Radical departures

the ether concept alone.

of

help

xi

i\ii:oi>rcTifW

had

row-opt had

observed

\i.

classical theory

had become absolutely necessary.

There were several outstanding difficulties in connec with anomalous dispersion, selective reflection and

selective

was

the

in

classic

that

evident

the

could

Such

an

be

not

satisfactory

assumption

meduim

assumption

some

of

had

naturally

It

theory.

electromagnetic

in the optical

discreteness able.

which

absorption

explained

kind

become rise

gave

of

inevit

to an

atomic theory of electricity, namely, the modern electron Lorentz had postulated the existence of electrons theory. so early as 1878, but it was not until some years later that the electron theory became firmly established on a satisfac tory basis.

Lorentz assumed that a moving dielectric merely carried its own polarisation doublets," which on his theory 1. The induced field proportional to K the to rise gave "

away field

K

near a

not

satisfactory

K

is

of all

explanation

s

those

which required

experiments

effects

value

moving

required

by

must

optical

1

*//**,

the

experiments of the

go back to Michelson and Morley s have seen that both parts of the beam

now

We

are situated in free ether

any

all

type.

experiment.

in

a

with

proportional to

the Fresnelian convection coefficient equal to

\VC

thus gave

theory

Lorentz further succeeded in obtaining a value for

1.

exact

to

naturally proportional

Lorentz

K.

to

dielectrics

moving

dielectric

moving

and

1

;

no material raeduim

portion of the paths actually traversed

by

is

involved

the beam.

Consequently no compensation due to Fresnelian convection

Til

lMM

I

IK OK

II

moving medium

of ether by

is

lU-.LATIVITY

Thus Kre-nelian

poi-sible.

convection compensation can have no possible application in this case. Yet some marvellous compensation has evidvuily taken "

absolute

In

"

Michelson

travelled

which has completely

place

the

m;i<ke.d

velocity of the earth.

the

by

parallel to the

and

Morley

experiment, the distance (that is, in a direction

s

beam along OA

motion of the platform)

distance travelled by the

2

is

//8

while

,

the

beam along OB, perpendicular

to

1

the direction of motion of the platform,

is

Yet the

%l(3.

most careful experiments showed, as Eddington says, that both parts of the beam took the same time as tested by the interference bands produced. It would seem that OA and "

OB could not really have been of the same OB was of length /, OA must have been of apparatus was

now

rotated through

90,

length

length

;

and

if

The

1 ft. 1

OB became

so that

The time for the two journeys was again that OB must now be the shorter length. The

the up-stream.

the same, so plain

meaning of the experiment is that both arms have a I when placed along Oi/ (perpendicular to the direc

length

and automatically contract

tion of motion),

when

1/J3,

motion).

This itself,

placed

O/

along

This explanation was Fitz-Gerald

does not

suffice.

"11(3

first

Hence the

form the

"

2,1

a is

this journey

is

fi/P,

a variable quantity

"

velocity of

of such an effect has ever light

is

:V/^

But the

r.

is

always

the

plat-

"

2//

absolute

is

platform

velocity of light with respect to

8

in

with respect to the ether

distance travelled with respect to the 2/.

enough

startling

this contraction to he

Assuming

and the time taken for

length

Fitz-Gerald."

given by

contraction,

real one, the distance travelled

a

to

(parallel to the direction of

the

been

platform.

found.

depending on

But no

The

trace

velocity

of

always found to be quite independent of the velocity

HI>Toi:l<

1^1

h\

ilic plat

.my

m in.

I

A

The precent <

further alteration

recount well, that

IVil:Ml

I

left

open

K, to adopt

i-:

lie

t

in to

in

two very

ad.-pt

*

loea

I

order

to

a

give

t

If

line."

a

mov

second becomes

real

moving system,

I

lie

velocity

remain

the concept of

"local

ft

of

the

time,"

the

of

explanation

satisfactory

1

We

c.

startling hypotheses, namely,

Fit/ (lerald contraction and in

The

time an

the measure ol

alter

light relative to the platform will always

must

solved

\>u

the measure of space.

conecpt of

the

XIII

dfffioolty cannot

ing clock goes slower so that one

second as measured

IIM\

Miehelson-Morley experiment. These results were already reached by Lorentz in the course

of further developments of his electron theory. Lorentx used a special set of transformation equations* for time which implicitly introduced the concept of local time.

But he himself it,

artiliee like

at

failed to attach

and looked upon

it

as

in

coherent

of

consequences

Lorentz established

(consisting men-iy

.

)

of

the

a

them

Relativity

of

physical as the

general

Theoremt

transformation equations) into a Universal Principle. In

a set of

and time, which served

demanded

circle

originality

single

while Kinstein generalised it addition Einstein introduced fundamentally of space

mathematical

his successful

in

these result?, and viewing

organised

principle.

The

projective geometry.

Einstein at tins singe consists interpretation

mere

a

in analysis or the

imaginary quantities

infinity

special significance to

any

rather

new concepts

to destroy old fetishes

and

wholesale revision of scientific concepts and thus opened up new possibilities in the synthetic unification a

of natural processes.

Newton had framed his laws of motion in such a \\-.\\ make then; quite Independent of the absolute velocity

as to

* See \

t See Note

4.

XIV

I

Uniform

of the earth.

could

RINTIPLE OF RELATIVITY

motion of ether and matter

relative

not be detected with the

help of

to Einstein neither could

According

it

dynamical laws.

be detected with the

Thus the help of optical or electromagnetic experiments. Einsteinian Principle of Relativity asserts that all physical laws are independent of the absolute velocity of an observer. For different systems, the form of conserved.

we chose the

If

all

physical laws

fundamental unit of measurement for

all

observers (that

assume the constancy of the velocity of light

we can

metric

a

establish

is

velocity of light* to be the

"

one

one

in all

is,

systems)

"

correspondence

between any two observed systems, such correspondence depending only the relative velocity of the two systems. Einstein

s

is

Relativity

application of the well

know nothing but

can

thus merely the consistent logical

known

physical

principle thai \ve

In this sense

relative motion.

it

is

a further extension of Newtonian Relativity.

On tion

this interpretation, the Lorentz- Fitzgerald

and

"local

time"

contrac

lose their arbitrary character.

Space measured by two different, observers are natur ally diverse, and the difference depends only on their relative Both are equally valid; they are merely different motion.

and time

as

descriptions of the

same physical

This

reality.

He

the point of view adopted by Minkowski. itself to

is

temporal rates

frame-work

"

is

in

in the space-time reality, relative

the

the

axes

of

reference.

measurement of lengths and

merely due to the static difference

in

the

of the different observers.

The above theory the

is

reduced to a rotation of

Thus, the diversity

"

essentially

be one of the co-ordinate axes, and in his four-

dimensional world, that

motion

is

considers time

of

Relativity

absorbed

praeticalh

whole of the electromagnetic theory based * See Notes 9

and

12.

on

the

HISTORICAL INTRODUCTION M;i\\\ell-|,orenU system of the

all

advantages

(

field

It

equations.

combined

Maxvvelliaa theory together

classic

>|

XV

The Lorentx assumption

with an electronic hypothesis.

of

polarisation doublets had tion

flu-

..f

theory

1

r

furnished a satisfactory explana re^nelian convection of ether, but in the new

his is

concept of

deduced merely as a consequence of the altered

relative

In addition, the theory of

velocity.

Relativity accepted the results of Michelson and Morley s

experiments as a definite principle, namely, the principle of the constancy of the velocity of light, so that there was

nothing

for

left

experiment.

the

in

explanation

But even more than

Michelson-Morley

all this,

established a

it

single general principle which served to connect together in

known

a simple coherent and fruitful manner the

facts

of Physics.

The theory

made

Relativity received direct experimental

of

Repeated attempts were Lorentz-Fitzgerald contraction. Any

in several directions.

coiifii -illation

to detect the

ordinary physical contraction will usual Iv have observable physical results

;

for example, the total electrical resistance

of a conductor will diminish.

and Rankine, Rayleigh and a variety

Fitzgerald

negative

of

different

contraction,

uniform rrforily

methods but

Whether

results.

//////

Trouton and Noble, Trouton Brace, and others employed

there to

r, x]><>ct

detect

to

invariably

if

ix

an,

can

the

with

ether

never

Lorentz-

the

be

or

same not,

detected.

This does not prove that there is no such thing as an does render the ether entirely super Universal compensation is due to a change in local fluous. ether but certainly

units of length and time, or rather, IxMtig merely different

descriptions of the at

same

reality, there

is

no compensation

all.

There was another group of observed phenomena which be fitted into a Newtonian scheme of

could scarcely

PRIXCIPLE OF

XVI

dynamic^ without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that mass v of an electron depended on its velocity. the "

"<>

early as 1881, J. J.

Thomson had shown

a charged particle increased with

now deduced

a formula for the

its

that

tlie inert in

variation

velocity, on the hypothesis that an electron

of

Abraham

velocity.

mass with

of

always remain

Lorentz proceeded on the assumption

ed a riyid sphere.

that the electron shared in the Lorentz- Fitzgerald contrac

and obtained a

tion

A

formula.

different

totally

very

measurements carried out independently b\ Wolz, Htipka and finally Neumann in 1913,

careful series of Biicherer,

decided

This

conclusively

"contractile"

in

favour of

Lorentz formula.

the

formula follows immediately as a direct

consequence of the new Theory of Relativity, without any

assumption as regards theVlectrical origin of the complete agreement of predictions

of

confirming

it

electron itself.

the

new theory

as a principle

The

must

considered

explanation, are

as

new theory

greatest triumph of this

winch had formerly required their

be

Thus

with the

which goes even beyond the

consists, indeed, in the fact that a large

for

inertia.

experimental facts

number

of

results,

kinds of special hypotheses now deduced very simply as all

inevitable consequences of one single general principle.

We the

have now traced the history of the development of or special theory of Relativity, which is

restricted

mainly concerned with optical and electrical phenomena. Ten years later, It was first offered by Kiustein in 1905. Einstein

formulated

his

second

Principle of Relativity. This

gravitation and has very

of

and

electricity.

theorv,

new theory little

is

the

(

Jcncralised

mainly a theory

connection with optics

In one sense, the second theory

is

indeed

a further generalisation of the restricted principle, but the

former does not really contain the

latter as a special case.

AI,

>i;u

s

iii

theon

is

re-tricted in

ol

EWDBC that

tlu-

movements.

cation to any kind of accelerated hi-

riOV

it

to uniform rect iliniar motion and has no appli

refers

only

-v.t

li

iNTHonn

Einstein

in

second theory extends the Relativity Principle to cases accelerated motion. If Relativity is to be universallv

e\vn accelerated motion must be merely rt lativf ct ii mutter and HI after. Hence the Generalised

true, ti.en nint in, i

ln

l

n

Principle of

that

asserts

Relativity

"absolute"

motion

cannot be detected even with the help of gravitational IRWH. All

vements must be referred

in

co-ordinate

there

If

axes.

is

any

to

definite

chaiiLjf

of

of

sets

axes,

the

numerical magnitude of the movements will also change. Bui according to Newtonian dynamics, such alteration in physical movements can only be due to the effect of ceitain 1

Thus any change

orces in the field.*

new

geometrical"

in

forces

the

of axes will introduce

which

field

are

quite

independent of the nature of the body acted on. Gravitation al

gravitation itself

geometrical"

<ifl/nl

i

/

.r/if.

;

fiif

Thus

new

tfantforatation

</

it

mav hecome elTect-^. at

bv referring framework"

"rectangular

But there Galilean

xtrictly equivalent to one

co-ordtunfey

and nopQwiltfo

all

"

possible

least

is

s\>tein

to

"transform

to a

new

of course have

may

,i>

more

I

set of axes.

all

ol

This

kinds of very

referred to the old Galilean

unarceleraled syslem of

no reason

away

small region^

For sufficiently

movements

complicated movements when or

<>f

ft it ilixtiiKinixh between the two.

gravitational space,

by a change of axes. famous Principle of Equivalence.

s

innal Jield of force in

introduced // t

forces introduced

Einstein

This loids to i/i

may

"

the.-e

A

same remarkable property, and be of essentially the same nature as

forces also have this

co-ordinate-."

why we should

liiHlainenlal than

look upt.n the

any other.

If

it

PUIXCIPLE OF BELATIVITY

XV1I1

found simpler to refer

is

motion

all

a

in

we may

to a special set of co-ordinates,

gravitational field certainly look upon

this special "framework" (at least for the particular region concerned), to be more fundamental and more natural.

We

may,

more simply, identify

still

this

framework

particular

with the special local properties of space in that region. is, we can look upon the effects of a gravitational

That

h eld as simply due to the itself.

local properties of space

The very presence

of the characteristics of space and time in

hood.

As Eddington

and time

of matter implies a modification

saj s

curvature of space-time.

"

It

its

matter does is

the

neighbour

not cause

curvature.

the

Just

light does not cause electromagnetic oscillations;

it

as

the

is

oscillations."

We of view. all

may

look upon this from a slightly different point

The General Principle

motion

matter, and as

all

ground of any

must ultimately be material take

the

Relativity asserts that

movements must be

sets of co-ordinates, the

to

of

merely relative motion between matter and

is

referred to definite possible

in character.

matter actually

present

It in

a

/*

framework convenient

field

as

the

If this is done, fundamental ground of our framework. the special characteristics of our framework would naturally

depend on the actual distribution of matter in the field. But physical space and time is completely defined by the "

framework."

In other words the

Hence we

space and time. is

see

"

framework

how //////*/ v//

"

itself

?

<

space and time

actually defined by the local distribution of matter.

There are certain magnitudes which remain constant by any change of axes. In ordinary geometry distance so that between two points is one such magnitude ;

&r 2 +Sy- +5?-

is

an invariant.

light, the principle of

that

In the restricted theory of

constancy of light velocity demands

Saf*+fy*+Sa*c*Sl*

should remain constant.

IIISIOI;K

The

*i

/m?nti

= -,/./

ftx*

2

ni

-////-

</\

oi

//, .

j-,, ,/.s-

=

:,

I

,,

...,

//!

rr,-

./.,,

vntoiM

this

we

+ fy,

8

shall

.v.j

functions of ./,,./,,

.r.p

+

is

xix

It

is

:ui

defined

by

extrusion of

tin-

is

the ne\v invariant.

to

got .

iiox

(

events

adjacent

transformed

are

i

_,/ : v+,.-V/-.

notion of distance and .*,

\i.

any a

set

of

new

Now

if

variables

quadratic expression for

= ;></,

i,.<v,

where them

s

are

# 4 depending on the transforma

tion.

The

and time in any region which are themselves determined

special properties of space

are defined

by these

//

s

by the actual distribution of matter in the locality. Thus from the Newtonian point of view, these # s represent the gravitational effect of matter while from the Relativity stand-point, these merely define the non-Newtonian (and incidentally non-Euclidean) spice in the neighbourhood of

matter.

We have seen that Einstein s theory requires local curvature of space-time in the neighbourhood of matter. Such altered characteristics of space and time give a satisfactory explanation of an outstanding discrepancy in

the observed advance of perihelion of Mercury.

discordance

is

almost

completely

The

large

removed by Einstein

s

theory. in an intense gravitational field, a beam of light be affected by the local curvature of space, so that to an observer who is referring all phenomena to a Newtonian

Again,

will

system, the beam of light will appear to deviate from path along an Euclidean straight line.

its

This famous prediction of Einstein about the deflection beam of light by the sun s gravitational field was

of a

tested during the total solar eclipse of

observed deflection

Theory

is

of Relativity.

May,

19 IS*.

The

decisively in favour of the Generalised

XX

I

KI Vril

would decrease

appear at

OK KKLATIVITY

however that the velocity of light

It should be noted itself

I.K

a

in

This

field.

gravitational

may

sight to be a violation of the principle of

first

constancy of light-velocity. the Special Theory unaccelerated

is

But when we remember that to

explicitly restricted

the

motion,

the

case of

In

vanishes.

difficulty

the

absence of a gravitational field, that is in any unaccelerated system, the velocity of light will always remain constant.

Thus the

validity of

preserved within

its

the

own

Theory

Special

has proposed a third

Einstein

is

completely

rextrictwl field.

crucial

He

test.

has

predicted a shift of spectral lines towards the red, due to an intense

gravitational

potential.

Experimental

difficulties

are very considerable here, as the shift of spectral lines

complex phenomenon. conclusive can

Evidence

is

a

Einstein thought that a

yet be asserted.

displacement of the

gravitational

is

conflicting and nothing

Praunhofer

lines

is

a

necessary and fundamental condition for the acceptance of his theory. But Eddington has pointed out that even if this test fails, the logical conclusion

while Einstein it

bulk,

atomic

is

law of gravitation

s

not true for such

small

would seem is

to be that

true for matter in

material

systems as

oscillator.

CONCLUSION

From

the

conceptual

stand-point

theiv

are

several

important consequences of the Generalised or Gravitational Physical space-time is perceived to be intimately connected with the actual local distribution

Theory of Relativity. of matter.

Euclid-Newtonian space-time

is

not the actual

space-time of Physics, simply because the former completely neglects the actual presence of matter.

continuum time

is

Euclid-Newtonian

merely an abstraction, while physical spnnthe actual framework which has some definite is

INTKOIirrnnN

MlST.HiU-.M.

curvature due

the

to

Thrnrv

of

mental

distinction

XXI

matter

presence of

Gravitational

funda

Relativity tlius brings out clearly the

between

actual

space-time

physical

(which is non-isotropk" and non- Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian continuum (which is homogeneous, iaotropio and a purely intellectual construc tion)

on the other.

The measurements

of the rotation of the earth reveals a

fundamental framework which may be called the inertial framework. This constitutes the actual physical universe. "

1

This

universe approaches Galilean space-time at a

great

distance from matter.

The to

properties of this physical universe

some world-distribution of matter or the

work"

may

be referred

frame

be constructed by a suitable modification of the

law of gravitation itself. In Einstein curvature of the inertial framework

s

The dimensions

of

referred to vast

is

quantities of undetected world-matter.

consequences.

theory the actual

"

"

It has interesting

Einsteinian

would depend on the quantity of matter vanish

may

"inertial

in

universe

it; it

to a roint in the total absence of matter.

would

Then

again curvature depends on the quantity of matter, and hence in the presence of a sufficient quantity of matter space-

time

may curve round and

will then reduce to a finite

the

surface of

light rays will

universe and

a sphere.

come

close up.

Eiusteinian universe

system without boundaries, In this

"closed

up"

to a focus after travelling

we should

see an

like

system,

round the

<f

(corresponding to the back surface of the sun) at a point in the sky opposite to the real sun. This anti-sun would of course be equally large and equally bright if there is no absorption of light anti-sun"

in free space.

In de Sitter

world-matter

s

is

theory, the existence of vast quantities of

not

required.

But beyond a

definite

Xxii

Rl.\CIPLK

I

OF HKL.VTIVITY

distance from an observer, time itself stands to

observer nothing can ever

the

still,

so that

"

"

happen

there.

All

highly speculative in character, but they have certainly extended the scope of theoretical physics these theories are

to

the

central

universe

still

problem

of

the

ultimate

nature

of

the

itself.

One outstanding of electric force

still attaches to the concept not amenable to any process of being by a suitable change of framework.

peculiarity

it is

"transformed away"

H. Weyl,

it seems, has developed a geometrical theory (in hyper-space) in which no fundamental distinction is made

between gravitational and

electrical forces.

Einstein s theory connects up the law of gravitation with the laws of motion, and serves to establish a very

intimate relationship between matter and physical spacetime.

Space, time and matter (or energy) were considered

The

to be the three ultimate elements in Physics.

restricted

theory fused space-time into one indissoluble whole.

The

generalised theory has further synthesised space-time and

matter into one fundamental physical reality.

and matter taken sejarately reality consist** of

are

Space, time

more abstractions.

a synthesis of

Physical

all three.

P. C.

MAHALAXOBIS.

II

I-TOKIC A

I,

INTRODUCTION

XX111

Note A. For example consider a massive particle resting on a circular disc.

appears

If \ve set the disc rotating, a centrifugal force

in the Held.

to a set of

On

rotating axes,

the other hand,

if

we transform

we must introduce

force in order to correct for the change

of

a centrifugal axes.

This

newly introduced centrifugal force is usually looked upon as a mathematical fiction as "geometrical" rather than physical.

The presence

of such a geometrical force

interpreted us being due to the adoption of

framework.

On

is

a

usually

fictitious

the other hand a gravitational force

considered quite real.

Thus a fundamental

made between geometrical and

distinction

is

is

gravitational forces.

In the General Theory of Relativity, this fundamental distinction is done away with. The very possibility of distinguishing between geometrical and gravitational forces is denied. All axes of reference may now be regarded as equally valid.

In the Restricted Theory,

all

"unaccelerated"

axes of

reference were recognised as equally valid, so that physical laws were made independent of uniform absolute velouitv.

In the General Theory, physical laws are made independent of

"absolute"

motion of any kind.

On The Electrodynamics

Moving Bodies

of

Bl El \STKIN.

A.

INTRODUCTION. It

known

well

is

moving

bodies,

with

aijrec

thai

we attempt

if

conceived

as

electrodynamics,

to

the

at

apply Maxwell

present

s

to

time,

we are

led to assymetry which does not phenomena. Let u* think of the between n magnet and a conductor. The

observed

mutual

action

observed

phenomena, in this case depend only on the motion of the conductor and the magnet, while

relative

to

according

made between bodies

and the conductor energy-value

is

is

produced excites a

and the conductor

produced

in

electromotive itself

is

the force

produced

example, the magnet moves

for

If,

at rest, then an electric held of certain

where a conductor

field

rest

is

which

magnet,

where the one or the other of the

the cases

motion.

is in

conception, a distinction must be

usual

the

the

in

neighbourhood

current exists.

in

But. if

be set in

motion

in

it

the

of

the

of the

magnet be

motion, no electric

at

tield

neighbourhood of the magnet, but an which corresponds to no energy in

in the

conductor; this

current of the same magnitude and the electric force,

those parts

being of course

both of these eas.^

is

causes an electric <:ime

career as the

assumed that the the

iclative

I

RINCIPLE OF RRLATIV1TY

2. Examples of a similar kind such as the unsuccessful attempt to substantiate the motion of the earth relative

to

the

"

"

Light-medium

us to

lead

the supposition that

not only in mechanics, but also in electrodynamics, no properties of observed facts correspond to a concept of absolute rest; but that for all coordinate systems for which the mechanical equations hold, the

equivalent electrodynamical and optical equations hold also, as has already been shown for magnitudes of the first order. In the following

we make call

these

(which we

assumptions

the Principle of Relativity) and

which

an assumption

assumption,

quite irreconcilable

with the

shall subsequently

introduce the further the

at

is

former one

first

that

sight is

light

which is vacant space, with a velocity independent, of the nature of motion of the emitting These two assumptions are quite sufficient to gu-e body.

propagated

in

us a simple

and consistent theory of electrodynamics

of

moving bodies on the basis of the Maxwellian theory for The introduction of a bodies at rest. Lightather" "

will be

proved

to

be

superfluous,

for according

the

to

conceptions which will be developed, we shall introduce neith er a space absolutely at rest, and endowed with special properties, nor shall we associate a velocity- vector

with a

point

in

which electro-magnetic processes take

place.

Like every other theory

3.

theory

is

enunciation of every theory,

between

rigid

electromagnetic of

these

electrodynamics, the

in

based on the kinematics of

bodies

(co-ordinate

processes.

circumstances

we have

is

An the

rigid bodies; to

system),

insufficient

cause of

t

he-

clocks,

and

consideration

difficulties

which the electrodynamics of moving bodies have at present.

in

do with relations

to

with fi^ht

ELECTRODYXAMH^

.HI

MoVfVO BODIES

"F

4

I.-KINEMATXOAL PORTION. Definition of Synchronism.

1.

a co-ordinate

l^t us hrivc

tonian

from another shall

always

which "

call

For

hold.

equations

it

be

will

New

which the

introduced

the stationary

If a material point-be at rest in

this

system

we

hereafter,

system."

this

then

system,

its

be found

system nan

position in this

in

system,

distinguishing

out by a measuring methods of Euclidean

and can be expressed by the Oeometry, or in Cartesian co-ordiuates.

rod,

we wish

If

the values of of time.

It

is

it

i.

f.i;i,-

material

point,

coordinates must be expressed as functions

be borne

always to

nt/h-ntultcrtt ilcfniKlnn lhi\-i

the motion of a

to describe

its

,><>//>,

/i

///

dike into consideration

in

mind that such a

has a physical sense n

l.h<>

/i

il.

/.v

t

only when

tiMf.

tiifttut

Jt

f

l>y

ice

have

to

fact that those of our conceptions, in

a part, are al ^nyx conception* of synchronism time For example, we say that a train arrives here at 7 o clock this moans that the exact pointing of the little hand of my watch to 7, and the arrival of the train are synchronous />///<>//

/i!<ii/x

;

events. It

may appear

definition

we

of

connected

that all difficulties

time can be removed when

substitute the position of the little

Such a

definition

ilelinr

timt-

stationed,

is

in fact sufficient,

with

in place of

hand

when

of it

my

the

time,

watch.

required to

is

exclusively for the place at which the clock

hut

the

definition

is

not sufficient

when

it

is is

required to connect by time events taking place at different

amounts

same thing,

station*,

or what

by means

of time (zeitlich werten) the occurrence of events,

to the

to estimate

take place at stations distant from the clock.

4

JMIINCII LL

Now of

with regard to this attempt; we can satisfy ourselves

the time-estimation

comes

light which

to

following

stationed

is

the clock

the

at

associates a ray of

him through space, and gives testimony

of which the time

event

the

with

coordinates

of

origin

who

Suppose an observer

the

in

events,

manner.

to

Khl.ATIVm

<)i

is

with

to be estimated,

But

the corresponding position of the hands of the clock.

such an association

has

position of the observer

know by result

this

provided with

AVe can attain

experience.

depends on the

it

defect,

the

to a

clock,

as

we

more practicable

by the following treatment. an

If

be stationed at

observer

estimate the time

of

events

A

with a clock, he can the immediate

in

occurring

A, by looking for the position of neighbourhood the hands of the which are syrchrouous with clock, the event. If an observer be stationed at B with a of

we should add

clock, as

one at

the

that the clock

of the

is

same nature

the time

can estimate

he

A,

of

events

But without further premises, it is compare, as far as time is concerned, the

occurring about B. not

to

possible

B

events at

We

with the events at A.

A-time, and a

B-time, but no time

This

(i.e.,

last

establish in

,time

by

common

from

A

to

B

light requires in travelling

to

A

at

A

time) can be defined,

an

and B. if

we

which light requires equivalent to the time which

is

is

reflected

A-time

t

.

A

clocks are synchronous,

B

from

a ray of light proceeds from

and

hitherto to

definition that the time

travelling

arrives

have

common

from

B

A

at

For

A.

at B-time

According it

to

A-time

to

t

t

example, towards B,

and returns

the definition, both

u.Y

as-ume

\\

without (points,

IKOm

KLI.t

Till,

N .V.MIO

of

that tin s definition

JKiVIM.

possible

of

:

B

be synchronous with the clock

A

is

synchronous

at

A

as

at

the clock

If

1.

is

any number

therefore the following relations hold

at A, then the clock at

IKHMKS

synchronism

inconsistency, for

any

involving

(

with

the

clock

at B.

the

If

:>.

clock

A and B

B

are

clocks

at

well as the clock at

both synchronous with the clock at C, then

the

an- synchronous.

Thus with the help of certain physical experiences, we what we understand when we speak of

have established

at different stations, and synchronous with and thereby we have arrived at a definition of synchronism and time.

clocks

at

rest

one another

;

we

In accordance with experience

shall

assume that the

magnitude 2

AB ,

\\ in

to

where

a universal constant.

have defined time essentially with

"e

a stationary system. the

Ou

account of

a its

clock at

as,

$ 2.

system, we call the time defined time of the stationary system."

On

rest

adaptability

stationary "

way

r is

in this

the Relativity of Length and Time.

The following

reflections are

based

on

the

Principle

Kelatmt\

and on the Principle of Constancy of the velocity of light, both of which we define in the following

of

way

:

The laws according

1. >\>tmis

those

alter

flian^s

are ;,

n

tu which the uuture of physical

independent .

referred

to

of

the

two

manner

co-ordinate

in

which \stem>

6

PRINCIPLE OF RELATIVITY

which have a uniform translatory motion

relative to each

other. 2.

Every

of

ray

co-ordinate

moves

light

in

system

"

the

with the same velocity

"

stationary

c,

the velocity

being independent of the condition whether this my of light is emitted by a body at rest or in motion.* Therefore velocity ==

interval of

where, by in

Path of Light -

=

Interval or time

,

we mean time

time,

as defined

1.

Let us have a rigid rod at

when measured by that

the

system at

axis

of

rest,

a

the

rest

;

a length

this has

measuring rod at

rest

;

/,

we suppose

laid along the X-axis of the uniform velocity parallel

rod

is

and then a

>,

Let us now enquire about the length of the moving rod; this can be obtained to the axis of

X,

is

imparted to

it.

either of these operations.

by

(a)

The observer provided with the measuring

moves along with the rod

to

be measured, and

rod

measures

by direct superposition the length of the rod just as if the observer, the measuring rod, and the rod to be measured :

were at (6) in

rest.

The observer

a system at

in

rod

finds out,

by means of clocks placed

being synchronous as defined 1), the points of this system where the ends of the to be measured occm at a particular time f. The

distance

rest (the clocks

between

these

two

points,

previously used measuring rod, this time is

a length, which

wt>

may

call

the

"

measured it

length of the

the

.by

being at

rest,

rod."

According to the Principle of Relativity, the length the found out by the operation ), which we mav call "

* Vide

Note

V.

OX THE EI.FOTRODYXV.Mlrs OF MOVING HOUIKS length of the I

length

roil in

the

I

he

the

he

callo<l

is

from

equal

the

to

found out by the second method, roil mounted of Ike from

the ffnyth

m<>i

ir></

This length

xf tlitn/irij xi/xft-m. !>;isis

is

>ystem

of the rod in the stationary system.

The length which

mav

"

moving

/

of our principle, and

<"V

&h<i

i>

1.1

estimated on

to he

find

it

/<>

If

different

(.

In

the

kinematics,

recognised

generally

we

silently

assume that the lengths defined by these two operations or in other words, that at an epoch of time (, jiial, a moving rigid body is geometrically replaceable by the s-imc body, which can replace it in the condition of rest.

Eelativity of Time. Let us suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A, of a rod,

and^B

the time

<>f

/.<?.,

happen to arrive iti

the time of

the

clocks

correspond

to

the stationary system at the points where they ;

clocks

these

are

therefore

synchronous

the stationary system.

\Ve further imagine that there are two observers at the two watches, and moving with them, and that these observers apply the criterion for synchronism to the two clui k-.

At the time

rollected

from B

time

.

/

t

at the

Taking

,

a ray of light goes out from A,

time

into

t

B,

and arrives back at

consideration

constancy of the velocity oi light,

and

,,

t

A

AB t= B c+v

.

the

we have

A

principle

is

at of

S

PRIXOn LE OF RELATIVITY

where

the

is

/

the stationary system.

with

watches

the

will

the

of

length

}

in

moving

meu-mv<i

rod,

Therefore the observers stationed not

find

the clocks synchronous,

though the observer in the stationary system must declare the docks to be synchronous, \Ve therefore sei that we can attach no absolute significance to the concept nism but two events which are synchronous

of

synchro

when viewed not be synchronous when viewed

;

from one system, will from a system moving relatively

to this system.

Theory of Co-ordinate and Time-Transformation from a stationary system to a system which

$ 3.

moves relatively to this with uniform velocity. Let

there

co-ordinate

be

systems,

I.e.,

the

in

given,

two

of

two

system

stationary

series

three

mutually

Let the X-axes perpendicular lines issuing from a point. of each coincide with one another, and the V and Z-axes Let a

be parallel.

measuring

rigid

and clocks

constant

initial

also

is

communicated time

t

point of one of

velocity

the other which

Any

and

number the rods

in each be exactly alike each other.

Let the a

a let

and

rod,

of clocks be given to each of the systems,

in

the

the direction

systems the

t)f

have

(/?)

X-axis

to the rods

and clocks

of the stationary system

definite position of the axes of the

K

in

the system (/).

corresponds

to

we always mean the time

We suppose by the

moving

in

By

in the stationary system.

that the space

measuring rod placed

a

moving system, which

are always parallel to the axes of the stationary system. /,

of

stationary system K, the motion being

is

measured by the stationary

the stationary system, as well as

measuring rod

placed

in

the

moving

ON THE ILlOTHODYNAMIOa OK MOYl.M; BODIES system, and

the time

thus obtain the co-ordinates

\ve

(j

,y, z) for the

and ($,rj,) for the moving system. Let be determined for each point of the stationary

stationary

s\>tcm,

t

system (which are provided with clocks) by means of the clocks which are placed in the stationary system, with the help of light-signals as described in 1. Let also the time T of the

moving system be determined

point of the

moving system

are at rest

relative to

the

method

of

between

signals

which

-, /)

(<,//,

these

manner described

the

in

value of

each

for

which there are clocks which

moving system), by means

the

light

which there are clocks)

To every

(in

fully

points in

of (in

1.

determines

the position and time of an event in the stationary system, there correspond-; a system of values (,r/,C T ) ; now the

ing

find out

to

is

problem

the

of

system

equations connect

these magnitudes. it is

Primarily

that on account of

clear

homogeneity which we ascribe equations must be linea of

to

the

time and

property space, the

1

.

If

we put

relatively at

values (.<

y

let us find

:=

.?

z]

which

out T as

purpose we have

not other than at

ous

rest in the in

the

rf,

rest in the

to

then

-it

clear

is

that at a

point

system K, we have a system

are

independent a function of (

of

time.

,y,z,(}.

For

of

Now this

express in equations the fact that T

the time

given

by the

clocks

system k which must be made

manner described

in

is

which are synchron

I.

Let a ray of light be sent at time T O from the origin of the system k along the X-axis towards .c and let it be reflected

from

that

place at time

moving co-ordinates and then we must have

of

let it

rl

towards the origin

arrive there at time T 2

+ T,)=T

1

PRINCIPLE OF RELATIVITY

10

we now

If

introduce

that T

the condition

of co-orrdinates, and apply

the principle

the velocity of light in the stationary system,

|

JT (

0+T

(0, 0, 0,

0,

0, 0,

{t+

^.+^Lv v c c

It

be noticed that

is to

we could

ordinates,

instead

select

,

of

a function

)

we have \ 1 / J

o, o, t

the

the

similar conception, being applied to

+C

I

V ).

origin

some other point

point for rays of light, and therefore holds for all values of {^y,z,t,},

A

}

+

=T(*

is

of constancy of

as

of co

the

exit

above equation

they- and

r-axis

gives us, when we take into consideration the fact that light when viewed from the stationary system, is alnays those

propogated along

we have

From tion of

where a

these

and

.

is

With

is

equations

From

t.

the help

always

it

follows that T

equations (1)

light,

these

v*,

of

the

we

is

a linear func

obtain

v.

results it

is

easy to obtain the

we

express by means of equations when measured in the moving system tf

propagated with

the principle

with

of

(,i?,)>

fact that

tion

with the velocity ^/c*

an unknown function of

magnitudes the

axes

the questions

constancy principle

the

constant

of light of

velocity

velocity in

relativity

c

(as

conjunc For a requires).

ON THE ELECTRODYNAMICS OF MOVING BODIES time T=O, if the ray we have ,

is

sent in the

direction of

11

increasing

Now the ray of light moves relative to the origin of with a velocity c r, measured in the stationary system therefore we have

Substituting these values of

we

t

the

in

equation

Jc

for

;

,

obtain

In an analogous manner, we obtain by considering the ray of light which moves along the ^-axis,

where

-

y

,

c

If for

we

.</,

where ff=

.

vi

/8

t

v*

^ of

v.

its

value x

/

v.i-

\

V

c*

/

substitute

().

t=4>

c

.,

tv,

we

obtain

(*-),

,

and

<

(f)=

=r-!=r^

Vc*v*

=

is

P

a function

PRINCIPLE OF RELATIVITY

12

/

we make no assumption about the

If

of

moving system and about the

tlu-

additive constant

then an

hand

have

now

(when measured in we have actually assumed,

velocity c

in case, as

moving system),

c is also

the velocity in the stationary system

adduced any proof

as yet

in

support

that the principle of relativity

tion

moves

to show, that every ray of light

moving system with a

in the

not

tt

side.

We the

of

added to the right

be

to

is

initial position

null-point

of

;

for

the

we have assump

reconcilable with the

is

principle of constant light-velocity.

Atatimer = / = o

a

let

spherical

wave be sent out

from the common origin of the two systems of co-ordinates, and let it spread with a velocity c in the system K. If (>

}

De >/>

a

*)>

reached

point

2

a.2

with the aid of transform this

+2/

our

the

by

+ ._ C 2^

wave, we have

>

let

transformation-equations,

and we

equation,

obtain

us

by a simple

calculation, 2

Therefore the wave

+77

with the same velocity

we show

2

+

2

=c

z

Ts

.

propagated in the moving system and as a spherical wave.* Therefore

is

c,

that the two principles are mutually reconcilable.

In the transformations we have go: an undetermined (v), and wo now proceed to find it out.

function

<j>

Let

us

introduce for this purpose a third

co-ordinate

1

system k , which is set in motion relative to the system X-, the motion being parallel to the -axis. Let the velocity of At the time t = o, all the initial the origin be ( r). co-ordinate

time are

t

points

coincide,

the

and

= o. We

of the system k

for

f

= =y = z = n, ,<

shall say that (./

co-ordinates measured in the system k * Vide

Note

9.

,

y

;

the /

)

then by a

ON THE ELECTRODYNAMICS OF MOVING BODIES two-fold

application of

the

13

transformation-equations,

we

obtain

-v>,

Since

the

relations

do not contain

between (/, y

time

,

z

,

t

},

therefore

explicitly,

and

K

etc.

(x, y, z, t)

and k

are

relatively at rest. It appears that the

systems

K

and k are

identical.

Let us now turn our attention to the part of the y-axis = 0, 77 = 0, = o), and (=o, = ], =0). Let ( this piece of the y-axis be covered with a rod moving with

between

>/

the velocity v relative to the system to

its

axis

;

the

ends of

K

and perpendicular

the rod having therefore the

co-ordinates

Therefore the length of the rod measured in the system

K

is

~T7~y

For the system moving with velocity

we have on grounds

of

Z

symmetry, Z

(

?

),

14

PRINCIPLE OF RELATIVITY

The physical

significance of the equations obtained concerning moving rigid

4.

bodies

Let

us

consider

when

spherical figure

R

radius

which

and moving a

sphere

rigid

(i.e.,

one having a

tested in the stationary system) of

at rest relative to

is

clocks.

the

system (K), and

whose centre coincides with the origin of K then the equa tion of the surface of this sphere, which is moving with a velocity v relative to

At time

A

rigid

measured

moving

t

o,

K,

is

the equation

is

expressed by means of

body which has the_figure of a sphere when the

in

condition

moving system, has therefore in the when considered from the stationary

system, the figure of a rotational ellipsoid with semi-axes

Therefore the y and z dimensions of the sphere (there any figure also) do not appear to be modified by the

fore of

motion, but the x 1:

is

V v.

1

;

the

For v = c,

all

dimension

shortening

moving

is

is

bodies,

shortened

the

in

larger,

the

ratio

the larger

when considered from

a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become

ON THE ELECTRODYNAMICS OK MOVING BODIES meaningless

;

our theory

in

c

the part of

plays

15 infinite

velocity.

It

bodies

that similar results hold about stationary

clear

is

a

in

when

system

stationary

considered from a

uniformly moving system.

Let us now consider that a clock which in

the

stationary

system

at rest relative to the

the time r

;

time

X-,

and

How much

T.

it

is

to be placed at

lying at rest f,

capable the

K

We

?

t

7i

|

and lying of

giving of

the

gives

the

origin

to be so arranged that it

does the clock gain,

the stationary system

is

the time

moving system

suppose

moving system

gives

when viewed from

have,

\,

and x=vt,

V ITherefore the clock loses by an

amount

^"2

P er second

of motion, to the second order of approximation.

From this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense 3 when viewed from the stationary explained in system. Suppose joining

it

the

clock

at

A

to be set in

motion

in the line

with B, then after the arrival of the clock at

B,

they will no longer be found synchronous, but the clock which was set in motion from A will lag behind the clock p1

which t is

had been

all

along at B by an amount $t -$, where

the time required for the journey.

16

PRINCIPLE OF RELATIVITY

We when

see forthwith

A

If

and

B

that the result holds also

A

moves from

clock

to

B by

we assume that the curved

If at A, there be

set in

motion one of curve

completed

till /

result obtained for a polygonal

we

line,

obtain

the following

two synchronous clocks, and if we them with a constant velocity along a

law.

in

also

coincide.

line holds also for a

closed

when the

a polygonal line, and

comes back

it

then

-seconds,

to

A, the journey being

after arrival, the last

men01

tioned

clock

will

be

behind the stationary one by

\t

~

From this, we conclude that a clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all seconds.

other conditions being identical.

Addition-Theorem of Velocities.

5.

Let a point move

in the

system k (which moves with

velocity v along the ;r-axis of the system

K) according

to

the equation

= where It

t/

|

is

and

w

1?

to

find

system K.

of equations in

= M ,T, = 0,

are constants.

required

relative to the

we

Y

out the If

motion

3 in the equation of motion of the point,

obtain

;=

wf + v 1

of the point

we now introduce the system

\ t,

y

c

/

""

ON THE ELECTRODYNAMICS OF Mo VI.Mi

The law first

parallelogram of velocities

of

We

order of approximation.

and

u

i.e.,

a

and

w.

=

r/ [(

o

+

/<+

17

.ODIES

hold up to the

can put

1C 1

tan"

between the

put equal to the angle Then we have

is

I

vw sin a \ / I

2 vw cos a)\

u=

v,

:

"1

j

cos

//(

,

velocities

-i

.,

It should

be

noticed

that

of the v-axis of the

If

From

this equation,

of

iv

into the

outer

has the direction

moving system, TT

velocities, each

w

and

r-

expression for velocity symmetrically.

f+"

\ve

which

is

see

that

by combining two than

smaller

c,

we

ojbtain

If we put v=c velocity which is always smaller than and w=c\, where x and A. are each smaller than c, <.

It

is

altered

also clear that the velocity

by adding

to

it

of

light

a velocity smaller than

case,

Fufo Note 12.

cannot

c c.

For

a %,

be (his

PRINCIPLE OF RELATIVITY

18

We

U

have obtained the formula for

for the case

when

and w have the same direction, it can also be obtained by combining two transformations according to section If in addition to the systems K, and k, we intro 3.

v

k

duce the system to the

parallel

which the

of

,

with

-axis

initial

velocity

moves

point

w, then between the

magnitudes, x, y, z, t and the corresponding magnitudes k we obtain a system of equations, which differ from

of

,

the equations in r,

we

shall

We

3, only in the respect have to write,

see that such

a

parallel

that in

transformation

place

of

forms a

group.

Wo two

have deduced the kinematics corresponding to our

fundamental

and we

shall

now

principles for the laws necessary for us,

pass over to their application

in

electro

dynamics.

*II.

ELECTKODYNAMICAL PART.

Transformation of Maxwell

6.

s

equations for

Pure Vacuum. On

the nature

of

lite

in

Electromotive Force caused by motion

a magnetic field.

The Maxwell-Hertz equations for pure hold for the stationary system K, so that

9 -

-?-

[X, Y, /] =

9<

JL dy

M

N

vacuum may

ON THE KLE( THODVXA.Mli

oi

s

BODIES

.\[()VlN(i

and

a

i

a

a

3-

6y

X where force,

If tions,

Y,

[X, L,

M,

N

Z]

are

the

j 3

Y

(1)

Z

components of

the

electric

are the components of the magnetic force.

we apply the transformations in 3 if we refer the electromagnetic

to

and

co-ordinate system

[X,

.-

moving with

-

velocity

these

equa

processes to the

r,

we

obtain,

- N),

and

a

al

X

The

?

/8(Y-

c

N)

j8(Z+

-M) <

principle of Relativity requires that the

Maxwell-

Hertzian equations for pure vacuum shall hold also for the system k, if they hold for he system K, i.e., for the vectors electric

of

the electric

and magnetic forces acting upon

and magnetic masses

in

the

moving system

k,

PRINCIPLE OF RELATIVITY reaction, the

which are defined by their pondermotive equations hold,

.

.

.

i.e.

.

.

6

_a

di)

3

M

N

a. 1

J oT

c

(X

,

same

.

Y Z ,

)

= 6f L

a 9*

...

(3)

y Clearly both the systems

of

equations

(2)

and

(3)

the system k shall express the same things, for both of these systems are equivalent to the Maxwellfor

developed

Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor $ (v), which depends only on v, and is independent of (

V>

>

[X

[L

it

Hence the

T)

,

Y Z ]=V

(r) [X,

ft

(Y-

?X),

M N ]=*

(r) [L,

/i

(M

+

?Z;,

,

,

,

relations,

Then by reasoning similar can be shown that ^(r) = l. .-.

[X

L [

,

.

Y Z ] = [X, ,

W. N

]

= [L,

ft

to

(Y-

that

f N),

j8

ft

(Z+ ^M)],

(X-

Y)].

followed

M

(/+

;

in

)j

r

.^Z),

(N--

Y)].

(3),

ON THE ELECTRODYNAMICS OF MOVING BODIES

21

For the interpretation of these equations, we make the Let us have a point-mass of electricity

following remarks.

which i.e.,

it

is

of

magnitude unity

a distance of rest in is

in

the stationary

cm.

1

be

of electricity

If this quantity

the stationary system, then the force acting upon

the

quantity

moving system the

force

system

of

upon

acting

three of equations

1.

it,

and

way

(1), (2), (3),

it

rest relative to the

at

moment

considered), then

measured

equivalent to the vector

is

following

be

electricity

(at least for the

at

But

equivalent to the vector (X, V, Z) of electric force.

if

K,

system

exerts a unit force upon a similar quantity placed at

(X Y ,

,

in

the

Z ).

moving The first

can be expressed in the

:

If a point-mass of electric unit

electro-magnetic

field,

pole

moves

in

an

then besides the electric force, an

acts upon it, which, neglecting the numbers involving the second and higher powers of v/c,

electromotive force

is

equivalent to the vector-product of

the

velocity

vector,

and the magnetic force divided by the velocity of light (Old mode of expression). 2. If a point-mass of electric unit pole moves an electro-magnetic field, then the force acting upon it

in is

equivalent to the electric force existing at the position of the unit pole, which we obtain by the transformation of the

field to

a co-ordinate system which

to the electric unit pole

[New mode

is

Similar theorems hold with reference force.

We

st-e

that

in

at

rest

relative

of expression]. to

the

magnetic

the theory developed the electro

magnetic force plays the psirt of an auxiliary concept, which owes its introduction in theory to the circumstance that the electric and magnetic forces possess no existence independent of the nature of motion of the co-ordinate

system.

PRINCIPLE OF RELATIVITY

22 It

is

further clear that the assymetry mentioned in the

when we

which occurs

introduction

treat of the current

by the relative motion of a magnet and a con Also the question about the seat of ductor disappears. excited

electromagnetic energy

Theory of Doppler

7.

In the system K, co-ordinates, let

which

seen to be without any meaning.

is

is

s

Principle and Aberration.

at a great distance

from the origin of

there be a source of electrodynamic waves,

represented with sufficient approximation in a part by the equations

of space not containing the origin,

Z=Z

sin

I

N=N

* J

:

sin

*

}

M

N ) are the vectors Here (X Y Z ) and (L which determine the amplitudes of the train of waves, direction-cosines of the wave-normal. (I, m, n] are the ,

,

,

,

Let us now ask ourselves about the composition of when they are investigated by an observer at

these waves, rest in a

moving medium

A

:

By

applying the equations of

and magnetic 3 and the equations of transformation obtained in for the co-ordinates, and time, we obtain immediately

transformation obtained in

6 for the electric

forces,

:

L/=L n sin*

ON THE ELECTRODYNAMICS OF MOVING BODIES where

,

n

=

1If an observer moves From the equation for w it follows with the velocity r relative to an infinitely distant source of light emitting waves of frequency v, in such a manner :

that the line joining the source of light and the observer with the velocity of the observer makes an angle of 4>

referred to

co-ordinates which

a system of

with regard to the source, then is

perceived by the observer

is

the

is

frequency

stationary v

which

represented by the formula

\

This is Doppler s principle for any velocity. then the equation takes the simple form

1

We =

for v

If

see that

If

4>=

+

contrary to the usual conception

v=oo,

c.

$ =angle between

ray) in the

the wave-normal (direction of the

moving system, and the

observer, the equation for

1

/

- cos e

line of

takes the form

4>

motion of the

24

PRINCIPLE OK RELATIVITY This equation expresses the law of observation

most gen eral form.

If

<=-,

the

equation

in

takes

its

the

simple form

We waves,

have

still

to

which occur

investigate

the

these equations.

in

the amplitudes in the stationary and (either electrical or magnetic),

cos

(1

If

From

8

the

moving systems

we have

I

)

c

this reduces to the simple

4>=o,

A

4>

amplitude of the If A and A be

form

=A

these equations,

it

appears that for an observer, c towards the source of

which moves with the velocity should light, the source

8.

appear infinitely intense.

Transformation of the Energy of the Bays of Light.

Theory of the Radiation-pressure on a perfect mirror.

Since

is

07T

equal

to

volume, we have to regard

the energy

of

light

per

unit

J

as the

energy of light

in

ON THE ELECTRODYNAMICS OF MOVING BODIES

A

the

moving

system.

between

ratio

"measured

the

when

25

*

would

_

denote

therefore

the

A.

of

energies

moving"

and

a

definite

"measured

light-complex

when stationary/

volumes of the light-complex measured in K and k Yet this is not the case. If /, m, u are the being equal. the

of

direction-cosines

the

wave-normal

of

in

light

system, then no energy passes surface elements of the spherical surface

the

through the

stationary

(.1-

which expands with the velocity

oi

light.

say, that this surface always encloses the

We can therefore

same light-complex. energy, which this

Let us now consider the quantity of encloses, when regarded from the svstem

surface

the energy of

the light-complex relative

the

to

k, i.e.,

svstem

A-.

Regarded

from

the

moving

system,

the

surface becomes an ellipsoidal surface, having, r

= 0,

the equation

If

at

spherical

the time

:

S= volume

of

the

S

sphere,

= volume

ellipsoid, then a simple calculation shows that

S

of

this

:

ft "

s -

If

the

E

OOS

<t>

denotes the quantity of light energy

stationary

4

system,

E

the

measured

quantity measured

in

in

the

PRINCIPLE OF RELATIVITY

26

moving

system,

which

by the surfaces

enclosed

are

mentioned above, then

E

A

2

8^ If

4>=0,

Vl-v*/c*

Q S

we have the simple formula

:

It is to be noticed that the

energy and the frequency according to the same law with

of a light-complex vary

the state of motion of the observer.

Let there be a perfectly reflecting mirror at the co-or =0, from which the plane-wave considered

dinate-plane

the last paragraph is reflected. Let us now ask ourselves about the light-pressure exerted on the reflecting surface in

and the

direction,

intensity of the light after

frequency,

reflexion.

Let the incident

A

light

be

defined

cos $, r (referred to the system K).

we have

the corresponding magnitudes i

v

cos

1

4>

A =A

cos cos

v

<*>

1

4>

=

=v ,

~c~

:

by the magnitudes Regarded from A-,

ON TKE KLECTKOUYNAMUS For the is

reflected

we

light

referred to the system k

=A

A

By means

,

cos

MOV1N(; BODIES

()!

when the

obtain,

=

<$"

=

cos

<

v"

,

=v

.

back-transformation to the stationary

of a

:

=A

-

A"

process

:

system, we obtain K, for the rellected light

A"

27

/

v*

V ~^ COS

,_ <t>

Cos4

"

1 l

+

>"

_

c_

1+

^ cos

2

1

$"

=/

c

r

"*

cos

+

<

C

"

/

-

(1+ \

__

=!/

(H) The amount

or energy

falling

upon

of the mirror per unit of time (measured

system)

is

.

87r(c cos

away from

A

""/8w

c cos

unit

surface

stationary

The amount

of energy going

mirror

unit of time

<

)

unit surface of the

(

the

in the

<I>"+r).

The

per

difference

of

these

is, according to the Energy principle, the unit work exerted, by the pressure of light If we put this eijiial to P.r, where P= pressure

expressions

amount of time.

of light,

of

is

two

]>er

we have ,,_.,

Aa

(cos V

*

1-

-

_/

|

PRINCIPLE OF RELATIVITY

28

As a

first

approximation, we obtain

-

P=-2

<*>*

4>.

8

which

with

accordance

in

is

and

facts,

with

other

theories.

All

problems

the

of optics of

method used

after the

here.

moving bodies can be solved The essential point is, that

and magnetic forces of

electric

which

light,

are

moving body, should be transformed to a

influenced by a

system of co-ordinates which is stationary relative to the In this way, every problem of the optics of moving bodies would be reduced to a series of problems of the

body.

optics of stationary bodies.

Transformation of the Maxwell-Hertz Equations.

9.

Let us

start

from the equations

M

,

:

1

1

?>L

_

ft

r>Y

7 1

"

c

dt

I/ pu c

l( c

_i.9 *

pu>

\

where

Y

\

-91 -9-? ~

dt)

\

4-9j?\ dt )

d=

= 9^I _9J Q.e

r

a.<-

~dt

"a.-

iaM_8Z_ax c

1

a^

_

J

,

6

Qy

dy

=

a-

a.

denotes

TT

times the density

oy of electricity,

of

and

electricity.

(?t,,

If

masses are bound

?,

tt r

)

are

the

velocity-components

we now suppose that unchangeably

to

small,

the

electrical-

rigid

bodies

UN THK KLLCTKODVNAMKS OF MUVINC BODIES

29

form the electromag electrodynamics and optics for

(Ion*, electrons), then these equations

netic

basis

these

transformed

equations to the

mation-equations the equations 1

s

bodies.

moving If

Lorentz

of

which

system in

given

in

and

-3

transfor

obtain

___

9Z. l

dr

9

67?

T

the

then we

(5,

_

f P I

*

the system K, are

:

,

C

hold

with the aid of

/,

8*

67;

=

Qr

J

8r

J- 87

dr,

where

-

-

Since the vector (/^

// >

of

velocity as can

be

velocities

in

the

electrical

easily

\

it

>

mass

from

seen so

T?

is

x 4

)

is

nothing

but

the

measured the

hereby

in the system A-, addition-theorem of

shown, that by taking

30

PRINCIPLE OF RELATIVITY

our kineinatical principle as the basis, the electromagnetic of Loreutz s theory of electrodynamics of moving bodies correspond to the relativity-postulate. It can be

basis

briefly

remarked here

important law

the following

that

follows easily from the equations developed section

manner

:

an

if

the

present

moves

charged body

electrically

and

in

in

any

charge does not change thereby, when regarded from a system moving along with it, then the charge remains constant even when it is regarded from in space,

if its

the stationary system K.

10.

Dynamics

of the Electron (slowly accelerated).

Let us suppose that a point-shaped the electrical charge

moves

the

in

following about If the in the

particle,

having

be called henceforth the electron)

electromagnetic its

electron

next

e (to

field

;

we

assume

the

law of motion. be at rest

"particle

of

at

any

time,"

the

definite

epoch,

motion takes

then place

according to the equations

=

n

w

d*i

Where

m

is its

(a-,

y,

.?)

=

z rf<-

are the co-ordinates of the electron,

and

mass.

Let the electron epoch of time. to

=

*

possess

the

velocity

v at

a certain

Let us now investigate the

which the electron

will

move

in

laws according the particle of time

immediately following this epoch.

Without influencing the generality of treatment, we can will assume that, at the moment we are considering,

and we

ON THK ELECTRODYNAMICS OF MOVING BODIES the

electron

the

at

is

origin

of co-ordinates,

with the velocity v along the X-axis

of

moment (^=0)

the

at

clear that

this

31

and moves

the system. electron

is

It

is

at rest

system A, which moves parallel to the X-axis with the constant velocity r. relative to the

From with

the

the

made

suppositions of

principle

from the system

it is

relativity,

the electron

k,

in

above,

combination

clear that regarded

moves according

to the

equations

dr*

in the

time immediately following the moment, where the

symliols (,

now

T,

,

rj,

that for

fix,

X Y ,

t

,

Z

)

refer to the system

.v=y-z=(\ T=g =

equations of transformation ^iven in

have

{

A\

If

we

=

=Q, then the (and (5) hold, and we i)

:

With

the aid of these equations,

we can transform

above equations of motion from the system

K, and obtain

Z

8

,-

<//

(A)

_

A-

:

1 <_

vn

J5

y

d*y (//"

I e_

,n

f

(

y_

r

the

to the system

PRINCIPLE OF RELATIVITY

32

Let us now consider, following the usual method of the longitudinal and transversal mass of a

treatment,

moving

We

electron.

write the equations (A) in the form

=eX=eX

"

=

?

M] and

let

us"

of

ponents electron, this

first

remark,

the

ponderomotive

and are considered

moment, moves with a

of the electron.

that

a

in

<?X

,

eY

force

,

?Z are the com

acting

upon the

moving system which, at

velocity

which

is

equal to

This force can, for example, be

that

measured

of a spring-balance which is at rest in this last If we briefly call this force as "the force acting system. upon the electron," and maintain the equation

by means

:

Mass-number x aeceleration-number=force-number, and if

we further

fix

that

the stationary system K,

we

obtain

accelerations are measured in

the

then

from the above equations,

:

Longitudinal

Transversal mass =

Naturally, when other definitions are given of the force

and the acceleration, other numbers are obtained f^r the * Vide

Note

21.

THK ELECTRODYNAMICS or MOVING BODIES

o.N

we see that we must proceed very carefully the different theories of the motion of the comparing

mass in

33

hence

;

electron.

remark that

\Ve

able material point

about the mass hold also

this result

for ponderable material

may

mass

for in our

;

made

be

addition of an electrical charge

into

sense,

a

ponder*

an electron

by the which may be as small as

possible.

Let

now

us

electron.

the

It

determine

K

dinates of the system

the

force

X, then

kinetic

with the

it is

clear

energy

initial velocity

under the action of

X-axis

along

the

of the

moves from the origin of co-or

electron

that the energy

electrostatic field has the value

drawn from the

Since

/<?X^--.

steadily

an electromotive

the

electron

only slowly accelerated, and in consequence, no energy is given out in the form of radiation, therefore the energy drawn from the electro-static field may be put equal to is

W

of motion. Considering the whole process of the energy motion in questions, the iirst of equations A) holds, we

obtain

:

For shows,

v

= c,

*\\

is

vel iritir.-

infinite!]

great.

\<vco!iiig

that

>

A>

of

our

light

former result can

have

no

possibility of existent

In this

consequence

expression

for

of

the

kinetic

arguments energy

mentioned

must

also

above,

hold

for

of

the

ponderable masse*. \Ye

can

now

enumerate

motion of the electron tion,

the characteristics

H\,ulaMe

for experimental verifica

which follow from equations A).

34

PRINCIPLE OF

From

1.

RELATIVITY

the second of equations

A)

;

follows that

it

and a magnetic force N produce equal deflexions of an electron moving with the velocity an

v,

force

electrical

Y=

when

_^

our theory,

we

Therefore

.

is

it

Y,

obtain

to

possible

see

that the

according to

the electric deflexion A,, by applying the law

A. This relation can be

the

measured by means of magnetic 2.

v

value which

the

acquired

p=(

is

directly

and

deduced for the kinetic

follows that

it

is

We

of

when the

xd<;=c

2

l

r

-ii

radius

of

.

curvature

path, where the only deflecting force

is

:

J

L0-? the

calculate

electron

the velocity v

P,

given by the following relation

J 8.

be

electric

fields.

From

is

can

oscillating

through a potential difference

which

and

by means of experiments electron

rapidly

energy of the electron, falls

,

:

c

tested

of

Am

t>

A,

because the velocity

of an

velocity

electron from the ratio of the magnetic deflexion

R

of the

a magnetic force

acting perpendicular to the velocity of projection. the second of equations A) we obtain

N

From

:

_d*y <

^ R

^

r

,.

c

H=

K ,

""^

.

\

These three relations are complete expressions for the law of motion of the electron according to the above theory.

ALBRECHT EINSTEIN [y/ s/turf

The name

<it

t/o/e.]

l>iot/i-a/j//i<

of Prof. Albrecht Einstein has

beyond the narrow pale of confirmation

the brilliant

now

spread far

scientific investigators

of his

owing

to

deflection of

predicted

light-rays by the gravitational field of the sun during the total solar eclipse of

May

But

to the serious

known from

the beginning

29, 1919.

student of science, he has been

/

and many dark problems in physics has been illuminated with the lustre of his genius, before,

of the current century,

owing before

to the latest sensation just

mentioned, he Hashes out

public imagination as a scientific star of the

first

magnitude. Einstein

began

is

a Swiss-German of Jewish extraction, and

his scientific career as a privat-dozent

in

University of Zurich about the year 1902.

migrated to the German University of Prague as

ausser-ordentliche (or

associate)

the Swiss

Later on, he

Bohemia

in

In 19!

Professor.

through the exertions of Prof. M. Planck of the Berlin University, he was appointed a paid member of the Hoyal (now National) Prussian Academy of Sciences, on a salary of

18,000

marks per year.

In this post, he has

only to do and guide research work.

Another distinguished of same was the Yan t ilofY, the eminent post occupant physical chemist. It

is

rather difficult to give a detailed, and consistent

chronological account of his scientific activities, so variegated,

which

gained

and cover such

him

distinction

Brownian Movement. in Perrin s

book

a

wide

field.

was an

The

work

investigation

An admirable account

The Atoms.

they are first

will

/

*

4,

on

be found

Starting from Boltzmann

s

PRINCIPLE OP RELATIVITY

86

theorem connecting the entropy, and the probability of a state, he deduced a formula on the mean displacement of This particles (colloidal) suspended in a liquid. formula gives us one of the best methods for finding out a very fundamental number in physics namely the number small

of

in one gm. molecule of gas (Avogadro s The formula was shortly afterwards verified by

molecules

number).

Perrin, Prof, of Chemical Physics in the Sorbonne, Paris.

To Einstein

due the resusciation of

also

is

Planck

s

This theory has not yet caught the popular imagination to the same extent as the new theory of Time, and Space, but it is none the less

quantum theory

of energy-emission.

scope as far

as

concepts are long time that the observed emission of light from a heated black body did not correspond to the formula which could be deduced from the older classical theories of continuous emission and iconoclastic in

concerned.

It

its

was known

classical

a

for

In the year 1900, Prof\ Planck of the Berlin University worked out a formula which was based on the bold assumption that energy was emitted and absorbed by propagation.

the molecules in multiples of is

a

constant

(which

gravitation), and v

is

accepted theories that s

quantity hv, where // like the constant of

the frequency of the light.

The conception was Planck

the

universal

is

so

in

radically

spite

of

different

the

great

from

all

success

of

radiation formula in explaining the observed facts

of black-body radiation,

it

did not meet with

much

favour

In fact, some one remarked jocularly that according to Planck, energy flies out of a radiator like

from the physicists. a

swarm

in

But Einstein found a support for the new-born concept another direction. It was known that if green or ultraviolet

light

of gnats.

was allowed

to fall

the plate lost electrons.

on a plate of some alkali metal, The electrons were emitted with

ALBERT EINSTEIN all

but

velocities,

From

the

there

investigations

but upon

its

is

generally a

of

Lenard and

made

curious discovery was

emission did not at

37

that this

maximum

limit.

Ladenburg, the

maximum

velocity of

depend upon the intensity of light,

all

The more

wavelength.

was the

violet

light,

the greater was the velocity of emission.

To

account

for this

assumption that the light (he

pulse

calls

a

it

Einstein

fact, is

propogated

in

made the

bold

a

unit

space as

and falling upon an

Light-cell),

individual atom, liberates electrons according to the energy

equation liv=- -ftnv^

where

A

is

(in,

mass and velocity of the

are the

r)

+ A,

There was law when

it

material for the confirmation of this

little

was

before

elapsed

first

proposed (1905), and eleven years established, by a set of

Millikan

Prof.

experiments scarcely rivalled for the ingenuity, care displayed, the absolute truth of the law. this confirmation,

law

electron.

a constant characteristic of the metal plate.

is

In recent years, light,

and other

now regarded and

formula

X

brilliant triumphs, the

has

and

results of

quantum

fundamental law of Energetics. rays have been added to the domain of as a

this direction also,

in

skill,

As

to

proved

be

Einstein

one of

the

s

photo-electric

most

fruitful

conceptions in Physics.

The quantum law was next extended by Einstein

to the

problems of decrease of specific heat at low temperature,

and here

also

his

theory

was confirmed

in

a brilliant

manner.

We

pass over his other contributions to the equation of

state, to the

chemical

problems of null-point energy, and

reactions.

The recent

experimental

photo

works of

88

PRINCIPLE OF RELATIVITY

,

Nernst

and

Einstein

s

Warburg seem to we are probably

indicate for the

genius,

that

first

through time having

a satisfactory theory of photo-chemical action. In

Einstein

191."),

made an excursion

into Experimental

Physics, and here also, in his characteristic way, he tackled

one of the most fundamental concepts of Physics. It is well-known that according- to Ampere, the magnetisation of iron and iron-like bodies, when placed within a coil carrying an electric current is due to the excitation in the metal of small electrical circuits. But the conception

though a very

fruitful one, long

of experimental proof,

remained without a trace after the discovery of the

though

was generally believed that these molecular be due to the rotational motion of free

electron,

it

currents

may

electrons within the metal.

process of magnetisation, a

It

is

easily seen that if in the

number

of electrons be set into

rotatory motion, then these will impart to the metal itself a turning couple. The experiment is a rather difficult one,

and many physicists

But

in collaboration

carried

successfully

tried

essential correctness of

Einstein

s

in

vain to observe the effect.

with de Haas, Einstein planned and out this experiment, and proved the

Ampere

s

views.

studies on Relativity were

commenced

in the

year 1905, and has been continued up to the present time. The first paper in the present collection forms Einstein s first

great

Relativity.

contribution

We

to

have recounted

the

Principle

of

in the introduction

Special

how

out

and disorder into which the electrodynamics and optics of moving bodies had fallen previous to 1895, Lorentz, Einstein and Minkowski have succeeded in of the chaos

building up a consistent, and fruitful

new theory

of

Time

and Space.

But Einstein was not special

problem

of

satisfied

Relativity

for

with the study of the uniform motion, but

ALBERT EINSTEIN tried, in a series of papers beginiiin^ it

to the case of

the present collection article

is

this subject,

Principles

of

this theory are

Einstein

is

The

to the

and gives,

extend

paper in

Annalen der Physik in his

Generalized

Relativity.

now matters

of public

now

to

last

a translation of a comprehensive

which he contributed

1916 on

from 1011,

non-uniform motion.

only

-45,

and

it

science will continue to be enriched,

in

own words, the The triumphs of

knowledge. is

to be

for

hoped that

a long time to

come, with further achievements of his genius.

INTRODUCTION. A* the present time, different opinions are being held about the fundamental equations of Electro-dynamics for 1 moving bodies. The Hertzian forms must be given up, for it has appeared that they are contrary to

mental

many

experi

results.

In 1895 H. A. Lorentz- published his theory of optical

and

electrical

in

phenomena

moving bodies;

this theory

uas based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been In his theory, Lorentz proceeds

ushered into existence.

from certain equations, which must hold at every point of then by forming the average values over Phy regions, which however contain sically infinitely small large numbers of electrons, the equations for electro-mag "

"Ather";

"

moving bodies can be

netic processes in

successfully built

up. In particular, Lorentz s theory gives a

the non-exietence of relative motion of the lumiiiiferous

"

Ather

;

it

shows that

gobd account of earth and the

this fact

is

connected with the covarianee of the original

when

intimately

equations

certain simultaneous transformations of the space

and

time co-ordinates are effected; these transformations have therefore obtained from

is

a purely

when subjected mathematical

cal considerations; tivity

;

1

;

iueare

l

The covarjance

transformations. equations,

II.

this

theorem

n,i.

No*

1

fart

i.e.

will call this re-t>

Note

_.

name of Lbrentz-

the-e

fundamental

Lomit/-traiisfunnation

based on any physithe Theorem of Rela

not

essentially -

l.

to the

the

of

on the 3

form of

Vide Nute

::.

the

I

differential

KIXCIPT.E OF RELATIVITY

the

for

equations

propagation of waves with

the velocity of light.

Now

without recognising any hypothesis about the con Ather and matter, we can expect these

nection between

"

"

mathematically evident theorems to have their consequences that thereby even those laws of ponder

so far extended

which are yet unknown

able media

this covariance

by saying

this,

when

we do not indeed and

rather a conviction,

express

possess

of

may

The

Relativity.

;

an opinion, but

this conviction I

ted to call the Postulate affairs here is

may anyhow

subjected to a Lorentz-transformation

be permit position

almost the same as when the

of

Principle of

Conservation of Energy was poslutated in cases, where the corresponding forms of energy were unknown. Now if hereafter, we succeed in maintaining

this

covariance as a definite connection between pure and simple observable nection

phenomena

may

be styled

in

moving

bodies,

the

definite

con

the Principle of Relativity.

These differentiations seem to

me

to be necessary for

enabling us to characterise the present day position of the electro-dynamics for moving bodies.

H. A. Lorentz

has found out

1

the"

Relativity

theorem"

and has created the Relativity-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.

A. Einstein 2 has brought out the point very clearly, that this postulate is not an artificial hypothesis but is a

rather

which

is

new way

of

comprehending the time-concept

forced upon us by observation of natural

pheno

mena.

The

Principle

of

Relativity

lated for electro-dynamics of i

FcieNote4,

has not yet been formu

moving bodies *

Note

6.

in

the

sense

IVTKODUCTION characterized by me.

3

the present essay, while formu

"In

fundamental equa-

lating this principle, I shall obtain the \\\\<

moving bodies

for

mined by

But

a sense which

uniquely deter

is

this principle.

shown that none

be

will

it

assumed

in

for these equations can

of the forms hitherto fit

exactly

with

in

this

principle.*

We

would at

tions which are

first

would correspond

fundamental equa

the

expect that

assumed by

Lorentz

for

bodies

moving But

to the Relativity Principle.

be shown that this

is

it

will

not the case for the general equations

which Lorentz has for any possible, and also for magnetic bodies but this is approximately the case (if neglect the ;

square of the velocity of matter in comparison to the velocity of light) for those equations which Lorentz here

But

after

infers for non-magnetic bodies. accordance with the Relativity Principle

is

this

due

way not corresponding

therefore the accordance tion

of

is

due

two contradictions

But meanwhile enunciation

fact

formula

that the condition of non-magnetisation has been ted in a

latter

to the

to the Relativity Principle; to the fortuitous

to

the

of

the

compensa

Relalivity-Postulate.

a rigid

in

Principle

manner does not signify any contradiction

to the hypotheses

of Lorentz

become

s

molecular theory, but

the assumption

Lorentz

the

of

theory must be introduced

s

shall

it

contraction

of

position

of

Classical

Relativity Postulate. of

mechanics

for

an

at

than Lorentz has actually dene. In an appendix, I have gone into

the

earlier

respect

easily perceivable

satisfying

clear that

electron

discussion

Mechanics with

Any

the

in

stage

of

the

to

the

modification

requirements

of

the

theory would hardly afford any noticeable difference in observable processes ; but would lead to very Relativity

* See notes on

8 aud 10.

4

I

lUN Oll LK OK RELATIVITY .

By

surprising consequences.

Postulate from the

outset,

created for deducing

Laws

down

laying

Relativity-

means

sufficient

henceforth

tin

1

have

the

been

of complete from the principle of conservation of form of the Energy being given in series

of Mechanics

Energy alone

(the

explicit foiras).

+

-,.

NOTATIONS. Let a rectangular system (s, //, ~, /,) of reference be The unit of time shall be chosen given in space and time. in

such a manner with reference to the nnit of length that

the velocity of light in space becomes unity.

Although

I

would prefer not

used by Lorent/,

it

different selection of symbols,

geneity will appear

from

denote

electric

the

vector

to

the

change

me

important to

appears

thereby certain

for

the

notations to

beginning.

very

force

the

E,

by

use

a

homo shall

I

magnetic c and the

induction by M, the electric induction by magnetic force bv MI, so that (E, M, e, ;#) are used instead of Lorentz s (E, B, D, H) respectively. I shall further

way which ?

.

.,

is

make

use

of

instead of operating with

where /denotes \/-*-\. use the

method

If

of writing with

based a

general

use

of

the

instead of

evidence

;

apparently a purely real appearance to real equations

;

on

imaginary

//,

this

z,

(/ /),

it),

I

essential will

(1, 2, 3, 1).

be

The

I expressly

emphasize symbols which have we can however at

if it is

of thr symlbols with indices, such ones as

4 only once, denote

(./,

certain

suffixss

advantage of this method will be, as here, tha^ we shall have to handle

any moment pass

operate with

indices,

a

investigations,

physical

(/), I shall

now

circumstances will come into

in

complex magnitudes

not yet current in

understood

have the

quantities,

that suffix

while those

S OTATIOXS

which have not at

all

O

the suffix 4, or have

it

twice denote

real quantities.

An (

r

>r

i

-.

individual ^ 3 4)

Further

system of values of (.*, y, ~, sna ^ b called a space-time point.

let

shall

e.,

c

<r

//.

while p denotes the density

matter,

electricity in space,

which we

/.

denote the velocity vector of matter, the magnetic permeability,

dielectric constant,

conductivity of

/)

and

.v

the vector of

some across

in

7

and

of

the

the of

"Electric Current"

8.

PUNOIPLS OF RELATIVITY

O

PART

I

2.

THE LIMITING CASE. The Fundamental Equations for By

using the electron

Atlier.

Lorentz in

theory,

his

above

mentioned essay traces the Laws of Electro-dynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler

whereby we require that

laws,

= O,

limitting

case

laws for

ponderable bodies.

e=l, /*=!,

=-*, /A=-/,(T

<r=o,

E

every space time

the

In this ideal limitting case At to e, and to m.

M

will be equal

point

for

they should constitute the

y,

(t,

~,

f)

we

shall

for

(.r,

have the

equations*

Curl

(i)

div

(ii)

I 0>i>P

>

(iii)

(iv)

div

Ps>

now

>u

gr=/ p

m=o write

(.r

xy

t

u,,pu y ,pu

r

.r

3

.r

4)

y,

z,

t)

and

ip)

,

convection

the components of the

electric

=

+-.*?

Pi) for (P

i.e.

e=

Curlr

shall 2

m

current pu, and the

\/l.

by

density multiplied

Further 1 shall wriU./23<./3

ltfl

2>

/I

4>./2

./34

l

for

m,, i.e.,

now

m

v

,

m.,

the components of *f

and

we take any two

(1,2,3,4),

ie

ie,,

m

,

indices

/=-/.., *

See note

ie..

i.e.}

(

9.

(h.

along the three axes; k) out of the series

THK FUNDAMENTAL EQUATIONS FOR ATIIEK

7

Therefore ./

,1

/4

-.

1

= =

~~./2

~V

I

s>

4>

f\ 3 ./ 4 4

=

~"./s i

= ~/2

>

y

2

./ 4 U

4>

Then the three equations comprised equation

multiplied by

(ii)

&i

= =

in

*vi ~~.A (i),

-

4

and the

becomes

A

8

x

?

8x 2

8xj

8

8

/i3

/4_2 8x 2

*/_4J.

8x,

On

i

1

8x n

the other hand, the three equations comprised in

and the

equation multiplied by

(iv)

(/)

(iii)

becomes

(B)

^4

,

/t.

8/4. 8x a

8x,

.

8x 4

. . "

8x 2

Sx,

By means of the perfect of

this

symmetry

equations

as

8x 3

method of writing

\ve at

regards

permutation

once notice

the 2nd

of the 1st as well as

with

system

the indices.

(1, 2, 3, 4).

It iv) in

is

well-known that by writing

the

the

equations

symbol of vector calculus, we at once

evidence an

invariance (or

rather

a

f)

to

set in

(covariance) of

the

PKI NT

8

I

I

M. ol

liKI.ATIVITY

system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the For example,

null-point.

round

axes e,

m

the

z-axis.

.r (

amount

an

.r

.r,

2

s

4

=

<

+

sn<

<>,

3

t ,

,

where

=

p,

p, cos

+p / 8 4, L,

^>

......

aud/".,o,

/ H=/t4

COS A

sin^>,

=

p2

p,

p

sin<^>

+/M

sil1

/

*/

=

2 4

"/I

-f

kh

.r

2

,

3

2

p

+ p2

SU1

4

/ =

.r

a

3

p

4

=

(h I k

3

^ +

4=/ Mi :

1,2,3,4).

would follow a corres

then out of the equations (A)

ponding system of dashed equations

(A

)

composed of the

newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equa and (B) concerning the suffices (1, 2, 3, 4), the

tions (A)

theorem of Relativity, which was found out by follows without any calculation at all. a purely I will denote by and consider the substitution i

*i

=

/=

*i>

*/ =

Putting

-

*>>>

x 3 sin

/

tan

^

ity

=

imaginary

\j/

a

+

,

cos/>

where

24 COS ^,./

/,,

.r

.r/,

where

,

+ .r 2 sn .r 2 .r, r, = ;r, cos =.r 3 .r 4 = 4 and introduce magnitudes p ;

,r

of the

keeping

<f>,

and introduce new variables

fixed in space,

.r/ instead of ^

we take a rotation

if

through

= ^3 ,r

4

COR lV cos

T

r

+

-"4

Hn

T

magnitude,

^

(1)

?

i//,

= ^ ^=

lo ^

Lorentz,

_

THE FUNDAMENTAL EQUATIONS FOR ARTHEtt

\Vi; shall liavo

where

i

<

q

cos

and

i,

<

=

i\f/

v/l~?

a

*

s

always to be taken

with the positive sign.

Let us now write x

= /,

,

,7/

y

2

x

i

3

saz

)

iit

x

(3)

then the substitution 1) takes the form

*= y=y,z=-

,,

=,

>t=-

>

v/l-^

(4)

-x/l-f*

the coofficients being essentially real.

now in the above-mentioned rotation round the we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and by t ^, we at once perceive that simultaneously, new If

Z-axis, <

magnitudes p

j,

p

2

,

p

p

3,

4,

where

/

i=Pn

(P

P

=P2>

P

s=Ps

cos

e t/

+

p 4 sin

and/

/

4

COS

12

A,

/3

~/32

+

=

p 4 cos

t^r),

.../ 34 where

i=/4i f

p 4

t^r,

P 3 sin ty

Sin

,

= -/41 sin V +/ 13 +/13 sin COS ^ + / in ^, / = =/ / =/ ^ + /42 COS ^ /I =/18i /** = -/*t

cos ty 4

3 4,

7>,/

t

3 o

3 2

t

4 2

g

4 2

2

Then the systems of equations

must be introduced.

in

(A) and (B) are transformed into equations (A ), and (B ), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures,

and we can then formulate the

last result as follows.

If the real transformations 4) are taken,

be takes as a ,KX (5)

P=P

new frame

of reference,

r-qrn.+l-] -

,

P

and

then we

=P

p.=pu.,

.i/

y

shall

z

t

have

PRINCIPLE OF RELATIVITY

10

(6) e,

qm,

e,

=

,

oe.+m,

,

VI

q*

Then we have

m

newly introduced vectors u

for these

(with components u x

ur

,

u/

,

ex

;

,

<?/,

mx

e/

,

,

m

e y

,

,

/), and the quantity p a series of equations I ), II ), III ), IV) which are obtained from I), II), III), IV) by simply dashing the symbols.

We

remark here that

of the vector

of the positive Z-axis,

product of v and

m

and

The equations

e, ey

+ t mV = ( e + im y =

<

6)

and

+qm,

are

components

a vector in the direction

is

v \=q, qe t

and

[vm~\ is the

+ m,,m

f

vector are the

+ge,

m[ve]. 7), as

they stand in pairs, can be

as.

e, -\-irn

If

i

ev

y ,

v

similarly

;

components of the vector

expressed

e,qm

e+ [ym], where

,

.

+") cos

(e.+t m,)

~e

t

i\ff

denotes any other real

mY)

(e y

i\j/

(e y

sin

+m

y )

i\[/,

cos ty,

we can form the

angle,

:

+ im ) sin + + (^ ^) (e y +tm ) sin + sin + (e / + m /) cos. sin + i^) + (e +im, cos. + cos.

"

<t>+(e s/

]/

cos. )

+tm y )

+

-}-im t .

following combinations t

+

sin

</>

y

(<f>

i\j/),

t

</>

<^>

(<t>

if

)

(<^

).

SPECIAL LORENTZ TRANSFORMATION. The rl 3 which is played by the Z-axis in the transfor mation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation

11

SPECIAL LORENTZ TRANSFORMATION

about this

law

So we came

last

axis.

a

vector

more general

a

to

:

Let

and

v be v

let

which

with

perpendicular to

is

components of Instead of

be introduced shortness, r

v,

we

r v , r-^

of ~y and

the following way.

shall

denote

v.

new magnetudes

y, z, t),

in

v,,

denote any vector

shall

and by

r in direction

vx , vy,

components

we

(.r,

z

y

(x

t

}

will

If for the sake of

written for the vector with the components

is

the

(x, y, 2) in

the

By

=q<l.

\

\

first

z

y

same

for the

r

system of reference,

vector with the components (x

second system we have

in the

}

of reference, then for the direction of

y,

T^r and for the perpendicular direction r

(11)

and further (12)

The notations with

that

cular to v

v,

T =r7 t

=

(/-?, r

ar.+t

v/ r

.

I

)

are to be understood

the directions in

q*

v,

the system

and every

(.r,

are

y, z)

the sense

in

direction

always

y~

perpendi

associated

the directions with the same direction cosines in the system (*

y,

O-

A

transformation which

is

accomplished

(10), (11), (12) with the condition

a

special

vector, the

of v the

y

z

}

We

Lorentz-transformation. direction

moment

If further p (x

Q<q<\

of

v

the

by means of will be

shall

call

called v

the

and the magnitude

axis,

of this transformation.

and the vectors u

are so defined that,

,

e

,

pt

,

in

the

system

PRINCIPLE OF RELATIVITY

12 further

(15)

(e

Then

it

+ V) = (e + m)

i

,

[u, (e

+ /m)]

. ,.

follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The

solution of the equations (10), (11), (12) leads to rc

(16)

2

q

q

Now we about

make a very

shall

1, 2, 3, 4,

instead of

(.c

1

u

(p

,

,

p u

y

t/

,

,

;

,

u

p

and u

u

vectors

the

the indices

,

.

we

so that

write

and p/, p 2

it )

,

important observation We can again introduce (.*

,

#2

,

,

p3

vz

,

,

#/)

instead of

p^

,

ip.

,

Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with

+ 1,

(17) is

.T

a 1

+.r 2

2

+.t s

a

2

+.

4

i.

e.

x*

1

e

x

+ y*+ z *-t\

transformed into *!"

On

+.

.

,"

+

.

+

s

O

-

*+y *+z -t 2

the basis of the equations (13),

(p l *+pS+P>*+ P <*)=P*(l-u t

MS

transformed into p s (l

)

>,-u

we

(14),

*.

shall

have

\-u,*,)=p*(l-u*)

v

or in other words,

A/l^T*

(18) p is

determiuant

the

so that

an invariant in a Lorentz-transformation. If

we

divide (p l

the four values so that It

by

is

the

a)

2 1

+u) a

f(

1

,

,

ps

wa

*+w 3

1

,

,

ps

,

p4 ) o>

o>,,

+to 4

f

apparent that these vector u and

4)

=

by this magnitude, we obtain

=

.

VI

_M .

*

(u,, u y

,

,, i)

1.

four values,

inversely the vector

are //

of

determined

magnitude

SPECIAL L011ENTZ TRANSFORMATION

from the

follows

<i

wj

01,,

(<>

(ID)

values

-1

oj a

o^,

o>

,

3

,

w.t

;

where

and positive and condition

*w 4 real

are real,

13

fulfilled.

is

The meaning the ratios of (20)

The

of

o (<>!,<,,

d

ilr,,

lf

<Ar

,

3

,

here

<>,)

is,

that

they

are

rfx 4 to

V_(d.,: l

denoting the displaeements of matter ^ 3 .i-J to the point (.c lf

differentials

occupying the spacetimc

.

,

adjacent space-time point.

After the Lorentz-transfornation of matter

vococity

same space-time point dx

dz

Now

it

is

components.

apparent that the system of values

quite

=

.(

is

as

^p, ^p,

-J,,

-^-,,

-J

y

(.</

dl

<ly

ratios

is accomplished the system of reference for the t the ) is the vector n with

in the ne\v

transformed into the values

=

*i

in virtue of the

i

"i

-I

a^Wi

3

.

of values

//

5

*4

=

o)

4

the

same meaning

transformation as the

the

after

lias

3

Lorentz-transformation (10), (11), (12).

The dashed system has got velocity

=W

for

first

the

system

got for n before transformation.

If in particular the vector

of the

v

special

Lorentz-

transformation be equal to the velecity vector u of matter at tin- space-time point (.e,, a?,, r 3 x 4 ) then it follows out of ,

(10\

(11), (12) that <o

1

0)

=:o,

a

=

0,

ti)

Under these circumstances space-time formation,

point it is

the invariant

>

f

as

has if

^/ \

the

3

=

0,

o>

4

=l

therefore, the corresponding

velocity

we transform

to

v

=u rest.

after the trans

We

may

call

as the re.-t-dcn-ity of Mlectricity.*

M *

See Note.

PRINCIPLE OF RELATIVITY

SPACE-TIME VECTORS.

5.

mation

and 2nd kind.

the hf,

Of we take the

If

Lorentz transfor

principal result of the

together with the fact that ;he system (A) as well

as the system (B)

covariant with

is

a

to

respect

rotation

round the null point, we obtain the general relativity theorem. In order to make the facts easi.ly comprehensible, it may be more convenient to the

of

coordinate-system

define a series of expressions, for the purpose of

the

ideas

a concise form, while on

in

I shall adhere to

the

of

practice

expressing

hand

the other

magni

using complex

tudes, in order to render certain symmetries quite evident.

Let us take a

linear

homogeneous transformation,

an

a

a 21

L

__

_

,

a

55

22

a 33

a 43

2

a^

+ #2 2 +

+ #4

2

2 3

+^ 4

2

a4 4

shall

be

,r

=//

all co-efficients

while a 4

real,

and

real

is

transforms

The operation

.

4

+1,

occurring once are

are purely imaginary, but

,

a3

33 42 is

,

a2

a 41

out the index 4

a

3

a 31

the Determinant of the matrix

a43

t

23

into

>o,

+#

2

^,

with

,,

+

2 2

called a general

a 42

,

and C

2

3

Lorentz

transformation. If

we put x

=

l

immediately there

mation of

2

a }

.

,

3

=z

aggregate

2

v

of valut^ o,

,

y

y,

~*

:, ./

,c

,

homogeneous

y

,

z

,

t

z

-

with

+t

4

= it

,

*,

.c

and

a positive

2

y*

to I,

;2

every for

is

due

to

real

+t 2 such

which

there always corresponds a positive

This notation, whirli

then

linear transfor

with essentially

)

whereby the aggregrate

transforms into

this

t

(*, y, z, f) to (r

co-efficients,

system

,c

occurs

t

;

Dr. C. E. Cullis of the Calcutta

University, has been used throughout instead of Minkowski s notation,

15

SPACE-TIME VECTORS last is

this

aggregate

The If

column of

last vertical

=

,i

2

a

:!2)

fl

+a.J4 8

4

l

34

=

the

of

continuity

has to

co-efficients

+

a

tf

34

-f-a

44

= i,

then

o,

fulfil,

= l.

2

44

and the Lorentz

to a simple rotation of

reduces

transformation

the

t.

y, z,

the condition

from

evident

quite

.r,

the spatial

co-ordinate system round the world-point. If i4

:

4

Ou a

a s4

l4 , fl

34

:

i4>

44 = v

other

the

a 34

rt 4>

I l) with real

with

which

44

and

zoro,

the

vertical

last

(a l4 , a 24 , a 34 , a 44 )

of

way fulfil the condition v,, we can construct the

,

the

can be

4

24

,

34 ,a 44 )

,

and then every Lorentzsame last vertical column be composed

to

supposed

Lor(?ntz-transformation, and a rotation of

the special

spatial co-ordinate

The

column,

with

transformatiou

we put

value

of

set

every

special Lorentz transformation (.16) with (a^

as

if

in this

of v,, v v

values

all

-v,:i

v

:

,

hand,

fl

not

are

S4

,

:

totality of

of

the

system round the null-point. all Lorentz-Transformations forms a

Under a space-time vector of the 1st kind shall group. be understood a system of four magnitudes Pi,p 8 ,p s p 4 ) ,

with the condition that it

to

is

thes; are

be

in case of

a Lorentz-transformation

replaced by the set p,

tlii!

v.ilue;

sab^tituting (p,,

p.},

oP p.,,

.

c./,

/, p

)

for

.-.,

p2

,

,

,

p3

,

(-,,

x j}

p4

),

where

obtaiael

.e/), .-

3

,

.-

4)

in

by the

expression (il). Besides the time-space vector of -

*3>

4)

of the

we

first

shall also

kind (y,,

make //.,,. // 3

,

y 4 ), and

combination

-* *

ys)

the 1st

kind (x lt

o;

2

,

use of another space-time vector

y.)

let

us form the linear

16

PRINCIPLE OP RELATIVITY

with six coefficients vectorial

/.,/,

v

method of writing,

Let us remark that in the

.

this can be constructed out of

the four vectors. Ci,as*,#a the constants

yi,y.,y s ;/ 3 ./si./n /i*. A* A* and and y 4 at the same time it is symmetrical

;

;

,

4

,

with regard the indices

(1, 2, 3, 4).

,r 3 4 ) and (y t y a y a y 4 ) simul taneously to the Lorentz transformation f21), the combina

we subject

If

tion (23)

is

whei e the solely

.r,,

,

,

,

,

,

to.

+/sl

y, -.r, y.)

(,-,

coefficients

/ 83 / 3 ,

on (/8S / 94 ) and the

We as a

changed

/

(24)

(aj lf

/,

/

x

2

(a,

,

<//-../

/ 14 /, 4 /84 ,

,

coefficients a lt ...a +4

shall define a space-time

system of six-magnitudes

/

2 3

t

/s

t

......

)

+

/,

depend

,

/34

2nd

kind

with the

,

when subjected to a Lorentz transformation, satis it is changed to a new system y^./ ...... 34J ... which fies the connection between (23) and (24). condition that

/"

I enunciate

in

the

manner

following

the

general

theorem of relativity corresponding to the equations (I) which are the fundamental equations for Ather. (iv), If

,--,

i/,

jected to a

(space co-ordinates, and time

z, it

is

if)

sub

Lorentz transformation, and at the same time

(convection-current, and charge density transformed as a space time vector of the st kind,

(/)/*,, pity, pit ,, ip]

pi) is

I

further

and

(m x

,

m

f

,

electric induction

time vector of the

*

)..-,

x

,,

(

:!nd kind,

and the system of

ie,, /) is

(magnetic force, transformed as a space if.)

then the system of c |inti

equations

(IV) trans forms into essentially corresponding relations between the

(1), (II),

corresponding system.

magnitudes

newly

>

is

(III),

introduced

,

.

Vector of the ,

ys

into

the

SPECIAL LORENTZ TRANSFORMATION

These words

:

more concisely

facts can be

the system of equations (I,

17

iu these

expressed

and

II) as well as the

system of equations (III) (IV) are covariant in all eases of Lorentz-transformation, where (pu, ip) is to be trans

formed as a space time vector of the 1st kind, (ni ie) is to be treated as a vector of the 2nd kind, or more significantly,

is

(pit, ip) is a space time vector of the 1st kind, (m a space-time vector of the 2nd kind.

-I

add a

shall

the conception

fe

of

ie)*

v more remarks here in order to elucidate

2nd

the

vector of

space-time

kind.

when

Clearly, the following are in variants for such a vector

subjected to a group of Lorentz transformation.

A (m,

space-time vector of the second kind

and

e) are real

may

magnitudes,

be

(m

where

ie),

called

singular,

and at scalar square (in e =o ie)*= 0) ie m time (m e)=o, ie the vector wand e are equal and perpendicular to each other; when such is the case, these

when the the

s

t

i

>;une

two properties remain conserved for the space-time vector oi the 2nd kind in every Lorentz-transformation. If

the

singular,

a

space-time

we

manner

the Z-axis,

of the

vector

2nd

kind

rotate the spacial co-ordinate system

that the vector-product i.e.

Therefore

m

jt)

(i- t

= o,

+i

e,-=o.

m,, )j(e t

\me\

not

such

coincides with

Then

+i

c x ) is

different

from +i,

and we can therefore define a complex argument in

is

in

such a manner that

= +t m, Vide Note.

<j>

+ i$)

PRINCIPLE OF RELATIVITY

18 If then,

by referring back

<A,

and a subsequent

rotation round the Z-axis through the angle

m

the

the

ev

a

=o,

invariants

2 m"

e* ,

(me)

whether one of them

or of opposite directions, or

equal to zero,

v

both coincide with the new

e shall

of

m = o,

of these vectors, whether they are of the

values

final

same

and

Then by means

Z-axis.

we perform

<,

Lorentz-transformation at the end of which

and therefore

we carry out

to equations (9),

the transformation (1) through the angle

would be at once

is

settled.

%

CONCEPT OF TIME. By

the Lorentz transformation,

we

are allowed to effect

In consequence no longer permissible to speak of the The ordinary idea absolute simultaneity of two events. changes of

certain

of this fact, it

the time parameter.

is

of simultaneity

rather

that six independent

presupposes

parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to Since we are

three.

accustomed to consider that these

represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.* In fact, the following

limitations

considerations will prove conclusive.

Let a reference system (x,y, somehow known.

(events) be

(x,,y

,

O

point

P

time

tt

than

at

(.f, t

,

y,

(let

the time

* Just

as

the time z) t>

a

be

for space time

points

a space point A be compared with a space

the time

at I.)

t

z, t)

Now

less

/,

if

and

if

the difference of

than the length

AP

i.e.

required for the propogation of light

beings

which

aro

confined

within

a

less

from

narrow region

surrounding a point on a shperical surface, may fall into the error that a sphere is a geometric figure in which ouo diameter is particularly distinguished from the rest.

CONCEPT OP TIME

A

to P,

and

if

q= -A

1,

then by a special

is

taken as the axis, and which

i

which

in

transformation,

<

AP

19

Lorentz

has the moment*/, we can introduce a time parameter

t

same value

t

4) has got the

(see equation 11, 12,

both space-time

now

events can

points (A,

),

and

P,

So

t).

which

==o for

two

the

be comprehended to be simultaneous. take

us

let

Further,

t

,

at

same time

the

= 0,

t

two

different space-points A, B, or three space-points (A, B, C)

which

not

are

same

the

in

or

the

AB

plane

difference

t

tt

(t

t

]

be

less

outside the

is

C, at another

>

and

space-line,

therewith a space point P, which

time

t,

and

compare line

let

A

B,

the time

than the time which light

propogation from the line A B, or the plane A B C) to P. Let q be the quotient of (t to) by the second time. Then if a Lorentz transformation is taken for

requires

in

which the perpendicular from

the all

(P,

A B C

plane

is

P on A

the axis, and q

the three (or four) events (A, I.), t)

B, or from

is

[B,

the t

),

P

on

moment, then and

(C, t.)

are simultaneous.

If four

space-points,

which do not

conceived to be at the same time

lie

in

one plane are

no longer per missible to make a change of the time parameter by a Lorentz transformation, without at the same time destroying the t,,

then

it is

character of the simultaneity of these four space points.

To the

the mathematician, accustomed on the one hand to methods of treatment of the poly-dimensional

manifold, of the

and on the other hand

to the conceptual figures

non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Kinstein which so-called

has been cited in the Introduction, has succeeded extent

in

presenting

the

nature

from the physical standpoint.

of the

to

some

transformation

20

PRINCIPLE OP RELATIVITY

PART

ELECTRO-MAGNETIC PHENOMENA.

II.

FUNDAMENTAL EQUATIONS FOR BODIES

7.

AT REST. After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case =!,/*= 1, a- = o, let us

turn

We

look for those

the

to

electro-magnatic relations

it

when proper fundamental data

us

obtain the following quantities at

every

y,

z,

the

:

t]

induction

magnetic

magnetic force tion current

is

occurs

tivity

M, the

m,

the

of

the

electrical

to

and

time,

as functions of

force

electric

the

E,

induction

electrical

for

are given

the

<?,

space-density

the

p,

(whose relation hereafter to the conduc known by the manner in which conduc

current

electric

vector

matter.

possible

place

and therefore at every space-time point (.r,

in

phenomena

which make

in

s

the process),

and

lastly the vector u, the

velocity of matter.

The

relations

in

question

can

be

divided

two

into

classes.

those equations, which,

Firstly

matter

of

is

given as a function of

when (.<-,

y,

r,

v,

the

a knowledge of other magnitude as functions of shall

I

call this first class of

velocity

lead us to

(,],

.,-,

y,

r,

t

equations the fundamental

equations

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector u as functions of *, y,

*,

0-

For the case of bodies at

=

o

the

theories

of

rest, i.e.

when u

(x,

y,

:,

t)

Maxwell (Heaviside, Hertz) and

FUNDAMENTAL EQUATIONS FOR BODIES AT REST the same

Loreoti lead to are

fundamental equations.

21

They

;

The

(1)

Differential

Curl

(i)

~- -

m

(in) Curl

E +

-

which

Equations:

constant referring to matter

C, (iV) div e

=

o, (ir)

Div

=>.

M=

o.

Further relations, which characterise

(2)

no

contain

:

the

influence

matter for the most important case to which

of

existing

we

limit ourselves

i.e.

for isotopic bodies

;

are

they

com

prised in the equations

=

where

=

a-

c

j

>

=

e

(V)

the ">

e

E,

M =

=

C

urn,

dielectric constant,

conductivity

^j * is

hre

<rE.

=

/j.

the conduction current.

By employing a modified form cause a I

latent

magnetic permeability,

of matter, all given as function of

symmetry

in

of writing, I shall

these

equations

now

to appear.

put, as in the previous work,

and write *,

st

,

53

,

*4

for C,,

C y C, ,

V_

1 p.

further /,,,/,,,/, ,,/,,,/,,/,.

form,,m,,m,

(e., e y

t

,

c. ),

andF ls ,F 31 ,F 19 ,F 14 ,F,.,F S4 for

M., M,, M,,

lastly (tin-

we

shall

letter/,

F

-

i

(E,,

E y E.) ,

have the relation shall

denote the

fkk = field, *

,./**,

the

F =

(i.e.

kll

F

current).

)t

k,

PRINCIPLE OF RELATIVITY

22

Then

the fundamental Equations can be written as

a/ sl

3/t, 3.c 4

and the equations

(3)

and

(4), are

a

a..,

3F,.

=

8F, a, 4

=

.

3

3*,

t

.

= + ^^-8 + 3F,,

are

o

THE FUNDAMENTAL EQUATIONS.

8.

"We

=

now

in a position to establish in a

unique way

the fundamental equations for bodies moving in any ner

by means

The

first

When moment,

man

of these three axioms exclusively.

Axion

shall be,

a detached region* of matter therefore

the

vector

"

is

is

zero,

* Einzelne stelle der Materie.

at

rest

for

a

at

any

system

23

THE FUNDAMENTAL EQUATIONS (j-,

y,

in

:,

time point hold

in

the

/)

motion

in

x, y, z,

the

case

hold between

may be supposed to be manner, then for the spacethe samo relations (A) (B) (V) which

neighbourhood

any

p,

possible

tt

when

matter

all

the vectors C,

entials with respect to x, y,

e,

z, t.

at rest, shall also

is

M, The

m,

E

and

their

differ

second axiom

shall

be:

Every velocity of matter

is

<1,

smaller than the

velo

city of propogation of light.*

The fundamental equations when

(f, y, z t

are

a kind that

of such

are subjected to a Lorentz transformation

it)

and thereby (m

(MiE)

and

ie)

are

transformed into

space-time vectors of the second kind, (C, ip) as a space-time vector of the 1st kind, the equations are transformed into essentially

forms

identical

the

involving

transformed

magnitudes. Shortly I can signify the third axiom as (/#,

1<?),

and (W,

second kind, (C,

iE] are

ip) is

This axiom I

:

space-time vectors

a space-time vector of the

of

first

the

kind.

the Principle of Relativity.

call

In fact thes^ three axioms lead us from

the

previously

mentioned fundamental equations for bodies at rest to equations for moving bodies in an unambiguous way.

the

According to the second axiom, the magnitude of the it is at any space-time point. vector In

velocity

<1

\

|

consequence, we can always write, instead of the vector the following set of four allied quantities 11

-

-

u

-

u,

u

x/n^T 11

* Vido Note.

PRINCIPLE OF RELATIVITY

24 with the relation

2= (27) Wl2+ W22+ w ,2+ W4 From what that

the

in

|

has been said at the end of a

of

case

4, it is clear

Lorent/.-transformation, this

set

behaves like a space-time vector of the 1st kind.

Let us now of

iix

our attention on a certain point

time

matter at a certain

= o,

point u tions

we have

then

the equa

at once for this point

of

(V)

(A), (5)

n

according to 16), in case

then

o,

there exists

a special Lorentz-trans-

<1,

\

|

u t

It

7.

(r, y, z)

at this space-time

If

(/).

formation, whose vector v is equal to this vector u (.r, y, z, on to a new system of reference (? y z I ) t), and we pass in

new

values

28)

new

the

therefore

considered, there

point

space-time

the

Therefore

transformation.

with this

accordance

the

0)^

= 0,

vector

velocity

=

o/ 2

arises

in

0, u/ 3 :=0, u/ 4

= o,

o/

as

the

for 4,

=f,

space-time

Now

according to the third axiom the system of equations for the transformed point (.r / z I) involves the newly introduced magnitude point

(n

p

as

is

,

C

,

transformed to

if

e

m E

,

M

,

,

with respect to (x

,

y

,

and

}

z

rest.

,

t

}

differential

their

when

u

=

o,

as

the

But according these equations must be

original equations for the point (x, y,

to the first axiom,

quotients

same manner

in the

z, t).

exactly equivalent to (1) the

differential

equations

from the equations the symbols in (./) and (B).

obtained

where bility,

(^

(#

),

(/?)

which

),

are

by simply dashing

and the equations

(2)

(V

(.-/),

)

e,

<?

^,

= o-

cA",

M = pm

,

C

= a/-

are the dielectric constant, magnetic

and conductivity

for the

the space-time point (t y, z

t)

system

of matter.

(./

//

:

t

permea }

i.e.

in

Now

we then

(/

C,

of

the-

A

///,

<,

,

obtain from the last

20

reciprocal

original variables

(lie

p,

,

Kor \TIOXS

means

us return, by

let

trausl onnation to

magnitudes

MM \II.\T\I.

II

Illl.

(.-,

y,

:,

/),

Lorentz-

and the

and (he equations, which mentioned, will be the funda I/)

mental equations sought by us for the moving bodies.

Now tions

from as

.-/),

and

4,

well

as

Lorentz-transformation,

A

backwards from am

the

equations

The

the equations, which

/>.

we obtain

must be exactly of the same form and as we take them for bodies ),

./)

/>

first result

:

expressing the fundamental electrodynamics for moving bodie?, when

differential equations

equations

of

written

p

in

and the vectors C,

same form

the rectorial

wav

curl

ni

n

K

VMH-! /

The

of writing,

= C,,

II)I div o

matter

In

equations.

we have

f=

lV)divM=o

/

velocity of

these

The

bodies.

moving

in

i

9M = U

-f

are exactly of

M,

K,

///,

>%

as the equations for

velocity of matter does not enter

III

be seen that the equa //) are covariant for a

},

\Ve have therefore as the

at rest.

the

B

}

to

it is

0,

the equations

the anxilliary

in

only

occut>

equations which characterise the influence of matter mi the Let us now basis of their characteristic constants e, <r.

/<,

transform these aiixilliary equations co-ordinates

(

,

Acconlin^ in

to

[//

tn

ponents of

[UP],

the

]),

r],

and

tin-

vector

in ^ /

component of

I,

the

the

i-

/;/

is

,

ni are the

multiplied

by

same

th"

as those of

.

/

On

of

(-omponent

same

us

same

but for the perpendicular direction f

the original

into

)

/.)

formula 15)

the direction of

(?+[H /

//,:,

\

r>,

(+[>

the

llial

>

n|

as that of

the

com

///i)and

other hand

(///

!

/

26

IMUXC

M

and

same

us

aud

<

the relation

= E

>/

c

(D)

M

relation

by

and

to

M

[Kj

the

in

and m

<!

+[>#],

the following- equations

(//>).

uliow

i

we have

/,

M-[ttE]=/t(/y/-[/^]), the

in

directions

perpendicular

to each other, the equations are to be

and

,

,

\livm

I.I

["M,],

///

=//.

For the components /

i;

r+[ W //,]=c(E+[//M]),

(C)

and from the

to

01

I.K

E +

stand to

shall

relation

From

II

multiplied

fll^:

v

Then the following equations follow from the transferin $ 4, when we replace

mation equations (12), 10), (11) q, r t

r-r-,

,

r

t,

,-

,.,

r

(

r ,

n

by

C +

|

\

,

C /

,

p

/

with the components C,

and C

in

,,

"Convection

\\ c /

=M

the

directions

This

current.

remark that

for

=1,

lead

immediately

C

bv means of a rcciprofid

which

in

convenient

be

it will

p,

C

.

v,

p

CM-

,

In cunsideration of the manner these relations,

C,,

C,,,

the

the direction of

vanishes for

last

//-=!

the

the

to H

/

E,

Liu cnt/.-transformatiun

=o

r

ecjuations

equations

,

with

that the fundamental

Ather discussed

:i

becomes

in

obtained here

fact

with

the

limitting

case

= 1, = !, ^o. /*

leads to

of

=E =M

//i

C

of

the

<r=o.

as vector; and for o-=o, the equation

equations

into

vector

|

|

[lerpendicular

T>

to

in

/

p

enters

o-

to call

C=p

it

i

;

in

the equations

FUNDAMENTAL EQUATIONS

THE FUNDAMENTAL EQUATIONS LORENTZ S THEORY.

1).

Let

LOKKXT2 THKORY

IN

now

us

see

how

fundamental

far the

assumed by Lorentz correspond

27

IN

equations

to the Relativity postulate,

In the article on Electron-theory (Ency,

as defined in sjS.

Math., Wiss., Bd. V.

2,

Art 14)

Lorentz

has

the

given

fundamental equations for any possible, even magnetised bodies (see there page 200, Eq" formula (11) on ,

XXX

page 78 of the same (part). )

(H-[//E]) = J+

Curl

(Ul

")

+

/

divD

-curl OD]. div

(1")

(IV")

Then (page

!)

curl

for

=

/.

E =

-^

Div

,

=

13

(V)

moving non-magnetised bodies, Lorentz pubs ^= 1, B = H, and in addition to that takes

:2:!3, :i)

account of the occurrence of the di-eleetric constant conductivity

Lort

o-

iit/ s

e,

and

according to equations

1),

1-],

II

;irt-

here

denoted

i>v

M,

E,

r,

m

while J denotes the :-onduction cuiTeiit.

The

three

last

here coincide with

would

be,

y.ero for

(/

if J

=

equations eq"

which have been

(II), (III), (IV), the

identified with C,

i>

/

/;

just cited

first

(),

(2U) Curl

i

II

-(.,K; =( j

>

f-

equation

(the current

-curl[l)i,

being

28

or

itfxcii T.K

1

and thus comes out

he

to

i;r.

i!

VTFVITY

in a different

form than

Therefore for magnetised bodies, Lorentz

(1) here.

equations do not

s

correspond to the Relnti vity Principle.

On

form

corresponding to the

relativity principle, for the condition

of non -magnetisation

is

as

the

other

to be taken out of

Lorentz

(M

[E]=/;/

Now

[<*]

-

(

)

when

(1) liere

i>.=

\,

[>!)]

by putting

11

not as

B = H,

=H

[D]

= B,

the d.flW-

transformed into the same form

is

;//

[//,]

=M

Pherefore

[K].

as

it

so

by a compensation of two contradictions to

that

happens

(D) in ^S, with but as (30) B

takes,

ential equation e<j"

tlie

hand,

the relativity principle, the differential equations of Lorentz at last agree with the for moving non-magnetised bodies relat.ivit v

If \ve

postulate.

make

use of

pat accordingly of (C) in (e

Il

=

(-

HI)

IJ-r-[",

for

non-magnetic bodies, and

(D

E)], then in conse(|iience

8,

_])

(

K+

r,,,B;)

= 0_E+r,,.

E]],

[;/,D

is. for the direction of n

and for

/

L^rent/

coincides with

i.r. it -

a perpendicular direct ion

in

comparison

Also

to

the

to

1

relativity

of

J,,

-I ;

same order

principle ftW

multiplied by

eijual

assumjition,

if

we

neglect

.

form for J corresponds ;

s

fi,

to the

of approximation, Lorentz s

conditions

eomp. (K) to

^fZ^i

the

or

^ 8]

imposed by

components of

^j^TT

the

that the components <r(E-f

respectively.

[n

B])

i

vjlO.

I

I

\UAMKXT\T.

ATIONS

i;OI

<>K

K.

29

COHEN*

YNDAMKNTAL EQUATIONS OF

E. COHN.

Cohn assumes the following fundamental equations.

K.

(31) Curl

(M-f

Curl [E

E])

(//.

=

^+u

div.

E+

_ + n div.

M)]=

.T

M.

ill

(32) J

where

=

,

r

I }

magnetism

objection

M. to

according to these, for

is

to be put

this

M

=

E.])

-fCU M], transform

>//

[/

M, <"],

our

to

vectors

If in the equations,

and not E-(t*. M), and and with forces, E,

of equations

the

/i

M+[T

magnetic

substitute for

ft.

svstem

= 1, =1,

induction do not coincide.

E and

/*(y//+[>

;

eoiHiderat ion, div.

and

M=

M],

The equations also permit the existence of if we do not take into account this

(induction).

An

cE-|>

and magnetic field intensities are the electric and magnetic polarisation

M

E,

(forces),

true

=

E,

M are the electric

1

E,

M,

p,

then

equations,

div. E, the

as

the

electric

we M, E

this

to

symbols

the differential

and

and

we conceive

E]

a glance

that

is

force

c,

equations

conditions

(32)

transform into J

= r(E+0 M])

,+ [*, (*-[**])] then

in fart

those

order It

the

required

=(E+[M])

equations of (John become the same as the relativity principle, if errors of the

by

are neglected in comparison to

may

1.

be mentioned here that the equations of Hertz

become the same as conditions are

those

of

Cohn,

if

the

auxiliary

.

50

I

lil.N

l

OF KM

I.K

II

I,

TYPICAL HKPKKSKXTATIONS OF THK

11.

EUND A MENTAL EQUATIONS. In

statement of

the

fundamental equations, our

the

leading idea had been that they should retain a

when

of form,

reactions

subjected

Xo\v we

formations.

and energy

a

to

have

to

eovariance

group of Lorentz-transwith

deal

ponoeromotive Here field.

the electro-magnetic

in

the very first there can be no doubt that, the settlement of this question is in some way connected with

from

the simplest forms which can be given to (he fundamental In of covarianee. equations, satisfying the conditions order to arrive at such forms,

fundamental equations

in

of

I shall first

a typical form which

all

puf the

brings

out

clearly their covariaucein case of a Lorentz-transformation.

Here to

I

am

using a method of calculation, which enables us

deal in a simple

the 1st, and

2ud

manner with the space-time

kind,

and of which the

vectors of

rules, as far as

required are given below.

A

system of magnitudes

arranged called a letter If

ft

in

xq

/;

horizontal

ii

kk

rows,

series-matrix,

and

formed into the matrix

and

//

will

vertical

be

columns

denoted

is

by the

A. all

the

quantities

are </,.,<

multiplied by (\

the

resulting matrix will be denoted by CA. It

the roles of the horizontal rows and vertical columns

be interchanged, we obtain

a

yx;;

series

matrix,

which

AI

known

will In-

1

!fiu>1etl

It

A+B

tl-.en

>

i

i

lOHS

ill

matrix oi A,

the transposed

;i>

a second

a,, k

x

jj

ij

series

x

the

denote

shall

are

If

2

-i,.\

aiul will

be

V A.

we have

members

i;i,i i;i

j>

matrix B,

y

matrix whose

series

-\-bhk.

we have two matrices

A=|a M

a

ki

a

where the number of horizontal rows of B, is equal to the of vertical columns of A, then by AB, the product

number

of the matrics

A

and B,

will be

denoted the matrix

. i

,;;f I

these

elements

bein^r

hori/onlal rows of

such a

point,

where S

is

formed

the

eombinat ion

the vertical columns

associative law

A

has l>)

-^ot

vortical

For the transposed matrix of

of B.

(AB) S = A(HS)

a third matrix which has -ot

rows as B (or

of

l>y

A with

u

many

the

For holds,

hori/ontal

columns.

C=BA,

we have C = BA

or RELAIIVITV

.K

We

3".

with

shall

have principally to deal with matrir-rs vertical columns and for horizontal

most four

at

As a

known

unit matrix (in equations they will be

the sake

of

for

shortness as the matrix 1) will be denoted the

following matrix (4 x 4 series) with the elements. (:H)

e 18

e I2

e,,

=

c 14

o

1

I

0100 00

1

01 1 !

For a

4x4

If det

A

+ o,

matrix, which

A

4x4

we may denote by A

is

so that

matrix.

a reciprocal

A -1 A = 1

y.,,

which the elements

,

/,

a

/,

fulfil

called an alternating matrix.

the transposed matrix il/

1

there

the

matrix

/,

the

A

then corresponding to

denote the

shall

elements of

./i/s>

in

A

Det

series-matrix,

determinant formed of the

df,

f

=

,

the

relation

Those ./.

= <

Then by /*

alternating matrix

(35)

;

//,

relations

/,,

/,is say will

that

be

TYPICAL

/>.

We

have a

shall

33

i;i:i iM>i:\Tm<)N.s

4x4

matrix

series

which

in

all

the

elements except those on the diagonal from left up to right clown are zero, and the elements in this diagonal agree with each other, and are each equal to the above

mentioned combination

in (36).

The determinant of /is

therefore the

the

of

square

| combination, by Det

A

4.

which

/we

shall denote]the expression

linear transformation

accomplished by the matrix

is

A=|

a,,. a,

n,

will be

2

,

2>

denoted as the transformation

a ls

"a

,

si

a,

+

a *4

A

By the transformation A, the expression

,+

.,-}-

for

///

^

where a Ai

/

i/,/

=a lt

,i4-

A

a sA

,*+/.

are the members of Jl

produrt of

is

l

changed into the

1-x

!

+u sA

t

a 3l

++*

matrix

series

A, the transposed matrix of

the tranrformation, the rxpn-sion

inu-

h:ue

quadrat ie

."/,

A A=

1.

i-

A

4lt

.

which into A.

cliaiiged to

the

is li

\\\

PRINCIPLE OF RELATIVITY

A

has to correspond to the following relition,

formation (38)

determinant of A) 1,

Det

or

From i.e.

the

if

follows out of (39)

it

that,

trans

For the

to be a Lorentz-transformation.

is

=

2

(Det A)

A= + l. the condition (39)

we obtain

A

matrix of

reciprocal

equivalent to the trans

is

posed matrix of A.

For Det

A

Lorentz transformation, we have further involving the index 4 once in

as

A= +1, the quantities

the subscript are purely imaginary, the are real, and

A

5.

/*

44

time vector of the

space

1x4

represented by the

*=

(41) is

|

to be replaced

A.

i.e.

-f

other co-efficients

>0.

= |

*

*,

by

ft

s

A

?/

*4

*,,

kind* which

s

|

Lorentz transformation

in case of a

=

*/

*../

?.>

A space-time vector of

first

series matrix,

|

.

f>

|

i

*s

2

the 2nd kindt with components

%

A;

|

/* 2 3

.

.

.

/ 34 shall be represented by the alternating matrix

/42

/,,

-A,

is to be replaced by tr;insformation [see the rules in

and

A"

referring

Det^

1

to

the

expression

(A/A) = DetA.

comes an invariant

[eeeq.

(:Jfi)

Sec.

in

/

5

(37),

Det*/

o

A (

in case of

23) (24)].

we have Therefore

a Lorentx.

Therefore

the identity

Det

V

be

the case of a Lorentz transformation

5]. * riiJr nut.. 13.

t

T

l

f/c

note 14.

M Looking back 1

en

fcBPJUU

\i

to (-M),

I-.N

we have

35

r.vnoNs

i

or the dual matrix

*A)(A- /A) = A- /VA-Det"

A- A = Dof

1

(A./

from which

exactly like the primary matrix/, and

time

/*

vector of the II kind;

dual space-time vector

is

i

1

/.

to he seen that the dual

it is

/

matrix/* behaves is

therefore a space

therefore kiiown as the

of/ with components (/

,

4>/

3 4,/34>)

OWso/ij). w and

0.* If

by w

then

combination

are

,v

(as

well

(1-3)

w

,1

two space-time as

by

-v

s v +10,,

i

//>

rectors of the 1st kind

be

will

)

s.j-Mfg

iV

+

3

understood

#>

tin-

4 ,v,.

In case of a Loreiitz transformation A, since (wA) (A-v)

=

this expression

10 s,

and

s

is

invariant.

If

= o,

x

tr

then

w

are perpendicular to each other.

Two a 2 x 1

space-time rectors of matrix

the

kind

lirst

(10,

gives us

.v)

.series

W,

Then

10:

10.,

IV.)

follows immediately that the

it

magnitudes (14)

.v

/._,

3

to.j *._,,

w3

w

-v,

-v

l

3

system of six 10

,

,

s.,io* s,,

behaves in case of a Lorentz-transformation as a space-time vector of the II kind. The vector of the second kind with the components (It) are denoted by [w, that

If

//;

.-eeond

v]=o.

[?,

Del"

written as

/

is

The dual

a space-time

kind,

/

/

A) =

/

/"

/

= \~ \

/.

of

s

["V

see easily shall be

]

vector of the 1st kind,

si^nilies a 1

of a Ijoivnt/.-traiisfonnation / into

vector

We

,v].*

,

[

.v].

l

/ "

.

A, /

is

x

1

A, f

therefore

series is

changed into become>

/ /"

transformed as

matrix.

a

of the

/

In case tr

= icA,

=(/ -AA~

>pace-time

/

vector of

PRINCIPLE OF RELATIVITY

36 the 1st kind.* of the

We

can verify, when w

* O, wf } + O, /*] = (m w

(45)

The sum on the

)

f.

two space time vectors of the second kind

of the

left

a space-time vector

is

2nd kind, the important identity

/ of the

1st kind,

side

the

be understood in the sense of

to

is

addition of two alternating matrices.

For example, for </=

|>

eo/J

*/*!

I

=0,

The

(a

*/*,

o, o, /.,

t

,

=o,

l

o;

a

*!/4S,

/,

2

,

/

t

,

;

w [

;

w 3 =o,

=:o.

I

</*

= =

a>/*]*

I

ta.t

=i,

#3

o, o,

#21

*/! o,

/,

2

I

>

,f i;>J.2l

.

fact that in this special case, the relation is satisfied,

establish the theorem (45) generally, for this has a covariant character in case of a LoreNitz

suffices to

relation

transformation, and

is

homogeneous

in (wj,

wa

o) ,

3

.

w t ).

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants e /x, a will be introduced. Instead cf the space vector u, the velocity of matter, shall introduce the space-time vector of the first kind

to

we

with

the components.

(40) where w 1

and

By F and / of tin-

In

2

iw.t

= wF.

a 2

+

co

2

3

+w

>0.

shall be understood

second kind 4>

+w

\vu

M

i\

].

n>

(lie

space

time vectors

ic.

have a space time vector of the

with components

3

.,

.

4jl

I

ln>

+ w .F^

idc note 15.

first

kind

I-YPICM.

The of

first

the

three quantities

space-vector

and further

O}*

(49)

&

write

4>

t

4

IrL+JJi

JH

-IfiLjEL

a

,

are the components

<,)

.

.

w"

an alternating matrix, 1

^

1

+w

a

4>

2

-|-<jJ

4>j

+w y

4>

3

4>

+oj 4

3

t

r=o.

we can

;

also

a +w.4> 3 ].

I shall call the space-time vector the Electric Rest Force*

Relations analogous

4>

vector w

to the

perpendicular [w x

<

(<,,

37

^1-1T~

^1

is

=w

is

=i

=

<

Because F

i.e.

I;H I;I;SK\TATIONS

of

the

kind as

first

holding between

those

to

<I>

wF,

Uphold amongst w/, v, m, u, and in particular normal to w. The relation (C) can be written as

E, M, is

{

C

}

o)/=e<oF.

The expression (w/)

but the

gives four components, fourth can be derived from the first three.

Let us now form

the

a

J*,*

( (

+"3/4,

!/*

",/,.,+

+*/) *4/)

*.=-(l/.*+Ji, ** = Of

i

*(*!/

these, thu

first

+/! three

+*|/g, !

,.

>!/,.

f = t

/

nil/

/

Fide note 16.

I

.

S)

"i

51)

kind

J

) J/

1st

1

+",/ 12 )

(iinpo;ients of the space-vector

and further (52;

vector

time-space are

^ = iw/*, whose components

*!=-* * =-

otf

irc

(

the "

)

.i\

//.

\

38

ri;i\(

these there

Among (53)

t1>V

(J

ty

>

l

KKI.VIIMTY

01

ii i.i.

is tlie

relation

+ M2 ty. + 0)^.1+ 2

i

which can also be written as 4

The vector * \Iay>ietic

is

4

c>

perpendicular to

M,

* 4 =o

+

"

^a

y

we can

;

~J""^

call

s)

the

it

rest-force.

Relations analogous to Ihese hold

iwF*,

4

=:i (u x fy l

the quantities

amon^

E, u and Relation (D) can

be

by the

replaced

formula {

We F and

f

WF -

D

the

from

and

_ $.

<J>

*J +

* we i

^i/.

e[<o.

,,,/

=_

relation (45) z>[o.

and (D)

(C)

1

,;

<i,,

*--/*.

*]*

1

i!

a

55) 5G;.

.>[

l

A

-on * 3 ] * 4 -w 4 * 3 ].

w /l

[o) 3

t

!

Let us uo\v consider the space-time second kind [$ *], with the components

* 8 * 3 -* 3 *

2

,

^/ 4

l

,

4>

i

xi/

_4>

i

*,* <J>

2

\J/

calculate

and (46), we have

*]

I

to

have

+ *[.*]* F 12 =( Wl $ -w * ) + / la =e(w * a -w * ) +

/= i.e.

[CD.

relations

up* = _

and applying the

F=

-o,F*= /4w/*.

}

can use

1

-* *

4

_4)^\I>

1

3

,

2

.

etc.

etc.

vector

^^j,-*,*, cj>

3

l/

+

<j>

Then the corresponding space-time vector kind w[*, *] vanishes identically owing to

t

\i/

3

of

for 1st

-,

J the

tir-t

c.(juation>

and 33)

Let us now take the vector of the

the

of

kind

the

.

etc.

;>)

TYPICAL

Then (58)

In

applying

=

[*.*]

n

is

also

(^-"O

2

normal

l

^

1

n + =t

(o

o>

+

+ $2.,)

etc.

(to^Qj +oj y Q 2

+a>

2

Q3 )

In case w=o.

o>.

n 4 =o. and

*.v =o,

*

*

a

which

fi.

r

the relation

*i I shall call

=(w ,O

>I

<P.

fi fulfils

we have * 4 =o,

wo

(1

[u>n]

(which we can write as

and

\TI<>\

i

^j^,

i>.

Tho vector

11

1

i:i-si;.\T

i:r.i

is

a space-time vector 1st kind the

Rest- Ray.

As for the we have

relation E),

This expression 8 and 4).

which introduces the conductivity s W 4-S ..) a 2 "r^s** +

a-

+W

( w i*i

u>S=:

us the rest-density of electricity

reives

(see

Then rl)=s+(t,w) represents a space-time vector of the axi>=

1, is

normal to ,>,

now conceive

Let us

cnrrent.

uf this vector as the vector, then the

C

-

I

P

Vll^

and the comiionent

C

current.

fiit,

which

c

kind,

lirst

the

co-ordinates of

~

u I

I

P

rest-

ui

//

-pace-

is

-1 "

l-"

conneeted

we denoted

the

three component

the direction

A/1

i-

which since

call

may

in a per])endicular direction

This space-\ector .1

=

in

1st I

of the

c)

//

(./

component

"

I

and which

in

with the

^

as the

is

(

.,

J

u

.

space-veefur

40

PRIN C[IT,K OK KKI.ATIVITY r

Now

by comparing with

<f>=

--o>F,

the relation

(li!)

cm

be brought into the form

This formula contains

time vector which Lastly,

four

from the

fourth follows

we

equations,

which

of

three, since this

first

the

a space-

is

perpendicular to w.

is

transform

shall

the

differential

equations

(A) and (B) into a typical form.

THE DIFFERENTIAL OPERATOR LOR.

12.

A 4x4

series

S=

matrix 62)

=

S,, S 12 S 1S S 14

|

S,

A

|

S S1 S a , S 2S S, 4

S 41 S 42 S 4;t S 44 with the condition that it is

to be replaced

a Lorentz transformation

in case of

by ASA, may

We

matrix of the II kind.

be called

a space-time

have examples of this

in

:

matrix f, which corresponds to the

1) the alternating

space-time vector of the II kind, 2) the product

fF

mation A,

it is

M) further

when

l

\vn

tlie

( MI

.

ui

s

by (Acu

.

spare-time veotiu-s of

element S AA

.

lastly in in

two such matrices,

of

replaced

which

=w n A

a multiple all

the

A

L

,

co

pace-time point

(.r,

/F A,

I

4 )

and (n i; O 2 kind, the

,

4x4

s

,

U4 )

aiv

matrix with

,

of

rest

>/,

by a transfor

I

the unit in

the

matrix of principal

4x4

series

diagonal

are

are xero.

\Yc shall have to do constantly with F

for

/A-A~ FA)=A~

llie 1st

elements

equal to L, and the

a

1

:, if],

and we may

functions

of the

with advantage

IH

III!

employ

the

Ixl

a

a

I

MJATOR LOR

KKUK.VIIAI. OI

series

formed of

matrix,

differential

symbols,

a 3

a

For this matrix

Then II

S

if

kind,

is,

S

a

use the shortened from

T shall

will

"

lor."*

a

space-time

matrix of the

be

understood

the

as in (62),

lor

by

a

a

a

or (63)

dy 3:

x4

1

series

matrix

where

K =

When system

K.

K<

|

+ *i* +

^L*

4

K,

K,

I

by a Lorentz transformation A, a new reference

(.K\

.<:

,

x

=

lor

,i-

s

4 ) is

introduced,

^

a

8."/

we can use the operator

a 8*,

8.V",

a a-,

Then S is transformed to S =A S A= S S is meant the 1x4 series matrix, whose element j

|

lor

*ri

Now we have

aS|4.oSjt.as. foi

1

t

as

i

*

a

the rule

ST

a

^ 3

.

a .*

3-

t

3. 8

3

3

3 3

*

i

we have symbolically

a -^ 3 .*

,

,

,

,

"T

a 3 j?

.--,

3

,

L $

lor

= lor

Vide note 17.

a W 3

3 3 A.

are

, i

a

3

3A ,

,

(x y

*

3^

87T

so bj

,

*

the differentiation of any function of

+ a

so that,

j

|

3

t

f)

42

PRINCIPLK OK RELATIVITY Therefore

lor

follows that

= lor

(A A-

SA) = (lor S)A.

S behaves

like a

space-time

S

lor

i.e.,

it

vector

the

of

first

kind. If L is a multiple of the unit matrix, then by be denoted the matrix with the elements

If #

is

8L

6L

6L

6L

8-i

8.I-,

9.t

9**

3

=il

+

+

.

9-c 2

8*1

|il 80,

+

V=lor

lor

i.e., lor * is

As = lor

A.

i...

o

these operations the operator lor first

plays

denotes

a space-time vector of the

8/i,

a O

,

+

a/,,

a O

,

"a

a/

a*

+ 8/ l4 0.

,

,

+

+

a/.,

._8At.. 9,

+

l^

+

(i

9-f.-,

the

part

kind.

the components

3.

4

*.

If / represents a space-time vector of the

f

-

an invariant in a Lorentz-transformation.

of a space-time vector of the

lor

will

we have

In case of a Lorentz transformation A,

all

L

a space-time vector of the 1st kind, then

lori

In

lor

8Z

second first

kind,

kind with

43

THK DIFFERENTIAL OPKRATOR LOR So the system of expressed in

differential

lor/=-,

{A}

and the system (B) can be expressed

{B}

log

the form

in

F* = U.

Referring back to the definition (H7) for that the combinations lor (for/), and

find

be

(A) can

equations

the concise form

vanish identically,

Accordingly

^

it

when f and F*

log

we

*,

lor (lor

F*

are alternating matrices.

follows out of {A}, that

&

+

-&

+

+ 1;:-

=

-&

>

while the relation 0>9)

lor

equations

F*)

(lor

in

B},

{

= 0,

only

that

signities

three

the

of

four

independent

represent

conditions. I

shall

now

collect the results.

Let w denote the space-time vector of the

first

kind

\

l_ W 2 (//

F

=

Vl-?<

2

/

velocity of matter),

the space-time vector of the second kind (M,

(M = magnetic

induction,

E = Electric

./the space-time vector af the second kind (/

s

= magnetic

force,

f= Electric

the space-time vector of the space-den>it\

//

a = conductivity,

(///,

/>)

Induction.

first

kind (C.

= electrical (p = dielectric constant, = manin tic ,

>E)

force,

(

>/>=

f

com

1

ip)

net ivity curn-nt,

penne:il)ility,

PEINCIPLK OK RELATIVITY

44 then

fundamental

the

processes in

for

equations

electromagnetic

moving bodies are*

lor/=v

{A}

{B}logF* = o {C} w/ T-V

*

i

=oF /

TiMf"

"X*

{B}^^=-*P. ww = are w,

1,

space-time

and

W F, w/,

vectors

+

kind are

first

which (O>*)<D

all

normal to

and for the system {B}, we have lor (lor

Bearing as

w/**,

o>F*,

of the

many

in

mind

F*)

= 0.

this last relation,

we

see that

sary for determining the motion of matter as vector

11

we have

independent equations at our disposal as arc mvt--

as

a

function

of

,

//,

,

f,

well

as

the

when proper funda

mental data are given. 13.

THE PRODUCT OF THE FIELD- VECTORS /F.

Finally let us enquire about the laws which lead to the determination of the vector as a function of (;,y,:,/.) u>

In these investigations, the expressions which are obtained

by the multiplication of two alternating matrices

/=

THK PRODUCT OF are of

much

importance.

8

Then (71)

S I1

L now

Let

KIELU-VKCTORS

/F

Let us write.

fF=

(70)

indices

T1IK

I, 2,

s ls

s 84

S 3,

S 33 -L

S..

S* a

S*.,

S 44 -L

+S,,+S 33 +S 44 =0.

denote the symmetrical combination of the

3, 4,

=

S .*-L

given by

F 13 +/3I P sl+ / lf p if +/14

/

Fi4

) Then we

shall

have

-

F

t;; In order to express (74)

in

a real form,

S= S 81

S 82 S,

Si

S,,

we

:;

write

X,

Y,

Z,

Y,

Z,

-iT,

Z

-zT

3

S2 +

X,

S 31

S 34

X..

>"

_,T,

-iX, -iT, -tZ, nrJJC.=?

r/^M,-^,,^

;;)

-m M r

r

+<.,E.-

y

E

v

-,.

;

T,

K.l

PRINCIPLE OF RELATIVITY

40 (75)

X,=tn,M,+e.E,,

Y,=wi,M,+?,E,

etc.

X,=e,M,-e

T,=m,E,-,E

etc.

M,,

r

t

li,=~ These quantities are

Z

ZJ

v ,

are

known

are

magnetic

known

Y

(X,, X,, X., Y., Y,,

as "Maxwell s

the Poynting

as

In the theory for bodies

all real.

at rest, the combinations

s

L

as

r ,

Z,,

T,, T,, T,

as the electro

Vector, T,

and

energy-density,

Stresses,"

the

Langrangian

function.

On

the

matrices (77)

hand, by multiplying the alternating and F*, we obtain

other

of,/"*

P*/*=-S 1I -L, S S1

-S lt

,

-S tl

S g2

L,

S 2S

.

S,

S1

S 41

-S 14 S 24

,

L,

S,,

,

2

.

S,,.

S 4S

S 4S

S 44

L

and hence, we can put

/F = S-L,

(78)

F*.r*=-S-L,

where by L, we mean L-times the unit matrix,

i.e.

the

matrix with elements .

|

L^<

|

Since here

,

(c 4A

=l,

e**=0,

SL = L^, we

A^A-

/*,

A-=l,

2,

3,4).

deduce that,

F*/*/F = -S-L) (S-L) = - SS + L, (

and

find,

since/*/

=

Dot -/. F*

F

at the intonating conclusion *

Vide note 18.

=

Det

*

F,

we

arrive

THE PRODUCT OF NIK M ELU-VKCTORS

=

SS

(79)

-

L

S

the product of the matrix

i.r.

*

Det

into

th*

elements

1

ex

a matrix in which

diagonal

the principal diagonal are

in

be

can

itself

the elements except those in the principal

zero,

47

* / Det F

pressed as the multiple of a unit matrix all

/

and have the value given on the right-hand

side

are

all

equal

of.

(7 J).

(

Therefore the general relations

+ S*.

S M S lt

(80)

S at

+6

A ,

S,*+S 4l S 4 =,

h, k being unequal indices in the series

S M S lA +S*, S,*+S*

(81)

S

Det * / Det for// = 1,2,

Now and

if

(73),

=L -

S S +S*, S 4 *

and

3, 4, a

F,

3, 4.

instead of F,

we

1, 2,

/

and

introduce

the

combinations (72)

the

in

rest-force *,

electrical

the

magnetic rest-force *, and the rest-ray tt [(55), (56) and (57)1, we can pass over to the expressions,

=

L

(82)

S

(83 j

i

c

*

+

p.

J

*

*7

4

c

-f-

(h, k

=

1, 2, 3, 4).

Here we have <|><t>

=

4i i

+t/+<l>

e AA

The in

a

:

,

=

+*,

,

1, e kk

right side of (82)

as

* *

=

^,

J-*!

1

+**+*

o (h=f=k).

well

as

Lorentz transformation, and the

L

4x4

is

an

invariant

element on the

PRINCII LK OF RELATIVITY

48

right side of (83) as well as

vector

second

the

of

S*

kind.

suffices, for establishing the

a

represent

A

Remembering

time

space this

fact,

theorems (82) and (83)

it

gener

= o, w, =o, s =o, l w = o, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) = ps// on the other hand. on one hand, and c = eE, to

ally,

But

for the special case

it

prove

w 4 =/.

M

the"

The

expression on the right-hand side

equals [-5-

^is

=

o,

?/?

o>

for this case

M-

(m

because (cm

c

+

<?E)]

<S>

^,

of

which

(81),

(em) (EM),

=

(EM)

/x

4 *; now referring

>

back

to 79),

we can denote

expression as Det

Since

f

*

/,

S.

=

and F

transposed matrix of

F/

(84)

the positive square root of this

S,

-

-

we

F,

obtain for S,

the following relations from

= S-L,/*

ThenisS-S=

|

F*

= - S-L,

S*-S,*

|

an alternating matrix, and denotes a space-time of

the

second

kind.

From

the

the

(78),

expressions

vector

(83),

we

obtain,

S

(85)

-

from which we deduce that (86)

<o

the matter

is

-

(c/t

1)

fa>,n],

[see (57), (58)].

(S

-

w (S -~S)

(87)

When

-

S~=

= = (e p S)*

o,

1)

O

at rest at a space-time point,

then the equation 86) denotes the existence of the

ing equations

Z y =Y,,

X,=Z,,

Y,=X,,

u=o,

follow

THE PRODUCT OF THE FIELD-VECTORS

4J

/>

and from 83),

T,=n

T,=n

if

2

x^t^n,, Y,=

T..=

,

f/

,n 2

.

:|

#,=^0,

Now

by means of a rotation of the space co-ordinate system round the null-point, we can make,

X =Z,=

Z,=Y.=o. .

According

t

to 71),

,

we have

and according to 83), T,>o. In vanishes it follows from 81) that

=

.

X,+Y,+Z.. + T,=o.

88)

and

X,=X,=

special

cases,

where Q

T, and one of the three magnitudes X,, Y,. Z. Det * S the two other.s = Det ^ S. If fi

if

-

r

vanish

let

O

^=0, then

we have

does

in particular

are not

from 80)

T X,=0, T Y,=0, Z,T s+ T T =0. r

s

t

and

if

f),

=0,

1= 0,

Z

r

=-T,

It

f

follows from (81),

(see also 83) that

and

-Z,=T, = The

Det

"S-f-e,*!),

space-t- me vector

of very o^reat imjwrtanpe

demons! rnte

of the

first

kind

K = lorS,

f89) is

V

a very

Accord in-

to

lor

for

which

we now want

important transformation

7*\ S = L+/F, and

S=lor L + lor/F.

it

follows that

to

PRINCIPLE OK RELATIVITY

50

The symbol in

lor

of f, on

Accordingly

lor

The

parts. (lor

/)

F,

The second

differential

process

78)

can be expressed as the sum

f

first

/

being regarded as a

part

is

that

part

x

1

4-

/F,

of lor

which

of

two

matrices

series matrix.

in

which

the

components of F alone.

the

upon

operate

the

the product of

is

part

lor

diffentiations

From

denotes a

lor

operates on the one hand upon the components the other hand also upon the components of F.

/F,

we obtain

/F=-F*/*-2L;

/F=

hence the second part of lor of

2 lor L, in

components

of lor

where

N

is

which the

F

thus obtain

f

6F,, ~^~

S1

= !,

2,

f

_

3,4)

By using the fundamental relations A) and transformed into the fundamental relation lor

(91)

In the identically.

part

upon the

the vector with the components

<//

is

F*)/*+ the

S = (lor/)F-(lor

-~-~ 8F 8F..

We

alone

(lor

differentiations operate

limitting

S=

case

B),

90)

*F + N. <

= 1.

/x

= l.

/=F. X

vanishes

PRODUCT OF THK FIELD-VECTORS /K

1HI.

Now upon the and referring back we obtain

57) of

basis of the equations (55) to the expression (82) for L,

the following expressions

as

51

and (56), and from

components

\,_ X

,92)

t

=-

*

for

Now

if

the

symbol

A=l,

we make

vector which

has

l_l **_.

**

2. 3, 4.

use

of (59),

11,, i! 2 , 11 3

as the

and denote the space-,//,

-

components by

then the third component of 92) can be

W,

expressed in the form

,93)

The round bracket denoting the vectors within

1A.

product of the

THE PONDEROMOTIVE FORCE.*

Let us now write out the relation in a

scalar

it.

more practical form

Vide note 40.

;

K=lor S =

.

we have the four equations

52

PRINCIPLE OF RELATIVITY

_

0# +_.. 9Z. _ QZ, _ v

/

1

07 \

(97)

K* _ -K-

T "

a

9 Ty

9T

.

9T

I?

,

<-t^i:

It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the spacetime point , y, :, /, it lias got, y, - components as the .

.<,

first

three components of the space-time vector

This vector Jiiids iti

is

perpendicular to w

;

the law of

Energy

expression in the fourth relation.

The establishment

of

this

opinion

i^

reserved for a

separate tract.

In the limitting case

=1, /*=!,

<r=0,

the vector

S=p(D, wK^^O, and we obtain the ordinary equations theory of electrons.

N=0, in

the

APPENDIX MECHANICS AND THE RELATIVITY- POSTULATE. would be very unsatisfactory, if the new way of at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of It

looking

Phvsics.

Now many in

authors say that classical mechanics stand postulate, which is taken

to the relativity

opposition

new Electro-dynamics.

to be the basii of the

In order to decide this

let us fix

our attention upon a spe

Lorentz transformation represented by (Hi), (H), with a vector v in any direction and of any magnitude

cial

but different from zero. For a

any

relation

special

-)>

<y<l

moment we

shall not suppose between the unit of length

hold

to

(1

we shall and the unit of time, so that instead of /, / write ct, ct and q/c, where c represents a certain positive ,

<y,

,

constant,

and

is

-7

The above mentioned equations

<c.

arc transformed into

,

c1

They denote. (>y>

as

s th

1

-)>

= oo,

The new tion

of a

</

we remember,

that

space-vector

y

If in these equations,

the limit c

q

then

we

e(|Uatiniis

-patial

spatial co-ordinate

(

kcej>ing

r

is

the space-voMor

)

constant

we approach

obtain from these

would now denote the transforma

co-ordinate \>t.

in

(

y

system (*, y, :) to another ) with parallel axt-s, the

54

IMUNCIl LK OF RELATIVITY

null

point in

velocity

second system moving with constant the time parameter line, while

of the

a

straight

We

remains

unchanged. mechanics postulates

a

can, therefore, say that classical of Physical laws for

covariance

the group of homogeneous linear

transformations of

the

expression

+* c=

when

Now

it

of Physics,

is

we

(1)

<x.

rather confusing to find that in one branch shall find a eovariance

laws for the

of the

transformation of expression (1) with a

value of

finite

c,

another part for c=oc.

iu

It

that according to Newtonian

evident

is

this covariance holds

c=oo and not

for

Mechanics, ==

for

}

velocity of

light.

May we for

f

= oo

experience,

not then regard

an

as

only

the actual

those

covariances

traditional

consistent

approximation

with

covariance of natural laws holding

for a certain finite value of

c.

I may here point out that by if instead of the Newtonian Relativity-Postulate with e=oc, \ve assume a relativitythen the axiomatic construction postulate with a finite c, .of

Mechanics appears

The in

to gain considerably in perfection.

time unit to the length unit

ratio of the

a manner so as to

make

the velocity of

is

chosen

light equivalent

to unity.

While now in

convenient to leave

and

want

I

the manifold of

t

as

to

the (y,

)

out

any possible pair

refered to oblique axes.

introduce geometrical

variables

of

-F

(

,

//,

-,

account, and co-ordinates

in

figures

may

/), it

to

a

treat

be

s

plane,

APl

space time null point

\

55

KXUIX

/=0,

-,

y,

(.r,

will be

0, 0, 0)

kept fixed in a Lorentx. transformation.

The

-y -z* +* = !, a

*

figure

...

/>0

(2)

which represents a hyper holoidal shell, contains the spacetime points A (j-, y, :, / = 0, 0, 0, 1), and all points A which after a Lorentz-transformation enter into the newly introduced system of reference as

The the

direction of a radius vector

A

point

A

(2) at

(.r

,

y

-.

,

,

= 0,

/

0, 0, 1).

OA drawn from

to

of (2), and the directions of the tangents to

are to be called normal to each other.

Let us now follow a definite position of matter in its all time t. The totality of the space-time which r, t/, correspond to the positions at points (

course through ,

different times

The task

/,

shall be called a space-time line.

of determining the motion of

prised in the following problem:

It

is

for every space-time point the direction line

passing through

To transform

matter

is

com

required to establish of

the

(*, y,

~,

space-time

it.

P

a space-time point

to rest

f)

is

equivalent to introducing, by means of a Lorentx transfor

mation, a new system of reference the

t

axis has the direction

OA

,

OA

(

y

,

,

,

tion of the space-time line passing through P.

=const, which is

is

to be laid

t

},

in

which

indicating the direc

through P,

is

The space

the one

which

perpendicular to the spuce-time line through P.

To

the increment

///

of the time of

P corresponds

the

increment

of

the

newly

the intesrral

introduced time parameter

/".

The

value of

56

PRINCIPLE OF RELATIVITY

when

calculated

initial

point

P

space-time

upon the space-time

line),

is

known

we

are concerned with at

position of matter

as the

which

fixed

was

introduced

the space-time of

Lorentz

by

of the

Proper-time

(It is a generalization of the idea

point P.

time

from a

line

to the variable point P, (both being on the

Positional-

for

uniform

motion.) If

we take

at time

t

body R* which has got extension

a

then

,

line passing-

the region comprising

through R* and

all

in

space

the space-time

shall be called a space-time

/

filament. If Q(x,

y

we have an z t)

=

ft

anatylical expression

intersected

is

filament at one point,

6(.\>

y,

?, /)

by every space time

so that

line of

the

whereby

*.\ -( a \

then the tolality of the intersecting points will be called a cross section of the filament.

At any point. P of such across-section, we can introduce by means of a Lorentz transformation a system of refer ence (/, y,

:

t),

so that according to this

.

8y

8..

The

direction of the

question

here

is

known

section at the point for

the

known

P

uniquely determined

/

axis in

as the upper normal of the cross-

and the value of r/J=/ J / ,hj ,f P on the cross-section is :

<L>

surrounding points of

as the elementary contents (Inhalts-element) of the

be

cross-section.

In this sense

cross-section

normal to the

point

and the volume of the body

/=i",

R"

/

is

to

regarded

as

the

axis of the filament at the

regarded as the contents of the cross- section.

R

is

to be

APPENDIX

we allow R

If

we oome

to converge to a point,

of an

conception

57 to

the

infinitely thin space-time filament.

In

such a case, a space-time line will be thought of as a principal line and by the term Proper-time of the filament

this principal line

of

Proper-time which is laid along under the term normal cross-section

understood the

will be

the

;

we

filament,

shall

upon the space which

understand

normal

is

the the

to

cross-section line

principal

through P.

We

now formulate

shall

the

of conservation

principle

of mass.

To every space the

quantity to a point

(.c,

known

a

R

:, /),

R

the

as

The an

at

y,

the volume of

R

mass at

approaches a limit mass-density

thin

space-time

where /x= mass-density of

(i.e.,

the

through

Now

(

t), is

,y,:,

p.(x,

t/,

the

at

is

positive

converges

which

:, t),

space-time

mass says

of

(.r,

y

t

: , /)

that

for /*//.},

of the

//.I

is

point

filament, the product

at the point

fila

=eontents

normal to the / axis, and passing constant along the whole filament. of the normal cross-section

the contents f/J.

the filament which

R

If

t.

the principal line of the filament),

cross-section

a

belongs

t,

then the quotient of this mass, and

principle of conservation

infinitely

ment

time

at the time

laid

through (,

^

=n\/ \_.

of

f) is

t/,

:,

*

=/*-2l. O

(4) </!

and the function

v=

iw 4

may

be defined as the 8

l

(5)

rest-mass density at the position

58

PRINCIPLE OF RELATIVITY

(xyzf). Then the principle of conservation be formulated in this manner

of

mass can

:

For an

infinitely thin space-time

tJie

fi/amenl,

product

of the rest-mass density and the contents of the normal cross-section is constant atony the whole filament. In any space-time filament, let us consider two crossand Q which have only the points on the

sections

Q."

,

boundary common the

inside

on

Q.

shall

filament

The

.

be

finite

a

called

Q

boundary, and If

to each other

is

then

to

the space-time lines

sic/tel*

space-time

Q

on

i

ti is

Q

than

and

Q

the lower

the upper boundary of the nickel.

we decompose a filament

filaments,

let

;

have a larger value of range enclosed between

an

into elementary space-time

entrance-point

of

an elementary

filament through the lower boundary of the nickel, there

corresponds an exit point of the same by the upper boundary, whereby for both, the product vdJ, taken in the sense of (4)

and

has got the same value. Therefore the difference

(5),

two

integrals jW/n (the first being extended over the upper, the second upon the lower boundary) vanishes.

of the

According

well-known theorem

to a

the difference

is

////

lor

vu

dfdydzdt,

the integration being extended over the

,

Kic/icl,

and (comp. (67),

-_ --

8vo>

D11

"

1

of Integral Calculus

equivalent to

+ 8 -

<i)

2

+ -8^3 +

If the nickel reduces

tin

whole

range of

15)

,

,

-3va) 4

to a point,

then

the

differential

equation lor vw = 0, * Sichel original terra

(6)

a

German word

is

retained as there

meanintr a crescent or a scythe. is

no snitnble Knli.sli equivalent.

The

\iTi\m\ which

is the-

-*

condition of cortinuity ,

r

.

a* Further

a^ the

/

,

a--

form

us

let

59

_

f<8<

integial

N=SISJvdlyd:dt whole range of the space-time

extending over the

We

(7) .tic/tcl.

decompose the sichel into elementary space-time filaments, and every one of these filaments in small elements shall

dr of

proper-time, which are however large compared

its

normal cross-section

to the linear dimensions of the

us

let

;

assume that the mass of such a lilament vdJ n =dm and

write

T"

r

,

l

for the

boundary of the

Then

the integral (?) cau be denoted

IJvdJn taken over

Now

of the upper

Proper-time

all

let

and lower

sichel.

</T=/(T

-T

by

dm.

)

the elements of the sichel. of the

us conceive

space-time

lines

a

inside

space-time nichd as points,

and

continual

let

material curves composed of material suppose that they are subjected to a

us

of length inside the sichel in the follow

change

The

ing manner.

manner

possible

to

entire c-urves are inside

the

on the lower and upper boundaries individual substantial points upon

be

while

*ic/tc(,

remain it

varied

in

fixed,

and the

are displaced in such a

manner that they al \va\s move forward normal The whole pit, cess may be analytically

curves.

sented by

means

t

,f

a

parameter

\,

may

.r

the

repre

.v/

7/t7.

Such a

be called a virtual displacement in the sichel.

Let the point values

to

nnd to the value \ = o,

shall correspond the actual curves inside the

process

any

the end points

+ S--,

.//

(.r,

r, /)

_//,

+ fy,

.-

in

+ 8-, f +

the /,

sichel

\ = o have the

when the parameter has

60

PRINCIPLE OF RELATIVITY

the value A

these magnitudes are then functions of (./, y, Let us now conceive of an infinitely thin spacetime filament at the point (j- y : f) with the normal section

of

;

A).

z, /,

dj n

contents

,

and

dJ n +&dJ K be the contents

if

of the

normal section at the corresponding position of the varied then

filament, of

mass

(v

to

according

+ dv

the principle of conservation

being the rest-mass-deusity at the varied

position),

(8)

In

(v

+ Sv)

(dJ

+ Sd J

)

=

= dm.

i/rfj

consequence of this condition, the integral (7) over the whole range of the sichel, varies on account

taken

of the displacement as

and we may

call

a

N + 8N

function

definite

this function

N + SN

of

X,

as the mass action

of the virtual displacement.

we now introduce the method

If indices,

(9)

we

shall

rf(a 4

of

with

writing

have

+8.- 4 )=cl,- 4

^

+ 2 !?* + O OA k r

k

d\

*= 1,2,3,4 ^=1,2,3,4

Now clear

on the basis of the remarks already

that

parameter

the value is A,

will be

of

N + SN,

when

the

made,

it

is

value of the

:

(10)

the integration extending over the whole where ^(r + Sr) denotes the magnitude, which

Jd ., + d$x by means of

(9)

and

t )

sichel is

_(dx s +d8x~)*

<(

deduced from

r ^(~d.i>

4

+dS

.>

APPENDIX therefore

:

= 1

We

now

shall

the

subject

value

of

the

1, 2, 3, 4.

= 1,

2, 3, 4.

differential

quotient (12)

to a transformation.

vanishes

z,

for

do.l t

,

general

----

O.

=o,

-

8-

as a function of (r, y,

A

for \

so in

A.,

= o.

*

Let us now put

then

Since each

the zero-value of the paramater

=

*

-

-

(

A

)

(/*

= !,

2, 3, 4)

(13)

on the basis of (10) and (11), we have the expression

(12):-

,1 B

for the Kic/i

-f,

\aiul

system

(8.r,

(a-,

8r 2

therefore

r

./

./-.,

,

&r 3 ,,

8 }

a

*

w

-i

is

-U

4 )

shall

)

,f,,

integration, the integral

on

4

the

boundary

(Jy

d: dt

of

the

vanish for every value of

are

nil.

Then by

partial

transformed into the form

rw*tf, "^v

,

8"w*,

~a^r Jj;

1 <hj

1

.

PRINCIPLE OF NEGATIVITY

6-1

the expression within the bracket

The

sum

first

equation

be written as

vanishes in consequence of the continuity

The second may be written

(6).

au

may

Qw

(l>

l

k

Qw

dcy

di

h

s

as

a<o

A

dc 4 ~

( l*>

d

k

whereby

is

meant the

differential

(dr k

dr\dr

dr

quotient

the

in

t/T

For the

the space-time line at any position.

direction of

differential quotient (12),

we obtain

the

final

in

the

Kiclicl

expression

dx

//-

tit.

d>/

For a

virtual

displacement

postulated the condition that the

we have

Hijipnsi d

points

to

be

advance normally to the curves i^ivinjf their actual motion, which is \ o; this condition denotes

substantial

that the

A

shall

is

to satisfy the condition

H

I

^i+

Let us now turn

^a

+w

our

3

3

+;*

attention

f= to

(15)

.

the

Mnxwellian

electrodynamics of stationary bodies, and and 13; then we find 1:1 us consider the results in

tensions let

8

in

the

that Hamilton

s

Principle can be reconciled to the relativity

postulate for continuously extended elastic media.

M PKXDIX At every space-time point (as in matrix of the ind kind be known

13),

let

a space time

s lt (16)

S=

where X, Y, For a

......

X

virtual

x

T, are

real

displacement

(with the previously the integral

,

applied

magnitudes. in

a

space-time siohel the value of

designation)

PRINCIPLE OF RELATIVITY

64.

methods of the Calculus of Varia

applying the

By

the

tions,

four

following

follow from this minimal

differential

principle

equations

at once

by means of the trans

formation (H), and the condition (15).

and

-,

(7,

= l,a

:

S4)

+*

components of the space-time vector 1st kind K lor S, X is a factor, which is to be determined from the

relation

tv^j=

K + (Kw}w

By

1.

the four,

summing

we

multiplying

obtain

X = Kw,

and (19) by w k and therefore clearly ,

will be a space-time vector of the Jst

normal

is

=K,+ x!t

.

=

K,

,,,onee

are

9

v

(19)

kind which

Let us write out the components of

to w.

this

vector as

X, Y, Z,-/T

Then we

arrive at the following

equation for

the

motion

of matter,

=X

(21)

(

|

ar \dr/

and

X

=T,

f+Y

On is

dr

;

,-

dr

\drj

=Z,

and we have also

+Z (Zr

</T

=T

,

:

I

=T

</T

<

.

(?T

the basis of this condition, the fourth of equations (21)

to be regarded as a direct consequence of the first three.

From

(21),

a material point,

we can deduce the law i.e.,

for

the

the law for the career of

thin space-time filament.

motion

of

an infinitely

65

APPENDIX Let

*, v, 2,

denote a point on a principal line chosen

f,

any manner within the filament. We shall form the equations ( 21) for the points of the normal cross section of in

the filament through s t

y t z, /, and integrate them, multiply ing by the elementary contents of the cross section over the If the integrals of the whole space of the normal section.

R y R R, and we obtain

right side be R,

R

is

now a

if

t

of the filament,

vector

space-time

R

components (R,

mass

be the constant ^

the

of

R. R,) which

y

m

time vector of the 1st kind w,

is

1st

kind with the

normal to the space-

the velocity of the material

point with the components

d.f

,r

>

(IT

We

may

call

this

dz

dy ~T

dt

.

T

*

~j

<ir

</T

<IT

R

vector

t/ie

moving force

the

of

material point. instead

If

of

ment which (

over the normal section,

integrating

integrate the equations

over

normal to the

is

/

axis,

s.4V,0, then [See (4)] the equations

are

now

tion

comes out

in

The

right side

is

multiplied by

(hue per unit 9

section of the

that cross

;

in

we

fila

and passes through

(2:!) art-

obtained, but

part ioular, the last

equa

the form,

to be looked li/ne

<>f

at

upon

the

x

the

material

amount of

u-ork

In

this

point.

PRINCIPLE OF RELATIVITY

66

we

equation,

the

obtain

energy-law

motion of

the

for

the material point and the expression

be called the kinetic energy of the material point. is always tit greater than tlr we may call the

may

Since

r

-

quotient

the

as

"

Gain

"

(vorgehen)

of the

over the proper-time of the material point and the law The kinetic energy of a then be thus expressed ;

terial point is the

time over

The

its

product of

its

metry

mass into the gain of the

(s&tsj), which

is

a^ain

shows the sym

demanded by the

sical

significance

seen

in

the

be attached, as we have already

to

is

case

analogous

of this

demand

for

of the first three equations

model of the fourth

;

relativity

a higher phy

postulate; to the fourth equation however,

ground

can

ma

proper-time.

set of four equations (22) in

time

in

electrodynamics.

symmetry, the are

On

the

triplet consisting

to be constructed after the

remembering

this circumstance,

we

are justified in saying, "

relativity-postulate be placed

If the

mechanics,

then

from the law I

of

the

whole

set of

at the

of

energy."

cannot refrain from

showing that

no

contradiction

assumption on the relativity-postulate expected from the phenomena of gravitation.

to

head

laws of motion follows

the

can

be

If B*(.>*, /*, z* /*) be a solid (fester) space-time point, then the region of all those space-time points B (./, i/, z, /), t

for

which

VIM

be called a

may

KXDIX "

"

Hay-figure

67

(Strahl-gebilde) of the space

lime point B*.

A

space-time line taken in any manner can be cut by this

figure only at one particular point

this easily follows

;

from

convexity of the figure on the one hand, and on the other hand from the fact that all directions of the space-

the

time

are

lines

concave side

only directions the

of

from B* towards to the

B* may

Then

figure.

be called the

light-point of B.

the point

If in (23),

the point

(t*

t/*

the relation

(:J#)

time points

ft*,

y

(

c* (*) be

z /) be

to be

then

\vould represent the loeus of all the space-

which are light-points of B.

F

of mass

to the presence of another material

experience a

fixed,

variable,

Let us conceive that a material point

may, owing

be

to

supposed

supposed

moving

force according to the

following

m

F*,

point

law.

Let us picture to ourselves the space-time filaments of F and F* along with the principal lines of the filaments. Let

BC F

;

be an infinitely small element of the let B* be the light point

light

OA

point

of

C

on the

figure (23) parallel

to

intersection of line

B*C*

and point

passing

F

in

line of

of

be the

F*; so that

of the

and which

composed

D*

finally

is

the

with the space normal

The moving

space-time point B kind which first

time vector is

B*C*, B.

through the

principal

C*

of B,

vector of the hyperboloidal fundamental

the radius

i.s

line

principal

further

force

point of to

itself

of the mass-

is

now

is

normal to BC,

the

space-

of the vectors

3 (

(21)

;;*(

another

j\

vector

of

HI)*

in the direction

)

Mutable

\;i.lue

in

of

BD*, and

direction of

B*C*.

PRINCIPLE OF RELATIVITY

68

Now

--

by

vectors

in

two

understood the ratio of the

J

It

question.

shows

once

to be

is

#

(

that this proposition at

clear

is

character

covariant

the

with

respect

to a

Lorentz-group.

Let us now ask how the space-time behaves when

the material

translate ry motion,

F*

i.e.,

F*

point

filament

the principal line of

of

F

a

uniform

the

filament

has

Let us take the space time null-point in this, and by means of a Lorentz-transformation, we can take this axis as the /-axis. Let y, z, t, denote the point of

a

is

line.

.*

B, let

Our

,

proper time of B*, reckoned

T* denote the

from O.

proposition leads to the equations

/ae\

<*

where (27)

In of the

,

c

*

_

2 +y* + ^ =(t-

(27), the three

consideration of

same

form

as the

equations for

equations (25) are the motion of a

material point subjected te attraction from a

fixed

centre

according to the Newtonian Law, only that instead of the time t, the proper time r of the material occurs. The jx>int

fourth equation

(2(>)

gives .then the

connection

between

proper time and the time for the material point.

Now point

(.r

for different values of

y

z] is

the eccentricity

an

ellipse

e.

Let

E

T,

the orbit of

the space-

with the semi-major axis a and denote the excentric anomaly,

T

69

A1TKND1X

the increment of the proper time for a complete description of the orbit, finally n~T initial

= 2tr,

so that

nr=E-e

(29) If

we now change

velocity of light

by

c,

sin E.

the unit of

7,

f

7

i

denote

the

1-fecosE

neglecting c~* with regard to

*="*L

time, and

then from (28), we obtain i*

Now

from a properly chosen

we have the Kepler-equation

T,

point

1, it

follows that

m * l-fcosE~l i^ssffiJ

i

1+t

from which, by applying (29),

(31) nt

+

=(\ 1 + ^-^ ac*

const

the factor

is ^

\ wr-f /

SinE. ac 2

here the square of the ratio of a certain

average velocity of F in its orbit to the velocity of f If now denote the mass of the sun, a the semi

w

axis of the earth

The law ed

and

of

orbit,

then this factor amounts to 10~ s

mass attraction which has bn-n

which

relativity

s

is

j>ostu!a<e

light.

major

foimnlated

would

in

signifv

just describ

accordance that

.

with

gravitation

the is

In view of the fact propagated with the velocity of light. that the periodic terms in (31) are very small, it. is not

possible to decide out of astronomical observations between

such a law (with the

Hnd

the

mechanics.

modified mechanics

Newtonian law of

pro|K>sed

attraction with

above)

Newtonian

70

PRINCIPLE OF RELATIVITY

SPACE AND TIME A

Lecture delivered

before

Naturforscher

the

sarnmlung (Congress of Natural Philosophers)

at

Yer-

Cologne

(21st September, 1908).

Gentlemen,

The conceptions about time and to develop before

physical grounds. is

itself,

and time

and some

Herein

for itself

old

hope

strength.

The tendency of space for

shall reduce

sort of union of the

I

experimental

conception

lies its

Henceforth, the

radical.

which

space,

you to-day, has grown on

two

mere shadow, be found consistent

to

will

a

with facts. ,

I

Now I want to show you how we can arrive at the changed concepts about time and space from mechanics, as accepted now-a-days, from purely mathematical considera The equations of Newtonian mechanics show a two remains unaltered when (i) their form

tions.

fold

we

in variance,

the fundamental space-coordinate

subject

any possible change of

position, (//)

system

to

when we change the when we impress upon

of motion, e., system in its nature itanv uniform motion of translation, the null-point of time We are accustomed to look upon the axioms no /

part. plays of geometry as settled once for

same amount nics,

.

all,

while

we seldom have

the

mecha

of conviction regarding the axioms of

and therefore the two invariants are seldom mentioned same breath. Each one of these denotes a certain

in the

group of transformations

for the differential

Wo

equations

look upon the existence of the mechanics. as a fundamental characteristics of space. off prefer to leave

li^ht

In-art,

tin:

second

conclude that

group

to itself,

first

We

and

of

group

always with a

we can never decide from physical

considerations whether the space, which

is

supposed

to be

APPENDIX at rest,

may

not finally be in uniform motion.

lead

groups

71

quite separate existences

Tlu-ir totally

mav

heterogeneous character

So those two

besides

each other. us

scare

away

from the attempt to compound them. Vet it is the whole compounded group which as a whole gives us occasion for thought.

We

wish to picture to ourselves the whole relation Let ( be the rectangular coordinates of i/, z)

graphically. space, and

are

/

<

,

denote the time. connected

always

Subjects of our

observed a place except at a particular time, or a time except

Yet

a particular place.

at

perception

No

place and time.

with

I

one has obserred

Tias

the

respect

dogma that time and space have independent existences. will call a space-point plus a time-point, i.e., a system values x,

;/,

-,

/,

part in

all

will be the world.

z, t,

//,

four world-axes with (onsi..ts of

The manifoldness

as a n-or Id-point.

possible values of r,

the

Now

chalk.

for the four-axes does not

cause

In

trouble.

wo

not to

order

shall

something substances. :,

this

t,

at

ric.it

\Vo direct

\

,

we

coordinates of this point

not

the

any subsequent time.

[t/.r, /It/,

a picture, so to speak, of the

the

if:].

perennial

a curve in the

to

to

time,

specify

simplv style these as ti-nrlil-pohil

in a position to

the time element corresponding to

substantial point),

place and

our attention to

point at

any

yawning gap

order

shall

and suppose that wo are

substantial

abstraction required

mathematician

every

In

exists.

perceptible

the

allow any

suppose that

either matter or elect

y,

drawn

any axis

;

The greater

,

of all

can draw

quickly vibrating molecules, and besides, takes the journevs of the earth and therefore gives

us occasion for reflection.

exist,

I

I

of

changes

recognise Let-

of

t/f

space

Then we obtain life-career

world

he

of

(as

the

the Karltl-linr,

the points on which unambiguously correspond to the para

meter

/

from

+ oo

to

oc.

Tho whole world appears

to be

72

PRINCIPLE OF RELATIVITY

resolved in sucli

my

if I

point

and

world-lives,

I

from

just deviate

may

say that according to

opinion the physical

my

laws would find their fullest expression as mutual relations

among

By

these lines. this conception of

folduess

/

=o

and

time and space, the (,#,

two

its

sides

If for the sake of simplicity,

and space

fixed,

then the

signifies that at

f

=o

/<o

and

^>o

falls

c)

mani-

asunder.

we keep the null-point of time named group of mechanics

first

we can give

the

., y,

and r-axes any

possible rotation about the null-point corresponding to the

homogeneous

linear transformation of the expression

The second group denotes that without changing the laws, we can substitute

expression for the mechanical

(.v-at,y-pt, z-yt} for

(.-,

According to

constants.

y, z)

where

(a,

p, 7 ) are

any

we can give the time-axis

this

any possible direction in the upper half of the world Now what have the demands of orthogonality in spaco

/>o.

to

do with this perfect freedom of the time-axis towards the upper half

To meter

?

establish this connection, let us take a positive para c,

and

let us

consider the figure

According to the analogy of the hyperboloid of two two sheets separated by ( = o. Let us

sheets, this consists of

consider the sheet, in the region of

l>o,

conceive the transformation of

r,

of variables

;

(,/,

y

,

z

,

t

)

.-,

y,

transformations.

the

in

and the

let us now new system

by means of which the form of

the expression will remain unaltered. of space round

/

null-point

Now we

Clearly the rotation

belongs

can have a

to this

full idea of

group of the trans

formations which we picture to ourselves from a particular

HKMHX

AI

tHUuforaialkm

and

.r-

=

]

remain

r)

Let

unaltered.

the upper sheets with the

of

/-axes, -

hyperbola -c-l

fig.

(y,

cross section

the

of

plane the

which

in

draw the

us

73

the

i.e.,

witli

f

half

upper

its

asymptotes

of (/////

1).

Then

A B B C

draw the radius

n

let

A

at

and

,

produce Let us now take Ox suring rods

OA

,

from

(.-,

i/ y

/)

-,

L-it us

|iif-iioii.

<

to

add

= 1,

is

.

mea again

and the transi

>o

one of the transitions in

i/z t} is

(

t

OA D

at

the unit

then the hyperbola

;

-

-

-axis

new axes with

-

(

the tangent

,

parallelogram

meet the

to

as

OC = 1, OA =

expressed in the fonii tion

C

B

also

;

OA

rector

us complete the

let

to this characteristic

transformation

any possible displacement of the space and time null-points ; then we get a group of transformation depending only on which we may denote by CJ r .

,

let

N"MW

it

:i])]icii

shrunk

i-n iie--

tlie

"leiv

")

that

siiii)>ly

state

iii

possible

but

10

1 <|irf-[;>.!

is

j^radu-

nnglc

lic-

nmsfninrit ion the

that

upwiirds,

ic

/-axis

and

in

(-in

-ind

m>iv

is Remembering this point the full group belonging to Newtonian Mechanics the n roup (i with the value of c=oo. In this it

r ,

thii

i

l)ol:i

.

i

(r hit

-x>,

a

(

ai-l

is

!

in:tt

mat hem

also for a

not

uroup

d

hemat ically

;

.

upon the thought

possess nn invarianre in

-

asymptot

manner

:i

dir;"-tiun

-oxiin:ites tn

imagination,

mena

ich

aiTairs, :inl sinre

iellin ible

the

-;i\i<.

one, and everv s

the h\ pei

figure that

the

limit changes in

i

ol

rdu the

into

-tnii^hi

:i

Miv

is

i

>

becomes zero

-

c

-

:uxl all\-

Thus

increase c to infinity.

as,

only ,

,

that for

where

,

b\ a

more i

iu-

n-e p!:iv

natural pheno the

group

c is lintte,

( <,,

but yet

74

PRINCIPLE OF RELATIVITY

exceedingly large compared to the usual measuring units. Such a preconception would be an extraordinary triumph for pure mathematics.

At

the same time

we shall

I shall

remark for which

substitute the velocity

of

light

c

in

of

value

For

can be conclusively held to be true.

this invariance

free

c, <,

space.

In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electro static

and electro-magnetic units of

We

can form an idea of the invariant character of the natural

for

expression

G,

Out

laws for the group-transformation

manner.

in the following

of the totality of natural

successive

system

electricity.

(., y, z, t)

phenomena, we can, by

approximations, deduce a coordinate by means of this coordinate system, we

higher ;

can represent the phenomena according 1o definite laws. This system of reference is by no means uniquely deter

We can change mined by the phenomena. manner corresponding

reference in any possible

meniioned group transformation G c bnt the natural laws will not be changed thereby. ,

For figure,

example, corresponding

we can

call

t

to

the

the to

system of the

abore-

expressions

for

above described

the time, but then necessarily the

it must be expressed by the maniThe physical laws are now expressed by t and the expressions are just the

space connected with foldness (,/

y

means of

,

:).

1

,<

same

as in

shall

have

?/,

:,

,

the case of

.-,

y,

z, t.

According to

this,

we

one space, but many spaces, the case that the three-dimensional

in the world, not

quite analogous

to

The threespace consists of an infinite number of planes. dimensional geometry will be a chapter of four-dimensional physics.

Now

you perceive, why

I said in

the

beginning

APPENDIX

75

that time and .space shall reduce to mere shadows and shall

have a world complete in

we

itself.

II

Now

the question

what circumstances

be asked,

may

changed views about time and space, are contradiction with observed phenomena, do

lead us to these

they not in

they finally guarantee us advantages for the description of natural

phenomena

?

Before we enter into the discussion, a very

important Suppose we have individualised

point must be noticed.

time and space to the

-axis

world-line

inclined

the

to

to a point

moving

a

to

correspond

moving uniformly

point

pond

manner; then a world-line

in a,ny

will

-axis

;

correspond to

will

and a world-curve

;

parallel

point

stationary

will

a a

corres

any manner. Let us now picture

in

mind the world-line passing through any world now if we find the world-line parallel

to our

point r,y,z,i;

to the radius vector

we can introduce

OA OA

of the

a

as

hyperboloidal sheet,

new

time-axis,

and

then then

according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned.

axiom

We

now

shall

Tin substance ,

!,<

<?</*//////

af

fiiiK-fircil to be,

space

introduce

fundamental

this

:

at

resf,

am/

if

ice

worfil

point

.flatifix\\

The axiom denotes that

Kiiifafj///.

in

vnr a

can

always

lime

and

world-point

tho expression ,.*,/(.*

shall

always

be

_,/.,-! _,///

or

positive

_,/;*

what

is

equivalent

to

same thing, every velocity V should be smaller than >hall

therefore

velocities

and

be

herein

the

upper limit

lies

a

deep

for

all

the c.

substantial

significance

for

the

76

I

KINTIM.K

01-

I!

At the first impression, the axio: quantity It is to be remembered that be rather unsatisfactory. will occur, in which the square a modified mechanics only <.

root of

this

will play

in

combination

differential

takes the

that cases in which the velocity

so

time,

no part,

of

imaginary coordinates

like

something

place

greater than c

is

geometry.

The impulse and

real

cause

inducement for

of

G

assumption, of the yroup-kranisforniatioii

r

is

tlie

that

the fact

the differential equation for the propagation of light in On vacant spase possesses the group- transformation G r the oth-3r hand, the idea of rigid bodies has any sense .

in

only

G

we have an

if

there

are

common

the

got

rigid

bodies,

be

defined

can

/-direction shells

to the

further

Gx

,

and

G

f

h; inl

that

a

hyperboloidal which Ins ,

by means of suitable the laborafory, we can perceive a

in

natural

in

two

the

by

groups

.

easy to see

is

it

that

consequence,

rigid instruments

change

Now

a system mechanics with the group x optics with G,., and on the other

if

uhcnomena,

in case of different orienta

with regard to the direction of progres-ive motion the earth. But all efforts directed towards this

tions,

of

and even

object,

of Michelson

supply

an

formed

a

invanance Lorent/

1

(

v

the

have

celebrate!

u iven

for

explanation

of

this

-vhich

hypothesis optics

i

or

~ ~

"

)

^

ll

- tn

results.

;roup shall

" >

result,

practically

ihe

substance

every

interference-experiment

negative

II.

order to L. rent/

amounts

to

an

Aeeordi

G,..

Mifi er

In

A.

a

the direction of

contraction

its

motion

7?

\ITI,.\|l|\

l

sounds rather phantastical.

hypothesis

in>

.rtion

For the

nut to be

thought of as a consequence of tin; resistance of ether, but purely as a gift from the skies, as a is

sort oF condition

always accompanying a state of motion. in our figure that Lorentz s hypothesis fully equivalent to the new conceptions about time and

is

show

shall

I

Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (y, z} and fix our attention on a two dimensional world, in which let upright space.

strips parallel to the /-axis

another

parallel

strip

state of uniform motion

extension (see

s:nti:il

we can take

strip,

in

O.\

IF

state of

and

rest

represent a

the /-axis

to

which has a constant

for a body, l).

fipf.

a

represent

inclined

parallel to the second

is

as the /-axis and x as the *-axis, then

/

ond body will appear to be at rest, and the first body uniform motion. We shall now assume that the first

body supp

is,-

cross

seutidn

\vlu-re

OC

is

1

to be at

PP

l.-u^th

this

/,

the -e .-oud strip has p;irall

-l

1

>

;nd

t.

our

the

<Tnss-soct,ion

:i

\\\-

strip

lielon^iii ^

Nov.

if

;in

01)

= OC

to

moves

(>;ls

x

it

=/

OC, and

v

OC when

/

,

measured

;se

iinifurinlx

ov

then (lie

nie::sui--ii

-iuc,.

.

if

th

Coordinates,

,-!,;,!

i|

"

the

i.e.,

/,

-axis

ifl

>

oilier

original

1

,

two bodies, we have a; Lorentz-ele ctr6bs, one of which In

iveii

If

length

when supposed to beat rest, has the means that, the cross section Q Q of

-axis.

tin-

no\v images of rest

the

upon the

the unit measuring rod upon the s-axig

the second bodv also,

same

has

rest,

of the first strip

(i

U

the

p:-.

rallel

=r/()C

.

therefore

if

we

extension

cross section

falcnhtion u ives that

,-

Now

.

to

that

(Id the

stick

of

the

of the -;i\is.

U(4 = / O1)

.

78

fRlNCiPLK OK RELATIVITY This

the

is

sense

of

Lorentz

hypothesis about

s

On

contraction of electrons in case of motion.

we conceive

if

hand,

the

second

and therefore adopt the system

P P

(*

,

then the cross-section

,)

of the strip of the electron parallel

regarded as

its

length and

we

to be at rest,

electron (

tlie

other

the

OC

to

the

shall find

to be

is

electron

first

shortened with reference to the second in the same propor tion, for it

is,

_ ^L_2H_P ~ QQ -OC -OC PP

P

CPQ

/

?

Lorentz called the combination local Li ne (Ortszeil) of the

of

I

(t

and

as the

,)

uniformly moving electron, and

used a physical construction of this idea for a better compre hension of

the

of

any other

valent,

Phys. 891,

electron,

been

has

electron

the concept of time was

service

shown

to

to

by was not arrived in

perceive

good as the time as equi

[Ann.

d.

1907] There be completely and un

natural

or Lorentz, probably because

as

A. Einstein

of

d. Radis...

ambiguously established concept of space

is

are to be regarded

i.e. t, t

the

1905, Jahrb.

p.

But

contraction-hypothesis.

clearly that the time of an

4-1-1

1

phenomena. at,

the

either

case

of

But the

by Einstein the

above-

mentioned spatial transformations, where the (./, / ) plane coincides with the plane, the significance is possible that the ^-axis of space some-how remains conserved in its position.

We can approach the idea of space in a corresponding manner, though some may regard the attempt as rather fantastical.

According to these ideas, the word Relativity-Postu which has been coined for the demands of invariance "

late"

in the

group

(r,

understanding

seems

of

to be rather inexpressive for a

the group

G

f ,

true

and tor further progress.

79

\PPENDIX Because

the

sense

the

of

postulate

the

that

is

four-

given space and time by pheno mena only, but the projection in time and space can be handled with a certain freedom, and therefore I would

dimensional

world

rather

to

like

in

is

to

give

Populate of the Absolute

name

the

assertion

this

worM"

"

The

[World- Postulate].

Ill

the world-postulate a similar treatment

By

of the four

determining quantities x,i/ z, t, of a world-point is pos sible. Thereby the forms under which the physical laws t

come

forth, gain in intelligibility, as I shall presently show.

Above

all,

much more

the idea of acceleration becomes

striking and clear. I shall again use the geometrical

Let us

call

any world-point

point."

The cone

consists

of

/<()

f\\e

,

two parts with

the other having

fore-cant

towards

O,

/>0.

O

as

The

method of expression.

as a

"

Space-time-null-

apex, one part having

first,

which we

consists of all those points

the second, which

we may

may

call

which send light the

call

aft-cone.

light

from

The region bounded by the fore-cone may be

called

consists of

O.

O

all

those points which receive their

the fore-side of O, and the region bounded by the aft-cone

may

be called the aft-side of O.

On

the aft-side of

hyperboloidal shell

O

F = rV

-p

(Fide

fig.

:>).

have the already considered

- r 2 -y*

-z"

= 1,

t>0.

80

HlMXCIl

RELATIVITY

Ol

I.K

Hie region inside the two cones will be ocoupibd by the

one sheet

f

hvperboloid

F= 9

where

can liave

which

hyperbolas

+

s+j/ all

^t^k-,

2 v-

are important for us.

dual branches of this hyperbola will be called the Such a hyperbolic liyperbola. with centre

"

of

a

as

for

/

and

o

approaches the velocity of light

t

=

represent

a

asymptotically

<x>,

<.

to the idea of vectors in space,

by way of analogy

If,

we

would

world-line,

=

Tnter-

branch,

0."

when thought motion which

The

positive values.

possible

upon this figure with O as centre, For the sake of clearness the indivi

lie

any directed length in the manifoldness -,//,:,/ a then we have to distinguish between a time-vector

call

vector,

directed

from

O

towards the

directed

space-vector

The time-axis can be

sheet

O

from

+F

1,

and a

/>0

F=

towards the sheet

1.

any vector of the first kind. Any world-point between the fore and aft cones of O, mav by means of the system of reference be regard --d either

than

as

synchronous

O.

with

O, as well as later or

Every world-point

nece-ssarilv

always

O, later thin O.

earlier,

Tin

to

parallel

on

every

limit f

the

fore

Let us decompose a (.r,//,/,/) ve<

vector

into its components.

If

Ins

been

to one of

tin-

surfaces

is

drawn

com

bet \\.v.i

In the

intentional!;-

iV.>in

O

t

"\\-ards

the directions of the two radius

tors are respectively the directions of the

OR

a

oa corresponds to

up of fcbfe wcdgc-shawd cross-section and aft cones in the taanifoldofess / = ().

this cvos-s-scc! ion figure drawn, drawn with a different breadth.

O

point on the nft side of

plete folding the.

earlier

of

fore-side

+F=l,find

of

a

vector

tangt-nt

IIS

APPENDIX at

tin-

point

R

surface, then

of the

called

normal to each other.

which

is

ponents

condition

the

(-, y,

:, /)

81

that

and (f

the

vectors

shall

be

with the com

vectors

are

z^ /j)

// L

l

the

Accordingly

normal

each

to

other.

For the measurement of vectors unit

the

measuring rod

to

is

in

different

directions,

be fixed in the following

a space-like vector from F = I is always to have the measure unity, and a time-like vector from

manner; to

O

to

F= 1,

-|-

/>()

Let us now substantive ;, 1}

we

to

always

have the measure

.

our attention upon the world-line of a

fix

point

then as

;

is

running through the world-point follow

the

the

progress of

(r, y,

line,

the

quantity

dx*

Vc*dt*

dz*

dy*

c

,

corresponds to the time-like vector-element (dr, dy, dz, dt}.

The any

integral

T=

initial

fixed

I

taken over the

dr,

point

P

(

to ,

world-line from

any variable

final

point P,

may be called the Proper-time of the substantial point at P upon the irnrlil-H.i*-. We may regard (r, y, :, t], I.e., "

"

components of the vector OP, as functions of the

the

"

"

proper-time

((iiotifMts,

of

(

,

/,

11

and

:,

T; let

(.r,

//,

(.r, //,

r,

/)

?, f)

the

with regard to

denote the

s(>cond

T,

then

first different ial-

differential

tlu

si-

may

quotients

respectively

82

UIXCIL

I

he

the

called

OF RELATIVITY

Lli

and the Acceleration-rector

Velocity -vector,

Now we

of the substantial point at P.

2

the

i.e.,

measure

is

Velocity -vector

in

Now

any

P

infinitely

world-line

centre

we have

M.

to the

at

,

vector

of unit

Accele

velocity-vector at P,

contiguous

P,

a

of

If

M

points

common

in

and of which the asymptotes fore-cone and an aft-cone.

be called the

may

(vide tig. 3).

then

time-like

)

as can be easily seen, a certain hyperbola,

is,

This hyperbola at

normal

is

the generators

are

the

is

O

z

yy

(

case, a space-like vector.

there

the

P

at

which has three with

,

in the direction of the world-line at P, the

ration-vector

and

X

f

t

have

"

hyperbola of curvature

"

be the centre of this hyperbola, an Inter-hyperbola with

to deal here with

P = measure

Let

of the

vector

MP, then we P is a vector

easily perceive that the acceleration-vector at

-

of magnitude

in the direction

of

MP.

P

If r, y, at

P

at P, and p

t

z,

=

are

nil,

then the

hyperbola of curvature

to the straight line touching the

reduces <x

world-line

.

IV In order to demonstrate that the

assumption

of

the

group Cr r for the physical laws does not possibly lead to anv contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this successfully

group.

made

in the

The

revision

case of

"

has

already been

Thermodynamics and

APPENDIX i or

Radiation,"* i

linally

For

"Electromagnetic

phenomena

and

",t

or "Mechanics with the maintenance of the idea of

mentioned province of physics, the ques

this last

may be asked if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-

tion

:

y

point (x,

how

}

where the velocity-vector

z, f),

we

is (f,

y,

z,

t),

when the system of any possible manner ? Now it is known that there are certain well-tested theorems about then

are

reference

is

to regard this force in

changed

the ponderomotive force in

G

the group us

lead

to

electromagnetic

where

fields,

is

undoubtedly permissible.

These theorems

the

following simple rule

{/ the system of

r

;

reference be changed in to

any way, then the supposed force is be put as a force in the new space-coordinates in such n

warmer, that the corresponding rector with the components *X,

T=

where

irork is

This vector

(?Y,

(

r1

-4-

\

a

X 1)\-

a

\v

I

t

rate

of

representing -recfur

<it

a

force

call

k, YAM-

tn-ii>ii<

x

the

at

P,

may

be

P.

P

will

be described

mechaincut

velocity-vector

Dynnmik bcwrgtor svRtemo, Ann.

d.

at

P

///</vv

as the

physik, Bd. JO,

1.

II.

edition.

(the c-

done at the world-point}, remains unaltered. always normal to the velocity-vector at P.

the world-line passing through

liiiii

l

-

t

substantial point with the constant

1908, p. t

X -f ~J Y + T- Z^ =

is

inuriiiii / oro

Let us

///.

Z, /T,

/

Such a force-vector, calk-il

t

MinkoNvski

;

the

ji:issn, o

refers to

paper (2) of the present

PRINCIPLE OF RELATIVITY

84-

iwpuhe-r.er.tor,

and m-times the acceleration-vector

P

at

as

of motion, at P. According to these law tells us how the motion of the definitions, following

the force-vector

a point-mass takes place under any

The force-vector of motion

is

moving force-vector*

equal

to the

:

moving force-

vector.

This enunciation comprises four equations for the com ponents in the four directions, of which the fourth cnn be deduced from the first three, because both of the above-

mentioned vectors are perpendicular

From

the definition of T, the

expresses

we

mass.

i.e., if

is

the

fourth

Energy-law/

Accordingly the

direction

be defined as the kinetic-energy of the

to

The expression

we deduct from

for this

simply

c z -times the

"

component of the impulse-vector in t-avis

to the velocity-vector.

that

see

of

the

point-

is

this the

additive constant me 2 ,

we

obtain the expression 4 mv- of Newtonian-mechanics upto

magnitudes of the order of

Hence

.

it

appears that the

But since the

energy depends upon the xyxlcm oj reference. /-axis

can be

laid in the

direction

of

therefore the energy-law comprises, for of reference,

thr>

This fact retains C

= oo,

for

mechanics,

the as

whole system its

of

axiomatic

any

the

construction

been

possible system

equations

significance even in

has already

any time-like MM S,

pointed

of

of

out

Newtonian by T.

Schiiix.t * Minkowski

Planck f St-hutz,

Mechanics, appendix, page

-\Vrh. d.

I).

fiott. Nuclir.

P.

(I.

1897,

G.">

of paper (U).

Vol. 4, 1906, p. 136 p.

110.

motion.

limitinir rase

R.

Al

KMMX

I

85

From the very beginning, we can establish the ratio between the units of time and space in such a nr.mner, thai the

of

velocity

\/~l

=

/

/,

,h

place of

-

into

,

we now

write

<ly

//,

.",

ilz

/)

*

+

ill

-

),

symmetry then

this

;

law, which does not contradict the worfit-

each

We

jjoxtnlafe.

(

+

a

-

in

If

unity.

then the differentia] expression

/,

= - (dx +

becomes symmetrical enters

becomes

light

in the

can

the

clothe

postulate in the mystical, but

"essential

nature

of this

mathematically significant

formula

^l

rH0 5 frm=i

The advantages are

the

by

the

illustrated

exerted by a point-charge

ing to the

formulation of

the

by nothing so strikinglv expressions which tell us about the reactions

world-postulate as

from

arising

Sec.

in

moving

Ma \\vell-Fjorentz

any manner accord

theory.

Let us conceive of the world-line of such an electron with "

Pr.

In order to

world-point to

(f), and let us introduce upon it the reckoned from any possible initial point. obtnin the lieM oau.-ed bv the electron at anv

the charge p r-(ime

P,

"

T

let

us

construct

the

tig.

4).

Clearly

this

I ,

(ridi

world-line of the electron directions are

tangent

normal

to to

all

time-like

tin s

According to

tangent.

vectors.

and

the

Let

definition

reckoned as th- measure of PU.

nnliinit-d

tin-

single point P, because these

at a

the world-line,

fore-cone belonging etits

let

r

of

At

be the a

P,

let

draw the the

measure of P,Q.

fore-cone,

Now at

us

draw from P,

us

rfc is to

be

the world-point P,,

Ob

PRINCIPLE OF RELATIVITY

the vector-potential of the

by the vector

in its three space

the scalar-potential

the

This

-axis.

PQ.,

is

represented

having the

magnitude

components along the

is

--axes

x-, y-,

;

represented by the component along the

is

A. Lienard, and K.

excited by e

field

direction

in

law found

elementary

out

by

"Wiechert.*

If the field caused

electron be described in

by the

the

above-mentioned way, then it will appear that the division of the h eld into electric and magnetic forces is a relative one, and depends upon the time-axis assumed ; the two considered

forces

force-screw

some

bears

together

mechanics

in

;

the analogy

analogy to the however, im

is,

perfect.

now describe the ponderonwlivc force which is by one moving electron upon another moving electron.

I shall

exerted

Let us suppose that the world-line of a second pointLet us passes through the .world-point Pj.

electron

M

determine P, Q, r as before, construct the middle-point of the hyperbola of curvature at P, and finally the normal

MN

upon a

"With

P

line

along PQ, the in

the

the

in

of reference

a

of

direction

z,

7 be

which shall

it

the

is

MN,

*

first

Lionard,

election

L

:

normal

<1.

the

acceleration-vector

e,

QPj.

system

The

QPj.

.

-,

at

is

t/-,

P,

t

?/-axis

automatically

Let

^-axes.

.<

,,

y l}

z

,, /,

Then the force-vector exerted

(moving

Eolnirnpi clcctriqne

Wirchert, Ann.

to

a

the /-axis will be laid

then the r-axis

be the velocity- vector at P,.

bv the

way

following

parallel to

is

establish

-axis in the direction of

determined, as 1, y,

P we

through

as the initial point,

T

Pliysik, Vol. 4.

in

any

10, 1896, p.

possible

:{.

manner)

Al

the

upon

><(<

1

election

.mil

at

manner)

o>>iblc

I

I ,

is

7 7"

y

following three relations hold

and fourthly

this vector

(likewise

>-,

moving

in

any

represented by

FJt

For the components

87

ENDIX

F

F

F., F, of the rector

,

the

:

normal

is

P,, a ltd through this circumstance

to the velocity-vector

alone,

its

dependence on

laxf relitcity-vector arises.

f/iis

If we compare with this expression the previous for mula * giving the elementary law about the ponderomotive 3

action of

electric

moving

cannot but admit, that inner

the

reveal

dimensions

;

but

charges upon each other, then we the relations which occur here

essence in three

of

full

simplicity

four

first in

com

dimensions, they have very

plicated projections.

In the postulate,

disharmonies

Newtonian

between

relations

reformed according to the worldwhich have disturbed tin-

mechanics the

mechanics, and I shall

electrodynamics automatically disappear. sider the position of the I will

this postulate.

describe

w,

exercised

by m upon

instead of

C;IM-

*

in

which

tlu-

A. Lorcntz,

;

a

and the expression

electron,

We

only

we

m

and

force-vector

moving

is

just tin-

have

to

is

saun write

shall consider only the special

acceleration-vector of

K. Sdiwar/.si-liild. II.

/#,,

ee^.

>//;;/,

attraction to

assume that two point-masses

their world-lines

as in the case of the

+

Newtonian law of

modern

now con

in

is

always zero

:

(iott-N aclir. 1903.

Ens/Uopidie dor Math.

\Vi.<.-i

iiM-haftm V. Art

14,

PRINCIPLE OF KKI.AT1VITY then

/

may

be introduced in such a manner that

m

regarded as fixed, the motion of

moving-force vector of

m

If

-

given vector by writing

to

alone.

instead

of

i

f

magnitudes of the order -j

m may

be

now subjected to the we now modify this

is

),

then

/

it

Kepler s laws hold good for the position MI at any time, only in place of the time write the proper time T, of m On the

that

appears

(r,,^, z ), have l} we v

l

basis

.

l

of

of to

this

be seen that the proposed law of attraction in combination with new mechanics is not less

simple

remark,

can

it

suited for the explanation of astronomical

the

Newtonian

phenomena than

law of attraction in combination

with

Newtonian mechanics. Also the fundamental processes in

world-postulate.

that the

equations

moving bodies are shall

I

deduction

of

also

these

in

for electro-magnetic accordance with the

show on a equations,

later occasion

as

by

taught

Lorentz, are by no means to be given up. The fact that the world-postulate holds without excep tion is, 1 believe, the true essence of an electromagnetic picture of the world

;

the idea

first

occurred to Lorentz,

its

picked out by Einstein, and is now gradu In course of time, the mathematical ally fully manifest. s will be gradually deduced, and enough consequent essence was

first

suggestions will

be

forthcoming

verification of the postulate find

it

uncongenial, or even

;

in this

painful

for

the

experimental

way even to

give

those,

who

up the

old,

time-honoured concepts, will be reconciled to the new ideas in the prospect that they will lead to of time and space, pre-established physics.

harmony between pure mathematics and

The Foundation

of

the Generalised

of Relativity

Theory

BY A. EINSTEIN. From Annalen der Physik 4.49.1916.

The theory which

is

sketched in the

following pages

forms the most wide-going generalization

what

is

this

latter

at present

"Special

The

known I

theory

Relativity

as

"

differentiate

theory,"

from

and suppose

of

conceivable

the theory of Relativity

it

"

;

the

former

to be

known.

generalization of the Relativity theory has been

made

through the form given to the special Rela was the tivity theory by Miukowski, which mathematician first to recognize clearly the formal equivalence of the space ranch easier

like

and time-like co-ordinates, and who made use of it in The mathematical apparatus

the building up of the theory.

useful for the general relativity theory, lay plete in the

"Absolute

already

which

Differential Calculus/

com were

based on the researches of (jauss, Riemaun and Christoffel

non-Euclidean

on the

manifold, and

which have

been

shaped into a system by Ricci and Levi-civita, and already applied to the problems of theoretical

B

for

of

our

this

purpose,

literature

is

not

so

that, a

necessary

study lor an

Finally in this place I thank

I

physics.

communication developed and clearest manner, all the supposed auxiliaries, not known to Physicists, which part

of the

have

in

in the simplest

mathematical will

be

useful

mathematical

understanding of this

my friend Grossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of paper.

gravitation. It

90

PRINCIPLE OF RELATIVITY

PRINCIPAL CONSIDERATIONS ABOUT THE POSTULATE o* RELATIVITY. 1.

Remarks on the Special Relativity Theory.

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics. Tf a co-ordinate

ferred to

it,

system

K

he so chosen that when

the physical laws hold in their simplest

these laws would be also

valid

when

referred

re

forms another

to

system of co-ordinates K which is subjected to an uniform translational motion relative to K. We call this postulate "

it

The is

Special Relativity

that

signified

the

By

Principle."

principle

is

the

word

limited

to

special,

the case,

K has uniform trandalory motion with reference to K, but the equivalence of K and K does not extend to the case of non-uniform motion of K relative to K. when

The classical late,

Special Relativity Theory does not differ from

mechanics through the assumption of

but only

through

light-velocity in

the postulate

of the

this

the

j>ostu-

constancy of

vacuum which, when combined with

the

special relativity postulate, gives in a

well-known way, the

relativity of

synchronism as

the Lorenz-transfor-

mation, with

all

well as

the relations between

moving

rigid

bodie

and clocks.

The modification which

the theory

of space and time

has

undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. According to the special relativity theory, the theorems of geometry are to be looked upon as the laws about any jwssible relative positions of solid bodies at rest,

and more generally the theorems of kinematics,

as theorems

which describe the relation between measurable bodies and

GENRRAI.1.41U TIIMMn

Consider two material

clocks. rest

then

;

jx>nd8

to

these

points a

of

independent

of

points

to these

according

RKT.ATIVITY

h

bodv at

a solid

their

conceptions

corres-

extent of length,

definite

wholly

91

kind, position, orientation and time of the

body. Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate

syetem a

;

then to these positions, there always

time-interval

and

It

place.

theory

tivity

of a definite length,

corresponds,

independent of time

would he soon shown that the general can

hold

not

fast to

this

rela

simple physical

significance of space and time.

About the reasons which explain the extension

2.

of the relativity-postulate.

To

the classical mechanics (no less than) to the

special

an episteomological

defect,

relativity theory,

attached

is

cleanly pointed out by E. Mach. Let by the following example two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another (and from all other masses) that only that sort of gravitational

which

We

forces

of

was perhai*

illustrate

shall

art-

these

the

to

it

from

;

into account which the

be taken

bodies

bodies

lirst

exert

UJKJII

each other.

one .another

is

of any The distance of |>art

The

invariable.

relative

motion of the different parts of each body is not to occur. But each mass is seen to rotate, by an observer at rest re lative to the

other mass

round

the.

connecting

masses with a constant angular velocity

motion for both the masses). surfaces of both

the

bodies

Now (S,

let

and

line of

(definite

us

S8)

think are

the

relative

that

the

measured

it is with the help of measuring rods (relatively at rest) found that the surface of S, is a sphere and the ;

then

uiface of the other

i*

an

ellipsoid

of rotation.

We

now

PRINCIPLE OF RELATIVITY

92 ask,

why

this difference

is

from

factory

between the two bodies

this question can only then be regarded

answer to

episteomological standpoint when the the cause is an observable fact of ex

the

thing adduced

as

The law

perience.

about

statement

An

.

as satis

of causality has the sense of

world

the

of

a definite

when

only

experience

observable facts alone appear as causes and effects.

The Newtonian mechanics any satisfactory answer. of mechanics hold true the

body S, which S 8 is at

The

is

does not give to this question it says The laws

For example, for a space

Rj

:

relative

which

to

at rest, not however for a space relative to

rest.

Galiliean space,

which

is

here introduced

is

how

ever only a purely imaginary cause, not an observable thing. It

is

thus

in the

of

clear

causality, in

placency, cause

that

the Newtonian mechanics does not,

actually fulfil the requirements but produces on the mind a fictitious com that it makes responsible a wholly imaginary

case treated

here,

for the different behaviour? of

B,

S a which

the bodies S,

and

are actually observable.

A satisfactory explanation to the question put forward that the physical system above can only be thus given composed of S t and S s shows for itself alone no con :

ceivable

cause to which the different behaviour of S, and

S, can be attributed. system.

We

general

laws

are of

The cause must thus

therefore led

motion

lie

outside the

to the conception that the

which

determine specially

the

forms of S, and S, must be of such a kind, that the mechanical behaviour of S, and S, must be essentially

by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under con

conditioned

sideration)

principal

are

then

observable

to be looked

causes

for

upon as the seat of the the

different

behaviours

KALISED THEORY OF RELATIVITY bodies under consideration.

the

of

of the imaginary

cause R,.

R and R a moving

spaces

in

t

another, there

They take the

Among any

9S place

the conceivable

all

manner

one

to

relative

a priori, no one set which can be regarded

is

advantages, against which the objection raised from the standpoint of the of cannot be laws The revived. theory knowledge again of physics must be so constituted that they should remain as affording greater

was already

which

valid for

We

any system of co-ordinates moving

in

any manner.

thus arrive at an extension of the relativity postulate. Besides

there

momentous episteomological argument,

this

a well-known

physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at also

is

least in the four-dimensionalregion considered) a muss a sufficient distance from other masses move uniformlv

a

Let

at in

K

be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K any mass at a sufficiently great distance experiences line.

an accelerated

motion

such that

direction of acceleration

and

position

its

Can any he

that

This

is

to be

an

actually

K

the freely

be

in

as

The reference-system

accelerated

in the

behaviour of

time region

material

its

com

physical conditions.

answered

explained

and the

acceleration

its

independent of

observer, at rest relative to in

is

is

negative

K

considered

the

;

moving masses

good a manner

in

then

,

is

?

above-named

relative to

K

can

the following way.

has no acceleration. there

conclude

reference-system

In the space-

a gravitation-field which

generates the accelerated motion relative to

K.

.

This conception is feasible, because to us the experience of the existence of a field of force (namely the gravitation field)

has

shown that

of imparting

the

it

possesses the remarkable property

same acceleration

to

all

bodies.

The

PRINCIPLE OP RELATIVITY

i>4

mechanical

same

behaviour of the

bodies

K

relative to

is

the

would expect of them with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can as experience

assumed that the systems

be

same legitimacy be taken

K

and

K

can both with the

as at rest, that

will be is, they systems of reference for a description of

equivalent as

physical phenomena.

From of the

these

discussions

general relativity

we

that the working out

see,

must, at the same time,

theory

create theory of gravitation ; for we can a gravitational field by a simple variation of the co-ordinate lead to a

Also

system.

we

see

of the constancy of

we

for

to

K

that

immediately

must

light-velocity

that

recognise easily

reference

with

"

"

must

the

the

principle

be

modified,

of a ray of light

path

be, in general, curved,

and constant velocity with reference to K.

a definite light travels with straight line

The time-space continuum. Requirements

3.

when in

a

of the

general Co-variance for the equations expressing the laws of Nature in general. In the

mechanics as

classical

well

an

immediate physical significance

any arbitrary point has the

that

projection

ascertained

the

Geometry unit

rod,

as its

t,

as the

X4

;

is

special

and space have

when we say

co-ordinate,

on

it

the

that

signifies

X,-axis

solid rod

according to the rules reached when a definite measur

can be carried

origin of co-ordinates along the st

X,

the point-event

of

by means of a

of Euclidean

ing rod,

.ri

the

as in

relativity theory, the co-ordinates of time

co-ordinate

Xi

signifies

times from the

.c^

axis.

A.

that

point having

a

unit

clock

be at rest relative to the system of adjusted co-ordinates, and coinciding in its spatial position with the

which

is

to

GENERALISED THEORY OF RELATIVITY

95

point-event and set according to some definite standard has

over

gone

.<-

4

=J

before

periods

the occurence

of

the

point-event.

This conception of time and

mind

SIWLCC is continually present

though often in an unconsci ous way, as is clearly recognised from the role which this This conception has played in physical measurements. in the

of the physicist,

conception must also appear to the reader to be lying at the basis of the second consideration of the last para

graph and

imparting a sense

we wish

show that we

to

these

to

it

aud

in

general

by more general conceptions in order work out thoroughly the postulate of general

to replace it

able to

to be relati

the case of special relativity appearing as a limiting

vity,

when

case

But

conceptions.

are to abandon

We

there

no gravitation.

is

introduce

in

tion-field, a Galiliean

a space, which

is

free

Co-ordinate System

K

from Gravita (<,

y,

s,

t,)

and

another system K (. y : t ) rotating uniformly rela tive to K. The origin of both the systems as well as their

also,

might continue to coincide. We will show that for measurement in the system K the above established rules for the physical significance of time aud ~-axes

a

space-time

,

space can not be it is

of

On grounds of symmetry round the origin in the XY plane

maintained.

clear that a circle

looked upon as a circle in the plaur now think of measuring the circum diameter of these circles, with a unit

K, can also be

(X Y ,

)

ference

of

K

.

Let us

aud the

measuring rod (infinitely small compared with the radius) and take the quotient of both the results of measurement. If

this

experiment

be

carried out

with a measuring rod

at rest relatively to the Galiliean system IT,

The

as the quotient.

relatively at rest as regards is

greater

than

*.

K

we would

ijet

measurement with a rod would be a number which

result of

K

This can

be

seen

easily

when we

96

PRINCIPLE OF RELATIVITY

regard the whole measurement-process from the system K and remember that the rod placed on the periphery suffers a Lorenz-contraction, not however when the rod

Euclidean Geometry therefore placed along the radius. does not hold for the system K the above fixed concep is

;

of

tions

which

co-ordinates

assume

the

of

validity

K

Euclidean Geometry fail with regard to the system We cannot similarly introduce in K a time corresponding to physical requirements, which will be shown by all similarly

K

prepared clocks at rest relative to the system to see this we suppose that two similarly made

one at the

arranged

and

centre

and one from

the

at

.

In order

.

are

clocks

periphery

of

the

circle,

K.

According to the well-known results of the special it follows that the (as viewed from K)

considered

the

stationary svstem

relativity theory

clock

at

placed

second one which

the

will

periphery

origin of co-ordinates

who

periphery by means

of

go slower than

The observer

at rest.

is

is

able to will

light

see

see

at the

the

common

the clock

the

at

at the

the clock

periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time in the his

way under

actully goes

than

slower

cannot therefore do a

consideration he

will

interpret

observation by saying that the clock on the periphery

way

that the

the

clock at

the

in

such

going of a clock depends on

of

rate

He

origin.

time

otherwise than define

its

position.

We

therefore

arrive

relativity theory time

defined

that

the

at

this

result.

In

and space magnitudes

the

general

cannot be so

difference in spatial co-ordinates can be

immediately measured by the unit-measuring rod, and timelike co-ordinate difference with the aid of a normal clock.

The means co-ordinate

hitherto at

system

in

our

the

disposal,

for

placing

time-space continuum,

our in

a

GENERALISED THEORY OF RELATIVITY definite

there

is

97

way, therefore completely fail and it appears that no other way which will enable us to fit the

co-ordinate system to the four-dimensional

world

in

such

we can expect

to get a specially simple

formulation of the laws of Nature.

So that nothing remains

a

that

way,

for us but

by

to

it

all

repaid

conceivable co-ordinate

as equally suitable for the description of natural

systems

phenomena.

This amounts to the following law: That in general, Laws of

which are covariant for all

unobjectionable relativity

till

from

co-ordinal e systems, that

the

standpoint

Because

postulate.

this

satisfies

among

postulate will be of

the

all

substitutions

there are, in every case, contained those, which to

all

motions

relative

of

the

co-ordinate

This condition

three dimensions).

which takes away the

last

of

is,

It ig

fraasfortudtioiU.

po*xi/j/<-

which

clear that a physics

are expressed ly mean* of

.\<if//re

equations which are valid for

general

general

correspond

system

(in

covariance

remnants of physical objectivity

from space and time, is a natural requirement, as seen from the following considerations. All our icell-sntjstantiatfd space-time of

propositions

amount

space-time coincidences.

consisted in last

case,

the

to

motion of material

points, then, for this

else

nothing

results

well-proved

determination

example, the event

are

really

encounters between two or more of

The

the

for

If,

observable these

except the

material

points.

measurements are nothing else than theorems about such coincidences of material

points, of our

of our

measuring rods with

other material points,

coincidences between the hands of a clock,

dial-marks and

point-events occuring at the same position and at the same time.

The introduction other purpose than coincidences.

IS

We

of a system of co-ordinates serves no an easy description of totality of such

fit

to the world our space-time variables

98 (.!

,u

./,

.c

s

such

4)

that

corresponds a system incident

.<

(./

,

s

,

r ,

.. (<.,

.:

s

introduce

four

any

4)

the

i.e.,

;

,

point-event

them, the equality of will

coincidence

all

be

still

coincidence

to

is

a

is

certain co-ordinate

If

.

(.

co

.<-

4)

3

%

l

is

cha

we now as co

an unique correspondence between

the four

the

eo-ordinates in

expression

two material

of

physical laws

there

is

Two

4 ).

,

same value of the

to the

functions

ordinates, so that there

dences,

of

by the equality of the co-ordinates.

racterised

system

any and every

to

of values

point-events correspond

variables

all

RKLATIVm

PRINCIPLE OF

allow

As

points.

us

to

another

to

system

the

the

of

purpose

coinci

to prefer a

present,

we get the

i.e.,

;

new

space-time

remember such

no reason

priori

the

of

condition of general covariance.

Relation of four co-ordinates to spatial and time-like measurements.

$ 4.

Analytical expression for the Gravitation-field. I

am

general

not trying in this communication as

Relativity- theory

possible,

with

a.

minimum

the

But

of axioms.

aim to develop the theory

in

reader perceives the psychological

such

a

deduce the

to

simplest

logical it

is

system

my

chief

manner that the

naturalness

of

the

way

proposed, and the fundamental assumptions appear to b most reasonable according to the light of experience. In this sense,

we

shall

now

introduce the following supposition;

that for an infinitely small four-dimensional region, relativity theory

valid in the special sense

is

the

when the axes

are suitably chosen.

The nature

of acceleration of an infinitely

tional) co-ordinate

system

is

hereby

to

be so

small

(posi

chosen,

that

the gravitational field does not appear; this is possible for au infinitely small region. X,, X,, X, are the apmtiaJ

GENERALISED THEORY OP RELATIVITY co-ordinates

X

;

the

is

t

time-co-ordinate

corresponding

measured by some suitable measuring clock. ordinates have, with a given orientation

immediate physical significance

These co

of the system, an

in the sense of the special

(when we take a

theory

relativity

99

rigid rod as our unit of

The expression

measure).

had then, according to the special relativity theory, a value which may be obtained by space-time measurement, and which is independent of the orientation of the local co-ordinate system.

Let us take

line-element belonging to

four-dimensional region.

dX

(r/X,

t

time-like,

To

^X

,

and

the

3

,

in

^X

4 )

d<

<

,

,

,

.

as the

/.

magnitude of the near points in the

infinitely

If ds*

be positive

belonging to the element call it with Minkowski,

we

the contrary case space-like.

line-element

nitely near point-events (I

two

considered, also

belong

i.e.,

both the

to

definite

infi

differentials

dx s d \, of the four-dimensional co-ordinates of If there be also a local system of reference. ,

any chosen

system of the above kind given for the case under consi /X s would then be represented by definite linear

deration,

homogeneous expressions

dX V =2

*2)

If

of the

we

form

a (T

d.r V<T

(T

substitute the expression in (1)

where g

will be functions of

.<

^

is

a

local

co-ordinates;

magnitude belonging to two pointnear in space and time and can be got by

definite

events infinitely

get

but will no longer depend

upon the orientation and motion of the for d*

we

PRINCIPLE OF RELATIVITY

100

The g

measurements with rods and clocks. be

so

that

chosen,

= rn

n "or

extended over be extended

all

over

w

s

are here to

the summation

;

and

to

is

sum

be

is

to

4x4- terms, of which 12 are equal

in

value? of

o-

so that the

r,

pairs.

From

the

method adopted

here,

usual

the

the case of

theory comes out when owing to the special behaviour of g in a finite region it is possible to choose the

relativity

system of co-ordinates

in

such a

that g

way

assume*

^

constant values

-1,

f

0-100 00-10 ooo+i

(*)

We

0,

0,

would afterwards sec that the choice of such a system

of co-ordinates for a finite region

From

the

considerations

is

in general not possible.

and

2

in

X

it is

that from the physical stand-point the quantities g .be

looked upon as magnitudes which

tion-field

We

with

assume

to

the

region considered, the special relativity

some

particular

gravita

chosen system of axes. that in a certain four-dimensional

reference

firstly,

describe the

clear,

are to

choice

theory

of co-ordinates.

is

true

The g

&

for

then

have the values given in (I). A free material point moves reference to such a system uniformly in a straight-

with line.

If

we now

time co-ordinates

intro:lu3J, .r,

.

...r 4

,

by u .iy substitution, the space-

then

in

the

new system g

s are

no longer constants, but functions of space and time. At the same time, the motion of a free point-mass in the ne.w

GENERALISED THEORY OF RELATIVITY

101

co-ordinates, will appear as curvilinear, and not uniform, in

law

which the

motion

We

I In-

nl

ini.fii.rr

as

of

We

nt<txx-])uinl*.

under the

on*-

be

will

motion,

innrJ,i<i

iwiependetd of the can thus signify this

influence of a gravitation

field.

appearance of a gravitation-field is con s. In the general nected with space-time variability of g the

see that

we can

case,

by any suitable choice of

not

make

axes,

special relativity theory valid throughout any finite region. s describe the thus deduce the conception that y

We

to

According

field.

gravitational

theory, gravitation thus plays an

tinguished from the others, as

in

forces,

much

the

specially

as the

general

relativity

as

exceptional role the

10 functions g

gravitation, define immediately the

dis

electromagnetic

metrical

representing properties of

the four-dimensional region.

MATHEMATICAL AUXILIARIES FOR ESTABLISHING THE GENERAL COVARIANT EQUATIONS.

We

have seen before that the

late leads

for Physics,

how

such

\Vr shall

must be co-variants

general

now

general

condition that the

co-ordinates

of

tion

the

to

.-

,

for arn

.

\ve

:

,

,

co- variant

relativity-postu

system of equations r

possible

have

substitu

now

equations can be

to

see

obtained.

turn our attention to these purely mathemati Ft will be shown that in the solution, the

cal pro positions.

invariant role,

given

ifs,

which

\vr.

in

<

tjiiat.ion

following

(:})

(iau-Vs

plays

a fundamental

Theory of Surfaces,

style as the line-element.

is

The fundamental idea of the general co-variant theorv With reference to any co-ordinate system, let

this

certain

:

tiling

be defined by a number of func which are called the components of

(tensors)

tions of co-ordinates

10?

PRINCIPLE OF KKI.ATIVITY

There are now certain

the tensor.

can

the components

be

when

co-ordinates,

rules

calculated

these

are

according to which

new system

a

in

known

the

for

of

original

and when the transformation connecting the two The things herefrom designated as systems is known. u Tensors have further the property that the transforma

system,

"

tion equation of their

ous

so that all

;

components are

and homogene new system vanish Thus a law system.

linear

the components in the

they are all zero in the original of Nature can be formulated by puttjng of a tensor equal to zero so that it is a if

equation

thus while

;

we

the tensors,

the components

all

general co-variant

we seek the laws of formation

also reach the

means

of

of establishing general

co- variant laws.

Contra-variant and co-variant Four-vector.

5.

The line-element

Contra-variant Four-vector.

by the four components is

denned

v

expressed by the equation

-=*

" <*>

The

d<

tion of

,<

<l>

*

;

we can

as

the

designate specially

thing which

is

",. .

and homogeneous func

look upon the

of

the

which

we

differentials

components of a

tensor,

as a contravariant Four-vector.

defined by Four quantities

a co-ordinate

the same law,

V.

T^

as linear

are expressed

co-ordinates

to

is

whose transformation law

d.-

system,

and

A

,

transforms

Every

with reference

according

to

103

i.KNKKAUSBO THEORY OK RELATIVITY

we may it

call

follows

a contra-variant

at once

sums (A 4 B

that the

\a

when

ponents of a four-vector,

From

Four-vector.

and B

(5. a),

com

are also

)

T

are so

;

cor

for all systems afterwards

responding relations hold also tensors introduced as (Rule of addition and subtraction "

"

of Tensors). Co-r<inanf-

We

call

four quantities

variant four-vector,

four

variant

From

this

vector

definition

when B

A

as the

for

of the

choice

any

^

(6)

components of a co-

A

contra-

= Invm-ianL

B

follows the law of transformation of

the co-variant four-vectors.

If

we

substitute

in the

right

hand side of the equation

the expressions

e -v v

for

B

we

get

inversion of the equation

following from the

B^

<T

_ 5

a ~

-^ V

A ,

=5

nO* B"

(5)

A A

<r

<7

As

in

the above equation B

and perfectly arbitrary, law is :

it

CT

are independent of one another

follows that the transformation

104

PEINCIPLE OF RELATIVITY

Remarks on

the

expressions.

A

the

paragraph

will

simplification

show that the

<>f

the

at

glance

mode of wriliny

tl/c

of

equations

indices which

this

appear twice

within the sign of summation [for example v in (5)] are over which the summation is to be made and that

those

It is therefore only over the indices which appear twice. possible, without loss of clearness, to leave off the summation,

sign

so

;

that

we

introduce

the

rule

wherever the

:

index in any term of an expression appears twice, it is to be summed over all of them except when it is not oxpressedly said to the contrary.

The

the co-variant and the contra-

between

difference

variant four- vector

transformation

the

in

lies

laws

[

(7)

Both the quantities are tensors according to the in it lies its In above general remarks significance. and

(5)].

;

accordance with

Ricci and Levi-civita, the contravariants

and co-variants are designated by the over and under indices.

6.

Tensors of the second and highei ranks.

Contravariant tensor

:

If

we now

v

products A^ of the components trava riant four- vectors

A^ =

f8)

A

*",

will

calculate

A^ B

v ,

all

the

of two

Ifi

con-

A^B*

according to (8) and

(ft

a) satisfy the

following

transformation law.

(9)

A"

=

6*

8

3-?

g-I

.-

A""

We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transfor mation relation (9), a contra variant tensor of the second

GENERALISED THEORY OF RELATIVITY

Not every such

rank.

tensor can be

But

vectors, (according to 8).

16 quantities

B

v

A^

v

can

,

be

built

105

from two four-

easy to show

it is

as the

represented

sum

in the simplest

for the

tensor of

proving

it

the

way

laws which hold

all

A

of

From

of properly chosen four pairs of four-vectors.

we can prove

that any

second rank defined through

it,

true

(9),

by

only for the special tensor of the type (8).

If is clear that Contravariant Tensor of any rank corresponding to (8) and (9), we can define contravariant 3 etc. comtensors of the 3rd and higher ranks, with 4 :

,

Thus

l>onents.

sense,

we can

it

is

look

clear

from

(8)

and

contravariant

upon

contravariant tensors of the

first

(9) that in this

four-vectors,

as

rank.

Co-variant tensor, If

the

for

ou the other hand, we take the 16 products

components of two

four- vectors

co. variant

A^ A

of

and

them holds the transformation law

8

Q u

r

V

v

^r =-97/67;,

<">

By means

of these transformation laws,

tensor of the second rank

we have

already

is

defined.

made concerning

the

co-variant

All re-marks

which

tbe contravariaut tensors,

hold also for co-variant tensors.

Remark It

:

is

convenient

to

treat the scalar

Invariant

as a contravariant or a co-variant tensor of zero rank.

14

either

]66

HllXCIPLK OF KELATIVITV

Mixed

We

tensor.

can

define a tensor of

also

the

second rank of the type

P-

P.

which

V

= A B

A

(12)

co-variant with reference to

is

with reference to

and contravariant

p.

Its transformation law is

v.

mixed tensors with any number of and with any number of contra- variant The co-variant and contra-variant tensors can be

Naturally there are co- variant indices, indices.

looked upon as special cases of mixed tensors. tensors

A

:

contravariant

or a

rank

called

or higher

is

co-variant

tensor of the second

symmetrical when any two com

ponents obtained by the mutual interchange of two indices

The

are equal.

we have

for

tensor

A^

or

any combination

A

symmetrical, when

is

^

of indices

(14)

A^W

(14a)

A

7*

=A p.V

Vfi

It must be proved that a symmetry so defined is a property It follows in fact independent of the system of reference.

from .or

(9)

remembering

9

i

-

.<

a-

-

9

(14-)

.T >p.v

e,,,

Q a ,i

T

e. v

a,,,

,

Q

,

T

8,;

,

/.vu.

.TO-

CKN Anti-til iniii

A 4th

Ktf

oh

ALIsKlt TIIKOllI

II

KI,.\TI

107

VITV

lri<-nl

contravariant or co-variant tensor of the 2nd,

rank

called

is

3rd or

when the two com

<iiifi*i/nnit<>trical

by mutually interchanging any two indices

ponents got

*"

The

are equal and opposite.

tensor

A.

1

or

A

thus

is

^

antisymmetrical when we have

A^ = -A^

(15)

(15a)

Of

the

16

components

A"

the

,

equal and

vanish, the rest are

four

components

opposite in

there are only 6 numerically different

pairs

;

so

A that

components present

(Six-rector).

we

Thus A!*""

rank)

(3rd

different,

that

see

also

lias

only

tensors

of

numerically

i)ii(/(iji/n-<iti<>/i

components

of

a

all

dimensions.

the

of

T

iix,n-x

tensor of rank

the components of

we multiply

4-

Multiplication of Tensors.

7.

,

components

A only one. ranks higher than the fourth, do

not exist in a continuum of

c

4-

tensor

and the antisymmetrioal tensor

Symmetrical

On/ft-

the antisymmetrical

a

:,

tensor of

:

\Vc

^ct from

the

and another of a rank rank (r-fc ) for which first with all the

components of the

components of the second

in

pairs.

Fur

example,

we

108

PRINCIPLE OF RELATIVIT*

obtain the tensor

kinds

T

from the tensors

=A

T

and B of different

The proof diately

of

from

(T

the

tensor

the

expressions (8),

transformation equations

and

B flV

UV<T

(10)

A

:

(12)

character

T, or

(18)

(11),

(9),

themselves

are

of

(10)

multiplication of tensors of the

;

imme

follows

the

or

(12),

equations

examples of

the

(S),

outer

rank.

first

Reduction in rank of a mixed Tensor.

From every mixed

tensor \ve can get a tensor which

is

two ranks lower, when we put an index of co-variant character equal to an index of the

and sum according

contravariant character

to these indices (Reduction).

for example, out of the

mixed tensor

the

of

We

fourth

get

rank

sv

A

,

the mixed tensor of the second rank

=A a8 a =(SA \

8

/

A

ft

and from

"

it

again by

a

reduction

"

aft

the

tensor

of the zero

rank

A= A.0 = A./* ft

The proof

that the

tensorial character,

aft

result of reduction

follows either from the

retains a

truly

representation

t,

I

109

THKOKY OF HKLATIVITY

NKI5AL1SK1)

combi

of tensor according to the generalisation of (1~) in

nation with (H) or out of the generalisation of (13).

Imier

a<ul

mixed multiplication of Tensors.

This consists with reduction. the

seconj.1

the

first

in

the combination of outer multiplication

From the co-variant tensor of Examples A and the contravariant tensor of :

rank

B

rank

we get by outer multiplication the

mixed tensor

B"

to

Through reduction according ting

i

=

indices v

and

o-

(i.e.,

put

the co-variant four vector <r),

/*

These

.v

we denote

and B

.

=A

= D

D

as the

Similarly

we

B

is

generated.

/"

inner

product of the tensor

get from the

tenters

A

and

A

v

B

ILV

through outer multiplication and two-fold reduction the inner product

and one-fold

A

B^

reduction

.

we

Through outer

get out of

mixed tensor of the second rank can

fitly

Pall

this operation

with reference to

the

a

indices

respect to the indices v and

o-.

D

A

multiplication

and B

= A

mixed one M and r

t

;

B for

the

.

it is

\Ve outer

and inner with

110

I

We

KI.Ni

now prove

I

l.K

I

a law,

KI,

I!

<>!

which

\TIVm

will be often applicable for

provingthe tensor-character of certain quantities. According

A

to the above representation.

B^

v is

when

a scalar,

A

P-V

and

B

We

are tensors.

/JLV

A

remark that when

also

an invariant for everv choice of the tensor

B^

B^

v ,

then

is

A (Of

has a tensorial character.

Proof

According to the above assumption,

:

A OT From

fJiV

o

o&

Substitution of this for

8 r

8

.

term

within

*

*/

/

B^"

in

the above equation ^i

- A

f

8-

This can be true, for the

we have however

9^

% r

, "

any

=A

B"

,

the inversion of (9)

A

for

we have

substitution

y

/

any choice of

the bracket vanishes.

B

only

when

From which by

referring to (11), the thtorem at once follows.

This law

correspondingly holds for tensors of any rank and character. The proof is quite similar, The law can also be put in the following from.

If

B^ and

C*

are

any two vectors, and

THEORY OF RELATIVITY

<ih\l,KAI.ISKI.

if

is

them the inner product

for every choice of

then

a scalar,

A

a

is

III

A

The

tensor.

co-variant

last

py

law holds even when there

the more special

is

formulation,

that with any arbitrary choice of the four-vector

A

the scala: product

we have the

B

additional

condition.

symmetry

C

B

*

B

v

a scalar,

is

condition

that

method

(A

the

satisfies

the

to

According

above, we prove the tensor character of

alone

which case

in

A

B"

v

+^v

},

given

from

which on Account of symmetry follows the tensor-character A This law can easily be generalized in the case of

of

.

co-variant and contravariant tensors of any rank. Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind of tensor

then

C of

when B

rank,

first

A

If the qualities

:

is

A

B

form a tensor of the

any arbitrarily chosen four-vector,

a tensor of the second rank.

is

v

If

for

example,

1

is

any four-vector, then owing

A n B

1

,

the

inner

both the four-vectors

product

C^

A

A

B

and

Hence the proposition follows

to the

v

tensor

C^ B

character

1

is

a

scalar,

being arbitrarily chosen.

at once.

few words about the Fundamental Tensor

//

U.V

The

co-variant fundamental

expression of the

tensor

In

the

square of the linear element <lx-

=7

df M*

1

/*

1. 1

invariant

PRINCIPLE OF RELATIVITY

112

AA

plays the role of any

vector, since further

g

y

chosen contravariant

arbitarily

=ff v

it

,

follows from

derations of the last paragraph that g

co-variant "

tensor

fundamental

some

of

second

the

rank.

Afterwards

tensor."

which

properties of this tensor,

a

is

the

symmetrical

We we

consi

call

it

the

deduce

shall

will also be true for

any tensor of the second rank. But the special role of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravita

tion

makes

which

it

clear that those relations are to be developed

will be required

only in the case of the fundamental

teusor.

The co-variant fundamental tensor. If

we form from the determinant scheme

minors of g

y

|

g

v

and divide them by the determinat ^ = v^

we get certain quantities f* g , which as prove generates a contra variant tensor.

j

we

|

the

y^ shall

According to the well-known law of Determinants

(16)

/

g

a

=8

/XO"

where

S

is

1,

or 0,

according as

of the above expression for

V

6

rf

V

rf.

2 ,

/*

=v

we can

V^

"

or according to (16) also in the form

0T ya tt7 Ja VT *a

*

.

(M

/

U.

V

or not.

also write

Instead

\

GENERALISED THEORY OF RELATIVITY

Now

according

to

the

118

of multiplication, of the

rules

fore-going paragraph, the magnitudes

fJL

p.<T

foims a co-variant four- vector, and of the arbitrary choice of If

we introduce

it in

d<

)

our expression,

and

according to

r,

so

it

its

defintibn

fact (on account

we get

For any choice of the vectors d

y*"",

in

any arbitrary four-vector.

d) this

is

a symmetrical

is

scalar,

thing in a

follows from the above results, that g

Out

contravariant tensor.

of (16)

it

and

is

also follows that 8

^ is

a tensor which

we may

call

the

mixed fundamental

tensor.

Determinant of the fundamental

According

to the

tensor.

law of multiplication of determinants,

we have

On

the other hand

So that

it

15

we have

follows (17) that

\9

v

\

I/"

114

PRINCIPLE OF RELATIVITY

of ruin me.

We see K first 0=

. |

1

v

the transformation law for the determinant

According to (11)

9

From

we

this

obtain.

=

dx U

6-c

V

6,v a,v

9, -/-

by applying the law of mutiplication twice,

GENERALISED THEORY OF RELATIVITY measured

with

solid

and

rods

llo

clocks, in accordance with

the special relativity theory.

Remark* on

the character of Hie space-time continuum

Our assumption that

an

in

small

infinitely

region the

special relativity theory holds, leads us to conclude that ds-

can always, according to (1) be expressed in real tudes fJX,...<JX If we call dr a the natural

"

"

.

element rfX t

</X

dX 3 dX 4 we

2

have

thus

magni volume tlr*

(18a)

Should Y/

g vanish at any point of the four-dimensional would signify that to a finite co-ordinate small at the place corresponds an infinitely

continuum

volume "

natural

it

This can never be

volume."

can never change

its

the case

;

we would, according to

sign;

so that g

the special

assume that g has a finite negative a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for relativity

It

value.

theory

is

the choice of co-ordinates.

however

If

clear that the this

(

g)

remains

positive

and

choice of co-ordinates can be so

finite,

made

it

is

that

We

would afterwards quantity becomes equal to one. of the choice of co-ordinates*

see that such a limitation

would

produce a significant simplification

in

expressions

for laws of nature.

In place of (18)

it

follows then simply that

=d

dr

from

this it follows,

remembering the law of Jacobi,

-

=1.

116

PRINCIPLE OF RELATIVITY

With

this choice of co-ordinates, only substitutions

determinant It

1,

would however be erroneous to think that a

signifies

We

do not seek

with

co- variants

the

lenunciation of the

partial

postulate.

determinant

its

simplify

relativity

Iran formations

the

but we ask

this step

general

laws of nature which are

thoj=e

to

regard 1,

what

:

are

having

the

general

we get the law, and then a expression by special choice of the system

co-variant laws of nature

we

with

are allowable.

?

First

of reference.

Building up of new tensor* with the help of the fundamental tensor.

and mixed multiplications of a

inner, outer

Through tensor

with

the

fundamental tensor,

tensors

of

other

kinds and of other ranks can be formed.

Example

We

:

would point out specially the following combinations:

A

<f

A

=g

and,

can

call

A

g

A aft

n

ap

the, co-variant or contravariant tensors)

(complement to

We

g

B

B

o

Q

A

a

the reduced tensor related to

A

117

GENERALISED THEOEY OP HELATIVITY Similarly a/3

,-V/AV

B

It

is

"

of y

plement

f*

remarked that

to be

,

v

for

ua Q y

we

vB

no other than the

S

9

*>

a

com

p-v

Equation of the geodetic

9.

"

have,

o~ J9

9 v

yQ

is

line

(or of point-motion).

At

the

"

element

line

"

ds

is

a definite magnitude

in

we have also between dependent of the co-ordinate system, two points P t and P 2 of a four dimensional continuum a which

is an extremum (geodetic line), i.e., one of the choice of a independent significance got

which

for

line

has

/>/?

co-ordinates. is

Its equation

(20)

From deduce

this

4

line

geodetic

we can

equation,

differential

total

this deduction

;

in

a

equations is

given

wellknown which

define

here for the

way the

sake

of completeness.

Let defines

X, a

be

a

series

function of

sou^ht-for as well as

We

of

the co-ordinates x v

;

This

which cut the geodetic line lines from P, to P 3 neighbourin<r

surfaces all

.

can suppose that all such curves are given when the value of its co-ordinates ^ are given m terms of X. The

118

PRINCIPLE of RELBTIVITY S

sign

to

corresponds

curve

geodetic

curve, both lying on the

Then

passage from

a

a

to

sotight-for

same surface

(20) can be replaced

a

point of

of

point

the

the contiguous

A..

by

u

(20)

dx fJiV

^

dx y

~9

l

~(j\

~rfX

But

LlL 2

a,

So we get by the substitution of 8w

in

remem

(20a),

bering that ii>

H(

v

^

-

/=

a

(8

-

i

v^

after partial integration, X.,

d\ k

J

=0

So; o-

o-

(20b)

a. / A

1.

1

AKIIALISED TIIEOKY OF RELATIVITY

From which

follows, since the choice of

it

fectly arbitrary that

k

(20c) are

the

equations of

(o-=l, line

geodetic

we have

per

since

;

2, 3,

along

4)

the

we can choose the

^,<?=/=0,

the length of the arc measured along the

as

A,

Then w =

geodetic line.

is

8.<-

Then

;

=0

a

geodetic line considered

parameter

should vanish

*

k^

119

l,

and we would get

in

place

of

(20c)

V

8**

8*

8.<-

8*

<r

a

_i Or by merely changing <ZV

(20d)

t

8

ar

V

-

_

the notation suitably, _

a -

g a(f

v

_

K"

-4-

(L-

~

(ft

-^

.

=0

where we have put, following Christoffel, (9i\

F/xv l

*

J

S

LO-

Multiply reference

finally (20d)

to

last the final

M

I

L

vcr "^

a* v O

with 0*

and inner with

T,

a~~

~

P-

v ~\

~ai

fi

(outer multiplication with respect

to

rr)

form of the equation of the geodetic

-

ds*

Here we have

put, following Christoffel,

we get line

at

PRINCIPLE OF RELATIVITY

120 10.

now

Formation of Tensors through Differentiation.

Relying on the equation of the geodetic line, we can easily deduce laws according to which new tensors can

For this be formed from given tensors by differentiation. purpose, we would first establish the general co-variant

We achieve this through a repeated If a pertain following simple law. curve be given in our continuum whose points are character ised by the arc-distances s. measured from a fixed point on

differential equations.

the

of

application

the curve, and

then

is

the fact that

if

further

be an invariant space function,

<,

d<j>

= 9* Qx

ds

i

a 5^

=

from

follows

as well as ds, are both invariants

Since

so that

The proof

also an invariant.

9*

|>

d.

,

also an invariant for

is

.

which go out from a point o] the vector d c any choice of

in

the continuum,

From which

.

all

curves

i.e.,

follows

/*

diately that.

A

=1* .

Vj is

a co-variant four-vector (gradient of

<).

According to our law, the differential-quotient taken along any curve

v= A

O r Q.,

M

iff,

d-

9,

we

.

^ d*

*

get

dr.

v

/*

<P

7^

x=

likewise an invariant

is

Substituting the value of

.

d

+ o

d** A4

.

<P

9f

1

for

imme

THKOKY

<;F.\K1UUSK1>

of

Here however we cannot we however It any tensor

OJ-

HKL.Vl

mTY

once deduce the existence

at

tiikc

that the curves along

which we are differentiating are geodesies, we pet from

it

dfx

--

by replacing

aa^_ 3* 9- v

= r I

From

according to

-

the

and

v,

\

tlie

<-

>

-9* 9- T

bracket

M

1

also according to (2o)

is

-

(

and (21

)

with svc iin-

J

T

and

r

JJ

interrliangeability of the differentiation

regard to ^ and that

^ v)

4

(

symmetrical

with

respect to

u.

)

t-.

As we can draw

;t

uvoilrt ic line in

x

point in the continuum.

ratio

arbitrary results of

of

is

any direction from any

thus a four-vector, with an

components. M

that

it

follows from Mir

7 that

(25)

a co- variant tensor of the second rank.

We

the result that out of the co-variant tensor

ot

is

A = of

"-^

we can

get

I>V

differentiation

2nd rank

V= 16

6/

a

have thus got tirst rank

the

co-\ariant tensor

PRINCIPLE OF RELATIVriY

\:l l

rhe tensor

call

W>

A

the

"

extension

"

>f

the tensor

ta>

A

Then we can

.

leads to

a.

\^

~~

is

when

the vector

uiso

combination also

this

A

not

is

we

order to see this

co-variant

i>

show that

four-vector

remark that

first

when

sum

the case for a

repvesentable

\L

of

and

are

<

four such

:

l

i/rd),

that

clear

In

a

This

scalars.

terms

when

tensor,

us a gradient.

easily

<j>(

are

<(*)

)...\f/(*)

co-variant

every

Now

scalar*.

four-vector

is

it is

however

j-epresentable in

the form of S P-

If for

example,

A

is

a fonr- vector whose components

are any given functions of x

we

,

have, (with

reference to

the chosen co-ordinate system) only to pat

= A,

in order to arrive at the result that

S

is

equal to

V-

In order to prove then that right side of (26 for

A

N t

we

A

substitute

we have only

to

show

in a tensor

A

.

P

when on

the

any co-variant four-vector that this

is

true for the

Til ROll Y

four- vector

S

.

For this

OF BRLATIVITY

latter, case,

howevtjr, a glance on

we have only

the right hand side of (26) will nhow that bring forth the proof for the case when

Now

1:

the right hand side of (25) multiplied by

which has a tensor character.

also a tensor (outer product of

~*

two

* is

four- vectors).

Thus we get the desired proof

for

the

of

four-vector,

l

-

\1/

is

f

Similarly,

Through addition follows the tensor character

S

i/

to

and hence

f->r

any four- vectors A

as

shown above.

With the help of the extension of the four- vector, we extension of a co-variant tensor of any can easily define This is a generalisation of the extension of the fouri-ank.

We

vector.

confine ourselves

to the case

of

the tensors of the 2nd rank for which

of

the extension the law of for

mation can be clearly seen. Aft already remarked every co-variant tensor of th^ 2nd rank can be represented as a sum of the tensors of the type

A fi

BV

.

124

PRINCIPLE OF EELATIVITY

would therefore be

It

for one

of extension,

we have

(26)

sufficient to

9

tensors.

with

B

third

to

According

the expressions

8A

are

deduce the expression

such special tensor.

-

(

)

(

r )

^

^

Through outer multiplication

and

A

2nd -with

the

we

the

of

first

the

tensors of

get

Their addition gives the tensor of the third

rank.

rank

-

A

/xv

A -

-

V"srr

A

where is

linear

first

put=A

is

y

.

B^,

J

av

The

differential co-efficient so that this

not

but also

tensors,

right

and homogeneous with reference

to a tensor

B

A

i.e..,

second rank.

only in

hand

side of (27)

A

to

.

and

law of formation leads

the case of a tensor of the type

in

the case

of

a summation

in

the case of any

We

call

A

co-variant

for

all

tensor of

the extension of the tensor p.va

It

is

(27)

clear

that (26) and

(extension of the

In general

we can

get

its

A

such the

A

.

/iv

(24)

are only

tensors of the all specinl

special

tirst

Ijtws

cases of

and zero rank).

of

formation of

tensors from (27) combined with tensor multiplication.

Some

A tn*<

f

,-.

special cases of Particular Importance. inuiJiai n

-ic

W<>

shall

It

miiia*

con<:rr,iui<i

deduce some of

first

28)

la.-t

,,

9

=

/V^ =-f/ v

follows From

foi-ni

tin-

lite fundamental h-mnms much xised

law of differentiation

ut fci -\\ards. According to the determinants, we have

Tlu-

125

KKLAT1V1KY

(JKXKH.M.ISKD TUKOliY Ol

^/

/xv

tho

first

V -

when we remember

that v ,/

ULV

,f

= ti

L ,

and there fun-

1

consequently;/

From

(28),

it

fly**

+

=

fa ,-/

U.V

g

v >j^

<l<i

folhnvn that

~t

^

i

(-r/)

6.

_,,/ -^

V 6,,

/"

^

Again, since ^

/

,

we

<j

vcr

-

01

a,-.

By mixed multiplication with

we obtain (Hwn^inir the mode

6-

havi-, Ity differentiation,

(/

(30)

(j

uf

nnd

;/

of wntinir the

126

I

KINTIPLE OF RELATIVITY

ftv

Id

=-

Pa

V

P d<

<

(31)

~ Q7~

V

^

;/

a/S

I

p.w

p.-

K^,

V

(32)

,

Tlio

expression

shall often use

If

we

get

allows a transformation

(31)

which we

according to (21)

we substitute

get,

By

;

remembering

this in the

second of the foi-mula (31).

(23).

substituting the

right-hand

side of (H4) in (29),

we

TUKOKV OF KEI.AT1VIIV

.lsKM

nf Let

u>

multiply

i

ii

(inner mult

ten-Mir if

of the firnt incjn^ci

)

.

).

;

to

(.

U

,a _ 8

;t

V r

.

-

th

(- .*).

;nul

)

lv

fandanumta]

side tnkes the form

ritrlit-lianrl

8 According

then

?

four^vcctor,

with the conttavariani

ijilicittion

th

1

eo^tfavoriani

///>

I"

lust

member

cun

the form

Both the second

The

member the

since

If

naming

the of

of

the

and the

i

other.

immaterial.

(B; cun then he united with

as well as

chosen, we obtain

This scalar

A

(B),

(A) cancel each

summation-indices

the

A

first of

are vectors which -an

V ,

is

be

trbi-

M

finally

9 a

vector

exj)rcssion

expression

we put

where A* trarily

of

meiuhei- of

laHt

(A).

members

first

the Di vcrym-

-"

ut

A

"

the contra variant four-

128

lUM

I

ttnfdiin,/

Tin- seron.l

/

fu,

and

t/tc

a/

in

iiHMiiiifi

up

iu a

an

is

indict tensor

\Ve ol)tain

vrry simple niainier.

aA

=

B

(36)

in the

antisymmetrical

mmetrical

-\

iv

)

:!>

vp.

p.*

built

(covarianfy fjouyvectof,

A

Hence A

v.

OK UKLATTVITY

ll i.l.

9A

-

/x

-

t>f

Jf

we apply the operation

tensor of

tht- sri-ond

I

jink

A

.

( 27)

;uid

ii

Six-vector.

on an nntisynimetrical

form

the equations

all

pr

arising from the cyclic* interchange of the indices

add

all

them, we obtain

/Aj

from which

it

is

ir

/x,

v.

<r,

and

tensor of thr third rank

=A

B

(.37)

a

4fiver

=

+A

A

easy to see that the tensor

is

-

Q

ir/j.v

V<TJJ.

aA

^

<T

antisymmetri-

eal.

<-/>

nf

tfir

S

V

If (27)

is

then a tensor

hand

6

multiplied by is

obtained.

<^

(

f]

Tho

mixed

tirst

side of (27) can be written in the form

A

a,

(r

multiplicjition),

member

of the right

we

If

TIIKOUY

\i I;M.ISI;D

t.t

rc,d:iee

/"

/

^.\

if

<>i

I

v

A"^. ,

/"<r

A

and

KMTI\

1

/"

\

1

v

J:><)

^A

r

/

hy J

M

rojthicc in llic ti-insforiiind first inoinliop /i

^

9-"""

,,,,,

9-V

9",

with

help of

tlio

here arises

t

t-ancel.

1

This

;xn

(^M-),

then from with

expression

tlie

right-hand side of (27)

seven terms, of which four

here remnifis

is

the expression for the ext elision of

of the second

rank

tensors

cont ravariant

em-responding

n

contra variant

extensions can also he formed

:

of

higher and

I

m

lower

ranks.

We

remark

that in the

extension of a mixed tensor

9A (:>

15\

/i

-tnd

A

:h

the tr (

~

reduction inner

9 17

"

~

i,

of (US)

(,

,)

V

with

^tr

r)

+

A

/

with reference

to the indices

v I

ft

ravariant four- vector

also form the

A

multiplication

V t

same way. we can

/

,

we

tret

a

con-

uixrii

I

On

tlu

;iccouiit

1

OK

i.i:

of

in;i,

\TI\ITY

svmiiirtrvv

flic

of

with

-

"

(

reference right

hand

tensor,

the indices

to

side vanishes

(3.

and

when A

which we assume here

a _ :

of tht

antisymrnetrical

member can

be

therefore get

a

I

j

7^~~8^7 p

(4)

This

we

o v

i

an

is

;

member

"

the second

;

transformed according to (29a)

A

the third

K.

u

)

is

the

expression of the divergence of

a

oontra-

variant six-vector.

of

Let us form indices a

If

tensor

and

we

A

t//f

/i.wr

the reduction of

<r,

=

ti

of

i/ie

T r/^

into

A

,

it

the

second rank.

(89) with reference to the

we obtain remembering

introduce p<r

t//ic<:il

last

(29;i)

term

the contravnriant

takes the form

T

A-. L -[">J ^ If further

A^*

7 is

symmetrical

it is

reduced

t<>

(,l

If

instead

\

i;

I

<>i

M.IM.D

A:

Til K( in

to (31) the last

UK

introduce

.

A

symmetrical co-variant tensor

owing

u|

\TI

in

\

IT1

1

>

i

way the

>imilar

;i

g a

^

member can

I

A"

then

.

</

take the form

A

-,j J

8r

f*r

In the symmetrical case treated, (41) can be replaced by -it

her of the forms

I

In.

(

which we snail have

$12.

\Ve

now

to

/-j A

make use

of afterwards.

The Riemann-Christoffel Tensor. seek

those

only

tensors,

obtained from the fundamental tensor alone.

It

is

)

found

ea-il\

.

Wo

^

which

can

be

\by differentiation

in

(27)

]>ut

instead

of

"

"

i-

A

tin-

fundamental t-nsor

//

and

i^et

from

IW it

PIMNCIPU. 01

m

l!i;i.\Tl\

a new tensor, namely the extension of the fundamental We can easily convince ourselves that this

tensor.

We

vanishes identically.

it in

prove

the following way;

we

substitute in (27)

[IV

8A

(^

Q

/ (^p

j:

r

f

i.e.,

P

\

J

the extension of a four- vector.

Thus we get (by

changing the indices) the

slightly

tensor of the third rank

_

^

_

/

a

/xtrr

.

-a-

P

)

r

/

ija,

"

T

_y

p

r

(a-

-

;

_

(TT

\

f

) n

(

_p a..-

6,

We r

use these expressions for the formation of the

A

A /AOT

Thereby

.

the

following

ten>ur

A fJ(TT

the fourth member, as well as tlie

in

I>.T(T

A ^;

cancel the corresponding terms in

to

terms

last

term in

<r,

and

T.

-

A /iOT

The same

=

A /ATO"

We tt

iirst

member,

member corresponding

within the square bracket.

symmetrical the second and third members.

/HOT

the

the

p fJUTT

is

These are

all

sum

of

true for the

thus get

A

.

p

I.I.M.I;

The

essential

hand nde

right

\i.i>i.n

thing of

(

this

in

result

is

we have only A

1:2)

differential C0refficientf.

or UKi,ATi\rn

TIII:OI;N

From

on

that ,

the

but not

its

A

the tensor-character of

fi(TT

A

and from the fact that A

,

B

an arbitrary four

p

flTO-

vector,

is

on

follows,

it

account of the result of

that

7,

a tensor (Kiemann-Christoffel Tensor).

is

fUTT

The mathematical follows;

of

significance

when the continuum

this

tensor

so shaped, that

is

co-ordinate system for which y

s

are constants,

as

is

there

Bp

is

a

all

/XCTT

[1.V

vanish. If \ve

any new

choose instead of the original co-ordinate system one, so would the

be no longer

constants.

s </

The

referred to this last system

character

tensor

of

B /XCTT

shows

us,

also

in

however, that these components vanish collectively The any other chosen system of reference.

vanishing of the Kiemann Tensor

is

thus a necessary con

dition that for -some choice of the axis-system y

taken case

a>

constant^

when by

a

In our problem

it

V can be

corresponds

suitable choice of the co-ordinate

to the

system,

the special relativity theory holds throughout any finite region. By the reduction of (1-3) with reference to indicc> to r

and

p,

we get

the covarinnt tensor of the second rank

1- J1-

iu.\(

]

I In-

ti/io/t

been remarked

ii

i.i;

01

KI.I.

c/io/ce o/

nivrn It has alrcadv

co-ordinoti-cs.

with reference to the equation (18a), that the co-ordinates can with advantage be so choseu that in

8,

= ! A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplifica

x/

It

tion.

is

a fundamental

S

tion,

true for the tensor

specially role

in

the

theory.

B

By

vanishes of itself so that tensor /it

R

which plays

,

this

B

simplifica

reduces

to

fJLV

.

/J.V

I shall

simplified

in

give

form,

the following pages all relations in the with the above-named specialisation of

the co-ordinates.

general

any

C.

It

is

then very

covariant equations,

if

easy to go back it

appears

to

desirable

the in

special case.

THE THEORY OF THE GRAVITATION-FIELD 13.

Equation of motion of a material point in a

gravitation-field.

Expression for the field-components

of gravitation.

A

freelv

moving body not acted on by

moves, according to the special straight

line

relativity

This

and uniformly.

also

external

theory,

holds

torcr-

along a for

the

for any part of the four-dimen generalised relativity theory sional region, in which the co-ordinates K,, can be, and are, so

chosen

that

//

s

have special constant values of

the expression (4).

Let us discuss this motion from the stand-point of any

K it moves with reference to arbitrary co-ordinal O-M -stem K, (as explained in 2) in a gravitational field. The :

;

la\\>

M.ISKD THEORY

(ii:\KI!

of

motion With

reference

of motion

geodesic

As

.

a

to

from

the

the law

K,,,

lino

defined

would

it

1:55

easily

straight is

geodetic-line

of the system of oo-ordinates,

m

\

reference

four-dimenfcional

is a

1

K, follow

to

\Yith

following consideration.

Kl. \TI

It

(>K

and tlm*

a

independently be the law of

also

motion for the motion of the material-point with reference to

K!

If

;

we put

v

we

get tho

motion

of

= -

J

=r T

\\ e

now make oovariant

the

in

relativity

the

as

legitimate,

when

If

oi

with

K,,,

is

defines Held,

gravitational

theory

there

</>

assumption that this

holds to

reference

also

when to

us

which

no relation

to be all the first

in

the

there

throughout a

contains only the

(1(5)

among which

/s-

equations

The assumption seems

region.

,

simplo

very

exists no reference-system

special

l

/

system

motion of the point

/(

"

,/X

general

dx

./

if.

__T

//

K

to

reference

by

jjiven

the

with

the point

^

T

finite

more

differentials of

the special case

K,, exists.

r

T s

vanish, the point

moves uniformlv and

in

a

/*"

straight

line; these

magnitudes therefore determine the

deviation from uniformity.

the gravitational

field.

They

are

the

components of

l- iC)

HI.M

I

ol

IIM.I.

Kill.

The Field-equation

14.

\Tlvm

of Gravitation in the

absence of matter. In the following,

matter

the

in

tion-field will

sense

we

be signified

everything besides the gravita :is matter therefore the term :

includes not only matter in the usual

but

sense,

Our next problem

field.

electro-dynamic

from

differentiate gravitation-field

that

is

also

the

seek the

to

Reid-equations of gravitation in the absence of matter. For we apply the same method as employed in the fore

this

going paragraph for the deduction of the equations of motion for material points. A special case in which the field-equations sought-for are evidently

the special relativity theory

constant finite

with

>vstem

K,,.

ponents B

reference

With reference

/J

the

of

be

the

to

a

is

that of

have certain

s

case

in the

certain

co-ordinate

definite

Tensor

s

a

in

to this system,

Riemann

These vanish then also

vanish.

satisfied

which </

This would

values.

region

in

the

all

com \:>

("equation

region

considered.

with reference to every other co-ordinate system.

The equations must thus be

in

of the gravitation-field free

everv case satisfied when

from matter

B^

all

vanish.

P.<TT

But

this condition

is

clearly one

which goes too

far.

For

by a material can never be transformed point in its own neighbourhood ,111;, by any olioiiv of a\-s, i.e., it cannot be transformed it

clear that the gravitation-field generated

is

t/

to a case

of constant

Therefore

from

BOW

matter,

H

it

it

i>

s.

clear that, is

deduced .v

ft

desirable

for

a gravitational

that

the

from the tensors

field free

symmetrical

1^r-

ten-

should vanish.

B

\\ethu-iget

1

fulfilled in the

(I

ILI8!

equations for

the

Hi

\TI\m

i!i:r.

quantities

1

137

which are

//

special case

when IV

V

all

we

see that

in

absence

Remembering

I)

I

(

come out

field-equations

to ihe special

TIIF.oin

I)

follows

as

vanish.

of matter referred

(when

;

co-ordinate-system chosen.)

er;

"

fan also he shown

It

tions

is

liesides

connected with l>

,

there

ohoiee

the

that

minimum

a

of

of

these

equa For

arbitrariness.

no tensor of the second rank, which

is

/"

can

lie

built out of

and their derivatives no

s

//

higher

/"

than the second, and which It will

is

them.

also linear in

be shown that the equations arising

mathematical

wa\

of the conditions

out

of

in

relativity, together with equations (Hi), givo us

loman law of attraction in

the

perihelion-motion

of

mercury

(the residual ellect which

the consideration of

view

is

all

that these are

cornctne-s

i.f

my

could

sorts of

t<>

the

Xe\\-

the

explanation

diseovertd not

by

disturbing of

of

for

factors).

the

the

Leverrier

be accounted

convincing proofs

theory.

purely general

approximation, and lead

as a tirst

second approximation

a,

the

by

My

physical

l-

i

s

I

vH5.

Hl N CIIT.I.

Or

Hamiltonian Function

Laws

for the Gravitation-field.

Impulse and Energy.

of

In order to show that the

field

the laws of impulse ;uid energy, write

it

in

\TIVITV

I!K1.

correspond to

equations

it

is

most convenient

the following Hamiltonian form

to

:

f

f

Hdr=o

8 I |

Here the variations vanish

at the

limits

of

the

finite

four-dimensional integration-space considered. It

is first

equivalent

consider

We

H

show that the form

necessary to

to

equations

(47).

as a function of

have

at

I

f

For

and

this

(47 n)

purpose,

I.-t

v ::.

(

>

<f

list

,

\

9^, A

3,

9 ,

",,A

6-

_

9 ,,

3 S

is

us

MI; \Usi.u

<.i

TIIKOIIS

bracket are of

different

cancel each other

and

our

for

thus have

that

so

ft,

,

which

is

and

/*

member

first

tin-

They

ft.

when they

symmetrical with

consideration.

)

ami change into one

si^ris,

the

only

1

terms within

the expression forSH,

in

f

multiplied by

last

\II\IM

the indices

another bv the interchange of

remains

IM-.I.

arising out of the two

The terms

fj

DI

are

respect

to

of the bracket

Remembering

we

31),

(

:

ra

ftft

i>

a

ft

Therefore f

9H 9

!

[

we now

cam

out thr

r f*P

8H

(48)

If

r

uv~

f

ft i

,r

r

9. /r

in

\ariation>

(

we obtain

IJa),

the system of equations /

a o O

(^ h )

an

/

~

(v

,

,<

which, nwinir as

was

If

to

the

o

a

relation.-

is

multiplied by

/

.

v

r ,

-

) I

/

(|s\

rr(|iiired to be proved.

(47b)

ii ,

eoineide

an =<>.

,

//

->

O

r

f/

with

(1-7),

14-0

i

JM.NTII

l.l.

KM, \TIVm

(M

aud consequently

r <

r

a

/

a.-

V

an

{

a

.,//

/ V

a.-;.

/ "

9"

a

8H we obtain

the equation

_a_ a.*

37.

**

L

Owinsr to the relations (48), the equations (4?) and fjifh

It

A

/"

to

is

be

noticed

that

,,/"V

L f

is

a

(:>4).

r-/^

not a tensor, so

that

(lie

<r

<iuatioii

(4 9) holds

only for systems

This equation expresses

and energy tiou of

this

in

tin-

equation

over

(40a)

r

or

which

\

laws of conservation of

-ravitation-field.

leads to the four equations

t

;i

In fact,

,/

i

in

= ].

pulse

the inte-ra-

tlncc-dimcnsional vohnne

V

\l.|i \I.1M-.I

(I]

where

..

,

,

.,

,

Sense.

laws

ol

\V- reco^ni-e

now

i/S

VITY

inward-

the

<>F

Kuclidean

the

in

this the usual expression For the

iii

conservation.

will

IJKI.ATI

the -urFaee-elemeut

to

energy-components I

(H-

direction-cosines

are the

>i

<>

drawn normal

MKollV

1

denote the ma^nituiles

\\ e

I

^

as the

oF the gravitation-field.

put the

Form which

(1-7) in a third

equation

-erviceable For a quick realisation of our object.

will be ver\

r 1

By

multiplxiii"

the field-equations (17) with

obtained in the mixed Forms.

ownm

to (:U)

^a

(

these are

that

a/"

;.__

,

r

V

9 ,._ -a., which

,

we remember

6

a r

,.<

IF

<j

f)"

r

a

/-

equal to

is

rr

"

] i>"

/r ^ r ^ -

v

V

r/

r"

or slight ly altering the notation equal to

6

The third member oF tin- expression cancel with the member of the field-equations (17V In place oF we ran, on account oF second term of this

second the

r\pre><ion.

the relations

put

(")H\

"

-

r

1

A

f

(

V

/

-j

/

\

.

ll

I

Therefore

XUI

I

I.K

OK I!M.\

|

|\

rn

UK- place of the equations (17),

in

wo obtain

a

General formulation of the field-equation

^16.

of Gravitation.

The

established in the preceding para-

lield-e<|uations

graph for spaces free from matter

V

the equation

now

to

matter)

s

<="

the

find

Poisson

-

is to be compared with Newtonian theorv. We have

of the

equations

V

Equation

which

= ^7r^.

2 <j!>

will

to

correspond

the density of

signifies

(/,

.

The

special relativity theorv

has led to the conception

mass (Triige Masse)

that the inertial

no

is

than

other

can also be fully expressed mathematically by energy. a symmetrical tensor of the. second rank, the energy-tensor. It

We

have therefore to introduce a

energy-tensor

~Y

energy components 10,

be

and

")())

(")!)

with

aiso

to tin

e|iiatioiis of

system it>

.neiL,r \

1

ir

like

the

of (he gravitation-field (equal

ion-.

eovariant

synimetrieal

how

teaches us

(corresponding field

our generalised theory

have a mixeil character but which however can

connected

equation

in

associated with matter, which

density of

gravitation.

(for

example the

total

i;ra\ itatin^

of the system,

tensors.

The

to introduce the energy-tensor 1

If

Sular->\

action,

oisson

s

equation) in the

we consider stem

v.ill

its

a

total

complete ina^-.

depend on the

ponderable as well

a>

a-

total

^ravitat ional.

CKVI .I; VI.KKD TIIKOUY or KKI.

\

can

\\\>

expi

!><

I

ct

/

energy-components of

hv

(s>cil,

putting

in

\Ti\m

l-l-"-

in

(")!),

^um

gravitation-field

energy-components of matter and gravitation,

hi-

\Ye thus

i.e.,

instead of (51), the teiisor-pijnation

i;et

"

T

where

of

place

alone the

T

1

s

(Laife

Sealar^.

The<-

;nv

tin-

general

tiehl-

M

(47),

we

gravitation

i)l

e(|tialions

must

It rLT\

working

Ljet.ltv

be

in

the mixed form.

l);iekvvards the

admitted,

that

analysis deduced

it

we have

\\a\-

ground this,

foi

a-

of the

the fore^oin^-

in

ner-\

should exert gravitating action

in

of

the

The stroii^e-! every other kind of energy. al ove eijiiilion however lies in

the choice id the

that

they

--IIIL;

the

energy

the

of

means

l>\

from the condition that the

the gravitation-field

same

introduction

this

of

plaei

svstem

-tensor of matter cannot he justified

Relativity- Postulate alone; for

In

(the

lead,

as

ini|iul>e-

correspond to the

shown afterward-

to

their

eon>crvatioM

and

e.|iiati"ii-

(I!

equations

<-..iisri|iirnce>,

the

of

the )

comjioneiit

which

ciieru\

and

i

M

a

>

.

of

total

\aetly

This shall

lie

1

vj

t

IMMXrlN.h. i|

I

The laws

17.

reduce

(5:2)

member on with

(or!)

\T|

VITY

of conservation in the general case.

The equations the second

i;i.I,

be easily so transformed that

can

right-band side vanishes.

the

reference

the

to

indices

and

p-

\\ c o-

and

subtract the equation so obtained after multiplication with .1

^

from

(5-2).

AYe obtain.

we

on

i>|)eratc

il

Now.

.

,., l>y

9

9

.

V .

2

I

lic

lead to eusilv ,r.

t

t

nst

8...u

..

and the third

.\]ifc>sions

>e-n

6. rr

which

r L

member cancel

by interchaii-in- the

on the one hand, and

ft

*P

and

A.

">

"""(

V

of

one

8*,

the round bracket

another.

a>

-nmniation-indices on the other.

can be u,

and

!

The M cond term Sr, that

u,.

TiiKui,^

u.is|.-.|i

i;

in

be transformed according

c;in

to

The second member leads

a.

a^

.

of the t-xprcssioii on the

Ifi

v

t-liand

first to

a

1

2

1 /

,//*

f/

a.

a.r

a--,

a,

I

,

(/

a

+ a..

The exprerokm

arising ont of the

of

tlic

ton-ether

So

flioiot-

of

and ni\r

axes. u>

that reroetnbering

The two

on account of

(">!)

we

incmlicr

last

the round brackel vanishes ftcooiding to

%

(

29)

others (

>!),

on

can

\\-itliin

aooonnt l>e

taken

the rxpression

li

(^-

9

a, ^a, ^<r i

V

A/^

,_

r

w

\ )

**

identical

19

t

(>!).

_ a

-

("):2a)

ll.

et

-

-idr of

\Ti\rn

1:1:1.

I

v.

1

I

1C,

I

Yoni

(5.)) jind

From the

that

the

lilN rUM.r.

field

follows

i!

(~>:l-,i}

\Ve

I

\TIYI

I

Y

hat

equations of gravitation,

conservation-laws

satisfied.

ItKI.

()!

see

of

most

it

impulse

(I .hi);

are

energy

following

simply

reasoning which lead to equations

also follows

it

and

onlv

the

>atne

instead

of

the energy-components of the gravitational-field, we are to introduce the total energy-components of matter and gravi tational field.

18.

The Impulse-energy law

for

matter as a

consequence of the field-equations.

If

we multiply

8 ^ (5-5)

with

.

we

L;-et

ill

a

w;iv

(T

similar to

15,

remembering

a/

=H

:/,.

~

a/"

the equations

or

(jf

^ =_ /"

(:>(>)

g-/

+

4 comparison with the above choice of

nothiiiLl hut

vanishes,

8..

:

Pof

1

I"

8

remembering

(57)

that

fV=

:

(Ml>)

shows

co-onlinates

these equations

that (

x

,/

the vanishing of the di\eruence

the eiier^ v-coiiiponents

ul

matlei.

= .if

1)

the

asserts

tensor

(.

K\

i.isi-

\

i;

i

IMivsieallv the

left-hand

iervation of impulse

when

hold

g^

are

a

ni

IIII.OIM

that

KI;I,

of

appearance

shows

>ide

D

alone the law of con-

hold; or can only

and energy cannot eon>tants

when

i.e.,

;

The second memhei

tation vanishes.

117

term on the

second

the

matter

for

\n\iTY

the ;in

i>

tield

of gravi

e\|H .^-ion for

impulse and euergj whic!ithe gravitation-field exe:ls pelThis comes out clearer time and per volume upon matter.

when

The

we write

instead of (57

9

C-

9

;

=-r:

of (17).

a

^

fi

ri-ht-liand side expresses the interaction of the energy

The

of the gravitatioiiftl-iifld on matter.

contain

gravitation

which are to be

^{

Form

the

in

it

thus

satisfied

when

the latter

material

the

the ei|iiations of

material

all

by

same

the

at

conditions

I

We

phenomena.

phenomena completely four

characterix-d In

i>

field-equations of

time

other

differential

of one another. equation;- independent

THE

D

The Mathematical once

enal)ie>

to

u>

auxiliaries

formulated

I

of

law>

llectro-dynainies

according

The generalised limitation

s

developed

according

^cnei-ali-e,

theor\ of relativitx, the physical

namies, Maxwell

PHENOMENA.

MATERIAL"

the

to

Kelativitx

the\

;

at

(Hydrodylie

already

special-retain ity-theory.

Principle leads us to no

po->il>ilitics

!

generalised

of matter a>

H

under

to the

hut

it

eiiahles

u>

further to

know

on all processes with exactly the influence of gravitation out the introduction of any new hypothesis. It is owinii to tins, that as

of matter tion>

are

1,111

a

to

he

narrou

regard- the physical

:.o

delinitt

iie:-e--ar\

>en>e)

introduced.

The

i|ite-tioii

nature

anni|ima\ he open

I

I

s

i;

i

IN (i

i

!,!;

whether the theories of the

i;i<;r,\TiviTY

(>!

Jonn a

The general

the theory of matter. teach

us

theory

it

no

new

by

the

basis for

sufficient

jo$tulate can

relativity

But

principle.

and

field

clectro-maguetic

gravitational-field together, will

building

tin-

up

must be shown whether electro-magnetism and

gravitation

Euler

19.

can

together

did not succeed in

what

achieve

the former alone

doin<r.

equations for frictionless adiabatic

s

liquid.

Let

//

and

p,

be two scalars, of which the

the pressure and the last the density of the

them

there

is

a relation.

denotes

first

fluid

between

;

Let the eontrawriant symmetrical

tensor

ff

il.r

,j

,j

/

be the contra-variant energy -tensor of

/)

the

To

liquid.

it

also belongs the covarianl tensor

/

/i

+

fir

>

well

It

^t

t

a>

we

the

/

,,

/

,

i><>

,/..

fl>

,- p ,/>

the mixed tensor

put the

^eiieriil h\

ri^ht-liand

drod\

side

ii;iinical

ol

^OSb)

ing tn the g --ne raided relativity theory.

Completely

solves

the

in

i^j/a)

we

ei|iialiuns of Kuler accord

problem of

This

motion:

in \>\

for

inciplc

the

four

i,KNKKAI,IM

equations //

and

f

(.")?a)

I

,l>

together

lltdin

01

IJKI.A

l\

I

ITV

.

between

the ^iveii equation

\vitli

1

II

and the equation

>,

-

p

V ./

,ls

are Millieient, with the

1

,/*

values of y

^iven

for

,,,

finding

out the six unknowns

(/X

If

tions

10

s

//

Now

//

it

(7) independent

wide

If

unknown we have

are

be

is

(">

$),

in

for

that

out

rinding

more than

is

the

the equ-

sufii-

equation (o?a)

is

so that the latter only represents

This

equations.

freedom

also to take

equations

noticed

in

l/S

number

so that the

,

already contained

the

l

There are now 11

(*)-i).

functions

cient.

I/S

the

indeliniteness

is

due

choice of co-ordinates, so

mathematically the problem

indefinite in the

is

sense

to

that that

of the gpace-fuuetiona can be arbitrarily chosen.

$20.

Maxwell

Let <f>

the

s

Electro-Magnetic

be the components of a covariant four-vector.

electro-tnagnetic potential

ing to

(. US)

field- equations.

the components

of the electro-magnetic

field

I-

;

from ul

it

let

iis

form accord

the eovariant

aeeordin^-

\<>

the

>i

\-vector

>\>tem

of

J50

\CIPI.K

i-ui

From

(")

.),

it

oi

\TI\ITV

I;KI,

follows that the system

9F

9

ol

equations

F,

a<

is

satisfied

(o?),

is

MM

which

of

anti-s\

the

left-hand

mmetrical

tensor

according

side,

of

third

the

to

kind.

This system (HO) contains essentially four equations, which can be thus written

:

((lOa)

a-.

This

of

system

eqaatioDfl

system of equations of

correeponds

Maxwell.

\Ve see

it

to the at

once

second if

we

put (

I

Y,

Instead of (OOa)

=

II,

we can

=

! ,<

therefore

write

K,

according to

the usual notation of three-dimensional vector-analysis:

an

.it

K=(

a/ div 1I=,

OBNERALTABD TIITOIM or RBT*ATIVIT\

The

litst

Maxwellian system is obtained onn n iven liy Miukowski.

lisation of the

\Ve tin-

by

1

>

I

genera -

a

t

introduce

the

(-out

ra-variant

six-vector

\

1

P

f|iu(i()n

/

and also a contra-variant four-vector electrical current-deusitv

in

vacuum.

\\liicli

.l

,

is

the

Then remembering

(40) we can establish the system of equations, which remains invariant for any substitution with determinant g to our choice of co-ordinates). I

It

we

put

11

(I

,

I

=

H

=

H

.,

K"

=

K S1

=

}

F"

which iuantities become

e(|iial

to

-

II,

.

M

!]

the special relativity theory, and besides

=

.1

we

Gjet

/.

)

instead of ((>

=

P

$)

r ,-ot

H-

= a/

*

in

the

ra>e

of

NMxni

152

The sation

p(|uations

remains true T/ti

(( ()).

Maxwell

of

iti

or

i.i:

VITY

\TI

and

((>:!)

s

1:1:1.

(<>.">)

give ihus in

field-equations

a

generali

which

vacuum,

our chosen system of co-ordinates. I In-

energy-components

Let us form

1

<>/

electTO-ntttffneftc fi

/<I

.

he inner-product

=

K

>:>)

K

-I

.

,<r <TJ>

According in

to (01) its

components ean

l>e

down

written

the three-dimensional notation.

/.

K

(.

K

four-vector whose components are equal

eovariant

is a

H].

cr

to the

negative impulse and

energy

to

electro-magnetic

field

per unit

of

ill"

bv

volume,

masses be

magnetic

K

the

free, that field

under the then

onlv,

time, and per unit

masses.

electrical is,

which are transferred oi

If

the

influence of

eovariant

the

electrical

the electro

four-vector

will vanish. (T

In order to get the energy

tro-magnetic

K

o,

field,

we

components

require only

the form of the equation

to

T

of the elec

give to the

equation

(">7).

(T

From

(!

})

and

we ((>">)

6

a,

get

first,

,, (

v

,,,

W

9 <"

e

"

riE.VElMUSED THKORY OF RELATIVITY

On

account

side

f

(60) the second

member

1

53

on the right-hand

admits of the transformation

ft

to

V

this expression

symmetry,

can also be written

in

the form

which can also be put

1

9

.<

in

the form

+

The

:in

I

first

F

ft ap

F

ft

2 ^v 6

/

/*

(

V

of these terms r-:m be written short lv

the second after

the form

ll

\

difl erentiation

^

Vft l

}

:i-~

can be transformed

in

NM\(

154 11

we take

all

I

ul

Li:

I

terms together,

the throe

we get the

relation

<

where

On

F

T"=-

(<Hk)

F

(Td

v

"

+

1

\

F

8

a/?

account of (30) the equation (66) become? equivalent

to (57)

K

and (57a) when

the help of

(61) and

Thus

vanishes.

the

energy -components of

electro-magnetic

(64)

we can

are the

s

With

show that the

easily

energy -components of the electro-magnetic

T

field.

field, in

the case

of the special relativity theory, give rise to the well-known

Maxwell-Poynting expressions.

We

have now deduced the most general

the gravitation-field co-ordinate

system

and matter satisfy for

which

\/

achieve an important simplification in

.

/

=

all

which

laws

when we use 1-

Thereby

a

we

our formulas and

calculations, without renouncing the conditions of general

covariance, as

we have obtained the equations through a

specialisation of the co-ordinate system C O variant-equations.

interest,

whether,

Still the question is

from the general not without formal

when the energy-components

gravitation-Held and matter

is

defined in a generalised

of

the

manner

without any specialisation of co-ordinates, the laws of con servation have the form of the equation (06), and the field-

equations of gravitation hold in the form (52) or (52a) such that on the left-hand side, we have a divergence in the ;

usual sense, and on the

energy-components

of

right-hand side, the

matter

sum

and gravitation.

of the J

have

\USKH TIIMilM

tiBNfcK

found out thai

this

is

indeed the ease,

eominnnicat ion of

that the

KKLATIVm

(>!

lint

I

I

am

of opinion

rather comprehensive work-

my

on this subject will not pay, for nothing essentially conies out of

E.

We

ne\\

it.

Newton

$21.

-V)

have

s

theory as a mentioned

already

special relativity theory

approximation.

several

case of the general, in which q

times

the

that,

upon as a

be looked

to

is

first

special

have constant values

s

(4).

p.v

This

signifies,

one

according

important

when

\

//

pared to

I)

second and

the

of

total neglect

to

what

if

approximation

differ

of

influence

been

has

before, a

said

gravitation.

We

we

the

consider

get case

from (4) only by small magnitudes (com

where we can neglect small quantities of the higher orders

asjK

(first

e.t

of

the approxima

tion.)

Further

it

assumed

should be

that

within

the

>pace-

,

time region considered, the word infinite

in

s

f]

at

infinite distances

(JIMIIL:

a spatial sense) can, by a suitable choice

of co-ordinates, tend to the limiting values (4); sider only those gravitational fields

/.<.,

we con

which can be regarded

as produced by masses distributed over finite regions.

\Ve can assume that this approximation should

Newton

s theory.

the fundamental

For

it

however,

equations

Let us consider the motion

equation

(4fi).

the components

it

is

lead to

necessary to

treat

from

another

of

particle according to the

a

<>f

|M>in<t

view.

In the case of the special relativity theory,

156

I

can take any values

This signifies that any

:

--

y +

appear which

can

vacuum

(r

(;;;;

we finally limit ourselves is small case when compared

it

/,-

signifies that the /(

<k

1

8oiilv

velocity

g

up

the

to the

components .I

r^ ui

f

lv<wx

can be treated as small quantities, whereas 1,

to

/-

2

ivf

Juiit*iio-

y

of light in

If

consideration of the

.>)

>%<

than the

less

is

<1).

velocity of light,

KELATivm

Lh OF

ltl.NCll

is

equal In

magnitudes (the second point of

to the second-order

view for approximation).

Now we

see that, according to the first view of approxi

mation, the magnitudes

A

at least the first order.

that in this equation

approximation, we

are all small quantities of

s

\~

glance at to

according

are

also show,

the second

take into

to

only

will

(4(5)

view of

account

those

terms for which /x,=v=4.

By we get

limiting ourselves only to terms of the lowest order instead of (46),

=

first,

f

\

i-

the equations

where

ds=<1s

:

=<//. t

or by limiting ourselves only to those terms which according to the

first

stand-point

approximations of the

are

order,

,/

,//

.x,

r

4

4

-i

-I

J-

first

niKoio

<;I.M.K.\I.I>I.I>

we

It

;tume

further

quasi-static,

matter producing the to

(relative

the

that

limited

is

i.e., it

KKLATIVITY

01

l">?

the gravitation-field

is

when the

the case

only to

is moving slowh we can neglect the

gravitation-Hold of

velocity

light)

differentiations of the positional co-ordinates on the ni;ht-

hand side with respect

to time, so that \ve get

<"

9</4

T

=-i a^

11171 ,fi.-

-idj lt

+

=

(,

,

a,

)

Tins the equation [of motion of a material point according to Newton s theory, where ff tt / t plays the part of The remarkable thing in the gravitational potential. i>

result

that in the

is

first-approximation of motion

material point, only the

component y tl

of the

of the fundamental

tensor appears.

Let us now turn

we have

case,

matter

is

density/*

to

mem her defined

exclusively of

matter,

>.<-.,

right-hand side of 58 necessary

h eld-equation

the

to re

that

the

or

second If

5M>)].

then

approximations,

In this

(5o).

energy-tensor of

narrow sense bv the

a

in

by

[(">Sa,

the

all

member

on the

we make the vanish

component

except

T On

the

left-hand

sidr

,

,

=

of

i*

~ T term

second

the

(r,:))

is

an

infinitesimal of the second order, so that the first leads to

the following terminteresting fur us

6 r^ i a-l J

4f

the approximation, which are rather

in

;

a e-.L

[>"!

a

4 +

r a. ,L

2 J

/u/

i

J

a

r^i

8^L

By neglecting all different iat ions with regard when ^=^=4, to the expression

.Jto

this leads,

_! / 6

2 .

/,

-^

(

6""

,6 +

a

-

6

>,,,

2

+ 6 ,

-

:/,,

v

x <7

"-

time,

158

i

The

ULLATIMIN

IM\( IIM.K OK

last of the

equations (53) thus leads to

VY,,=^-

(68)

The equations (67) and (68) Newton s law of gravitation.

together,

ai>e

equivalent to

For the gravitation-potential we get from (67) and (68) the exp.

(68a.)

whereas the Newtonian theory for the chosen unit of time gi ves

%

I

.

67xlO~

gravitation-constant.

*

(6!)

=

wliere

s

SirK

:

K

denotes

the

usually

equating them we

tret

= Ks7xlO-".

Behaviour of measuring rods and clocks in a Curvature of light-rays.

$22

statical gravitation-field.

Perihelion-motion of the paths of the Planets. In order to obtain

Newton

a

mation we had to calculate only nt iits

//

theory as //,.,,

a

first

approxi

out of the 10 coni|X)-

of the gravitation-potential,

for that

is

the only

component which comes in the first approximate equal of motion of a material point in a gravitational field.

We

M-e

however,

that the

other

should also differ from the values- given the condition

/

=

1

.

components in (1-) as

of

ion>-

</

required by

flF.N

ERALISF.i) T1IKOHY OK

HKLATIVJTY

l">y

For a heavy particle at the origin of co-ordinates and generating the gravitational field, we get as a first approxi mation the symmetrical solution of the equation :

S

is 1

or 0. according as p

=

o-

or not and

/

is

the quantity

... On

account of (68a)

\VP

(70a)

where to

M

verify

.

have .

=

denotes the mass generating the that

this

solution satisfies

field-equation outside the mass

It

is

easy

approximately

the

field.

M.

Let us now investigate the influences which the field M will have upon the metrical pro|>erties of the

ef mass Held.

Between the lengths and times measured

the one hand, and the differences in co-ordinutes

locally

on

on the

d.r V

other, \ve have the relation

For a unit me:i8iiring rod, for example, axis, we have to put

placed

the

./

then

=

-1.

i/.r,=,f.-

-!=</

s

=,/.,

/-".

4= o

parallel

to

100

PKixriPi.i.

If the unit

the equations

i;i;i.\Ti\m

01

measuring rod

on the

lies

axis, the

tirst

of

r (

From both

O) gives

these relations

follows as a

it

first

approxi

mation that (71)

The

,/.

= !-

.

4

when

unit measuring rod appears,

co-ordinate-system, shortened by the the

through the presence of place

tangential (/x

we

2

can

|x>sition,

=

1.

we then

=</.(

magnitude when we

_,

for

=</,

co-ordinate-length

in

-!=. /,

=o

1

,

.i.

!!

.!.,=.

.

3

=O

/!;

= "I

gravitational field has no influence upon the length

we

of the rod, \vhon

put

it

tangentially

Thus Euclidean geometry docs tational

lield

O\MI

that one and the orientation

extension. ex]>ected

in

the

same

can

first

serve

is

the

much

too s

field.

we

if

approximation,

as

measurement of earth

the

hold in the giavi-

rod independent of

its

(fi!))

small

to

conceive

position

measure of

But a glance at (70a) and difference

in

nol

the

and

same

shows that the l>e

noticeable

surface.

AVe would further investigate the rate of going of i

a

example

get

i71a.)

The

its

get

we put

if

</.",_

in the

Held,

gravitational

radially in the field.

it

Similarly

its

referred to the

calculated

mil -clock- which

is

placed

in a statical

Mere we have for a period of the cluck

gravitational

;i

field.

:i

\Ki; VI.ISKH TIIKiHiY

Kil

OF KKI.ATI VITY

then we h:ive

/,=! + I

"

f

Therefore the clock goes slowly what the neighbourhood of ponderable masses. this that the spectral lines in the light

it

is

place*!

in

It follows

from

to us

from

coming

the surfaces of big stars should appear shifted towards the red end of the spectrum.

Let us further investigate the path of light-rays statical gravitational Held.

According

vity theory, tin* velocity of light

is

in

given by the equation

thus also according to the generalised relativity theory is

given by the

e<|ii:it

>9

If the direction,

V*J

>.,

(/

the ratio d

=0

it

the velocity,

a

:

: ,

ihe ejii!ition (7:J) gives the magnitudes

and with

it

ion

*

(73

a

to the special relati

<l,<\

//-,

is

gi\Mi.

jl

PR1NTIPLK OK KEJ-A.T1VTTV

.->

in the sense of the

Kuclidean Geometry.

vVe can easily see

that, with reference to the co-ordinate system, the rays of light

must appear curved be

the

If

a

of

propagation,

light-rays

]

direction

we

(taken

curvature

^-?

in case

perpendicular

have, from

in

Ihe

*s

//

plane

are not to

II uygen s (y,

)]

the

constants. direction

principle, that

must

suffer

a

.

A Light-ray

Let us Hnd out the curvature which a light-ray suffers it at a distance A from it. If we goes by a mass

M

when

use the co-ordinate system according to the above scheme, then the totsil bending B of light-rays (reckoned positive when it is eonca.ve to the origin) is given as ;i sufficient

approximation bv

where

(?.

)

and (70) gives

, ,

\

bemlgra/ing the sun would suffer whereas one coming by Jupiter would have deviation of about ray of light just

ing of a

1

:i

7",

-0:2".

IM

KD THKOKV OF KKLVTIVITY If

of

we

calculate the gravitation-field to a

and

approximation

with

the

it

of a material particle of a relatively small

mass we get

a deviation

greater order

corresponding

path

(infinitesimal)

of the following kind

from the

Kepler-Newtonian Laws of Planetary motion. The Kllipse slow rotation in the direction of Planetary motion suffers ;i

of motion, of

amount S

7r (

7. )

per revolution.

.<=

i

," "

T"c*(l

In

the

Formula

this

velocity

eccentricity,

The

of

ft

light,

T, the time

)

signifies the

measured

in

semi-major axis, the usual way,

f,

c,

the

of revolution in seconds.

calculation gives for the planet Mercury, a rotation

of path of

amount

43"

per century, corresponding

suffi

what has been found by astronomers (Leverrier). found a residual perihelion motion of this planet of They the given magnitude which can not explained b\ the

ciently to

I>e

perturbation of the other planet-.

NOTES Note

The fundamental electro-magnetic equation

1.

Maxwell

of

media

for stationary

an>

:

<mrlH=I

D-//K According to Hertx and Hoaviside, these require modi fication in the case of

Now change

-

)

due

to

motion alone

to

motion

div

//,

\\

/ is

the vector

Hence equations

H=

curl

~

there

is

of

velocity

(1)

+v

\\

and

+enrl \Uu\

the

moving bodv and

//.

and (2) become div D-fcurl \ ect.

[Dv]+pv

(1-1)

and

K= which gives

O

+n

a

given by

[Rw] the vector product, of

I-

bodies.

moving

known that due

in a vector /?

( _ \ o

where

it is

-|-

linally, for p

div l)=:r curl

div o

H

-j-

curl \Vrti.

and div H

111-

= O,

Vwfc,

Dl

)

/

,H

)

166

PKINCIPT.K OK ItET.ATIVrn

Let us consider a beam travelling along the ./--axis, with apparent velocity (i.e., velocity with respect to the Hxed ether) in medium moving with velocity n, = n in the /

same

direction.

Then

if

the

-

A

f

to

proportional

and

electric

vectors

magnetic

are

(x-vt)

e

we have

,

= -/Ar, |- = =0, f-=/A, |j 6* 8dy

=

,/,

;,.

=0

5."

Since

D=K E

B=

and

H, we have

/A

.

/x

Multiplying

(;-.)

Hence

K

(i

-//) .

making

by

(1 23), /A

H,=cE

2

.

(1-22)

...

(2-23)

(2 28) 7

(f

v

...

=C 2

2 )

=c s /^ =

=

5f

.

+ w,

Fresnelian convection co-efficient simply unity.

Equations (1 21), and (^ ^l) may bo obtained more simply from physical considerations.

According both

electric

medium

itself.

to Heaviside

Now

to the ether, the rate

5D

and

and magnetic

at a point

of

Hertz,

the

polarisation

which

change of

is

real

the

is

tixed

electric

seat

of

moving

with respect

polarisation

is

NOTE* matter moving

Consider a siab of along-

the

electrostatic

= o,

then

./--axis,

even

there will be some-

.

-^Of

this

term

Doing

this

a

to

a

for

is,

field

we immediately

of

which

in

polarisation of

we

Hence

.

O

<:

of

rate

temporal

purely

velocity n,

Held

in the

motion, given by u r

its

with

stationary

change

or

must add

a

in

that

polarisation,

the body due to

167

arrive

at

change

equations

(T21) and (2 21) for the special case considered there.

Thus the Hert/-Heaviside form of

field

equations gives

unity as the value for the Fresnelian convection co-efficient.

how

this

entirely at variance with the observed optical facts.

As

It has been is

a matter of

shown

I

Larmor

fact, 2

that

historical

lias

shown (Aether and

order to explain experiments of the

A

short

summary

bearing on

introduction

not only sufficient but

is

I/ft

in the

this

is

also necessary, in

Arago prism

of the electromagnetic

question,

Matter)

type.

experiments

already been given in the

has

introduction.

According tion

is

carried

outside

Hertz and

to

the

in

the total polarisa

Heaviside

medium away by it. Thus the a moving medium should

situated

itself

and

is

completely

electromagnetic

effect

be proportional to K, the

specific inductive capacity.

Rvic/mtil is

showed

rapidly rotated

the magnetic

(the

effect

when

l^itl that

in

dielectric

outside

is

a charged condenser

remaining

stationary),

proportional to K, the Sp.

Ind. Cap. /, ,;/////,-//

if

the

(Annaleu

dielectric

is

stationary, the effect

.lev

IMiysik 1SSS,

rotated is

1890)

found that

while the condenser

proportional to

K

1.

remains

PRINCIPLE OF RELATIVITY

168 Kickenwalil

1903, IVOV) rotated

(Annaleu der Physik

together both condenser and

and fouud that the

dielectric

magnetic effect was proportional to the potential difference and to the angular velocity, but was completely independent of K.

This

of course

is

with Rowland

quite consistent

and Rb ntgeu. Blotidlot

of air

in a

current

this

f-ubt s

of

due <>f

charged

JOl)

with

a

passed

current

(H =H,=0).

velocity

the relative motion

to

A

induction.

with

up

But Blundlot

/

If

the

along

,

of

+ a,

of plates at .*=

pair

/i=D,

density

set up along the matter and magnetic

= KE =K.

H

y

/c.

failed to detect

with a cylindrical

experiment

above

the

contradict the

set

of

repeated

made of to its own

condenser

ebouy, rotating in a magnetic Held parallel axi^ He observed a change proportional to not to K.

Thus

will be

n.

any such effect. A. intsnn (Phil. Trans. Kuyal Sot-. 190i)

//.

the

moves

air

1

Held II,,

magnetic

an electromotive force would be

.F-axis,

r-axis,

Reudus,

(Coinptes

steady

electro-magnet ic

K

I

and

experiments

equations, and these must

llcrt/.-lleaviside

be abandoned. [P. C.

Not

Lur

2.

Ti

ii

itxfoi m/i/

\rrsuch einer

Lorentz.

optibfhon Ki

iii:

-

C lu

theorie der elektri-chen

pages

Lorentx. null-effect.

to explain

on

Theory

l:r---Dii.

the

of

.MK-ctrnns

i30,also notes

wanted

to

I

it/.gerald

hypothesis

(hat

<

>.

explain

In order to do 50, tin-

uud

ijinn^fn im bewegteii Korpern.

(Leiden Lorent/..

M.]

i<nt.

an

it

(English

86, pages :;is.

the

was

eilition), :;-:s.

(Vfichelson-Morley

obviously

COUtrACtlOD. electron

1895).

necessary

Lorentx, itself

worked

undergoes

169

NOTES contraction for

He

when moving.

introduced new

variables set of

moving system defined by the following

the

equations.

and for

used

velocities,

v.*=p*v,

With known

+

l i

r

=(3r,,

v,

,

l r

=(3r, and p =p/fi. l

the help of the above set of equations, which the

as

Lorentz

showing how the

is.

he succeeded

in

as

a

contraction

Fitzgerald fortuitous

of

consequence

transformation,

results-!

of

compensation

opposing

effects."

It should be observed that the is

not

with

identical

addition

Kinsteinian

of

velocities

also the expression for the It

is

true

that

remain practical/ tion, but they

1/

Lorentz

transformation

the

is

relative"

different as

quite

density of electricity.

Max we 11- Lorentz

Held

equations

unchanged by the Lorentz transforma changed to some slight extent. UUP

ar>-

marked advantage of the Einstein transformation in the fact

The

the Einstein transformation.

that the

preserve exactly the

Held

same form

as

a

of

equations

con^i-t>

moving system a

stationary

resin-Han

convection

those

of

M -torn. It

should also be noted that the

; l

coctKcient comes out in the theory of relativity as a

consequence

of

quite independent

Kin-tern o!

any

-

addition

of

direct

and

velocities-

[P.C

Note

is

electrical theory nf mattrr. If.]

3.

Lorent/.,

181, page 513.

Theory

of

Klectrons

(English

edition),

170

I

H. Poiucare, Sur

la

matematico

del circolo

RELATIVITY

MI

ttlNrill.F

Rendiconti

electron,

dynamujue

di

Palermo 21 (1906). [P. C.

Note

Lorentz of

lielaticity

Theorem

showed that

the

4.

electromagnetic

unchanged bv

the

M.]

Kelativiiy-Principle.

Maxwell-Lorentz system remained practically

field-equations

Maxwell

of

Thus the

transformation.

Lorentz

laws

electromagnetic

<ai<l

and Lorentz can

be

independent of the manner in which they arc referred to two coordinate systems which have a uniform translatory motion relative to each other." "

definitely proved

"

(See

be

to

Electrodynamics of Moving Bodies," page 5.) Thus electromagnetic laws are concerned, the

so far as the

principle of relativity ant he proved lo be fntf.

But

it is

not

known whether

true in the case of other

this principle will remain

physical

laws.

We

can always

Thus proceed on the assumption that it does remain true. laws in such a it is always possible to construct physical retain their form wln-n

way that they

The ultimate ground

coordinates. cal

the

laws

in this

principle

way of

is

referred

to

moving

for formulating

physi

merely a subjective conviction that is

relativity

no a priori logical neccs^it

\

universally

t!:;it

of

(so

should

it

There

true.

be

so.

is

Hence

ar as it is applied to Relativity other than must be regarded electromagnetic phenomtMin as a jjostiiliil.t.-, which we have u.-Mimed to be true, but for

the

Principle

I

)

which

\\v

cannot

the generalisation

is

adduce any definite proof, until after made and its consequences tested in

the light of actual experience. [P. C.

Note See

"

5. Kli-ctrodynamic> oi

Moving

Hodies,"

p.

5-8.

M.j

VOTES

Note

6.

/

Equation! Cartesians

fymi/f aim

/>/,/

:md

(/)

171 l/////-0jrj/r*

in

become

(//)

when

l-

onii.

expanded into

:

8*1 ^ o y

a,

8*, ao

oT o "

,

/>

,_.

T

a^

8 MI, o a.--

f.jj

pj, p^ for

We

9 ^ 8r

8>", ~"

>s~"

a//

Substituting pi(

;c,

.

t

r

av^""

,,

,

=/>-

.r

.-i-

2,

/>"yi

,

,

3>

pit

:

ipi

)

j

for

.<,

where

;/,

r,

and

/^= \/

ir l

t

get,

a

a^i,

s

a^j

~

"^

?

8

"8;r,

a?

9m,

:

,-9j\

ar,~ aa-,~*9^ and multiplyinG: a/

Now

*

,

(2 1)

a

by a/

-

,

/

\v<>

get

1

-

substitute

=/a 711

*

=

=/S 1 =

/ ""/I

I

S

a

id

/V,=/4I= *

,

=/ = 4 *

/ it /,

;

and

p,

,

172

HI? I

and

\ve irct finally

\CTPT.K

OP RELATIVITY

:

8./.

8/S4 88

a/,,

i

d^t-^+iSt

^ ...

a/,,

(3)

+* * ,

8/, 4

5

8/41

,

s/*.

,

a/..,

[P. C. M.j

Note Page

One of

On

9.

the Constancy

refer also to

i:>

of the

page

6, of

Einstein

two fundamental Postulates is

Relativity

s

/,//////.

paper.

of the Principle

the velocity of light should remain

that

constant whether the source

is

moving

or stationary.

It

a radiant source S move with a velocity should always remain the centre of spherical wave-

follows that even u, it

of the Velocity of

if

expanding outwards with velocitv

At

first

it

sight,

may

c.

not appear clear

velocity should remain constant.

theory of Ritz, the velocity should become source

o!

velocity

light

moves

why

the

Indeed according to the c

+

it,

towards the observer

when the with

the

n.

Prof, de Sitter has given an astronomical

deciding

suppose there about

argument for Let two divergent views. a double star of which one is revolving

between

the

is

these

common

centre

u<

of

gravity

in

a circular orbit.

VOTES

Let the observer

in the

lie

173

plane of the orbit, at a great

distance A.

The

emitted by the star when at the position

light

by the observer after a time

will be received

c

when

the light emitted by the star be

after a time

received

B

to

is

2

T+ is

A

Then

-Al

approximately

l

Now

.

that

impossible

Kepler

s

if

Law.

In

-^-=- are not onlv c

a

much

larger.

A/c = 33 then

T

binaries,

T

and

the

B

^

ha If- period

be comparable

the

A

to

is

T, then

it

should

observations

the

same if

=o.

a>

satisfy

to

c is

not

sec,

T=b

an annual parallax of

they

of

stars,

T, but are mostly

////

;<=!()()

The existence

fact that

therefore a proof thai

<>nler

from

B

to

most of the spectroscopic binary of

will

the real half-

and from

*-

years (corresponding -

be

observed

the

For example,

i/^A/c.-

T

Let

.

position

while

,

it

u

c

period of the star.

at the

+

A

da\s, -1"),

the Spectroscupio

Kepler s Law is by the motion of

follow

ad octed

the source.

In

a

Freun

llich

occurs

in

the criticisms of memoir, replying and (liinthiek that an apparent ecoontripity the motion proportional to / being the

later

t

^

"A,;*,

",i

17

rillNCll LE Ot

t

maximum

value of 7/,,=c

the velocity of light emitted being

t>

,

+

IIELATIVIIY

/t

X",

=

Lorentz- Einstein

/=!

Ritz.

But

Prof, de Sitter admits the validity of the criticisms.

he remarks that an upper value of k

may

be calculated from

the observations of the double sar /3-Aurigae.

The

7r

parallax

=

A /

e

OU",

>

=

= 110

005, w n

For

this star.

T=3

/f-w/sec

96,

05 light-years,

is

<

-OOe.

Fji an experimental proof, see a paper by C. Majorana.

Mag., Vol. 35,

Phil.

p.

1C,:}.

[M. N.

Note

10.

It

the

is

Urtl-tlcnuili,

volume density

/>

/

p

\

([

n -}

is

ing volume rest,

S.]

Muctricity.

<>f

moving system then

a

in

the corresponding quantity in the correspond

in the fixed

and hence

it is

system, that

termed the

is,

in the

system at

rest -density of electricity.

[P.

Note 11

(page

SpaM-limc

Ir

<}/<>

r^

a li.]

17).

of lk:

//>*/

uml

I/

.svv.W khnl,

As we had already occasion to mention, Sommerfeld in two papers on four dimensional geometry ( vWr,

hag,

1

Annalen der Physik, Bd. 2, p. Tli) and Hd. :i: ,, p. 640), translated the ideas of Minkowski into the language of four ;

M

dimensional geometry. Instead of inkowski s space-time vector of the first kind, he uses the more expressive term four-vector,

thereby

making

it

quite

clear

represents a directed quantity like a straight line, or

a

momentum, and

direction

time-axis.

of

space-axes,

that a

has got 4 components, three

and

one

in

it

force in

the

the direction of the

r

VOTKS

The representation is much more

I7. )

two straight

of the plane (defined by

In

difficult.

lines)

three dimensions, the

can

he represented by the vector perpendicular to Hut that artifice is not available in four dimensions.

plane itself.

For the perpendicular to a plane, we now have not a single but an infinite number of lines constituting a plane.

line,

This difficulty has been overcome by Minkowski in a very manner which will become clear later on.

elegant

Meanwhile we

the

offer

from

following extract

the

above mentioned work of Sommerfeld. (Pp. 755, Hd.

of the six-vector (which is the

space-time vector of the of

case

d. Pliysik.)

a

same thing let

kind)

2n<l

Minkowski

HS

us take the

s

special

piece of piano, having unit area (contents),

and

form of a parallelogram, bounded by the four-vectors

the //,

Ann.

Z-2,

In order to have a better knowledge about the nature

"

Then the

passing through the origin.

,

this

of

piece

plane

on

//,

/,,

w,,

projections

the r

a

of

plane four

the

projection

of

given

hv

the

vectors

in

the

is

.r//

combination

Let

us form in a similar

this plane

<.

Then

but are connected

six

1>\

manner

all

component*

Further the contents

<

|

)

i

Let

us

plane

4|

<j>*

normal

of

independent

of the piece of

sum

of

a

is

to

the squares

of

plane

In fact,

=*,

now on

all

the following relation

be defined as the square rout of the these six ipiantitio.

the six component are not

the to

+#,+#,++#. other hand take the

$

;

we

can

call

+<*>*.-

ea-^e

this

of the unit

plane

the

176

PRINCIPLE OF RELATIVITY

Complement

Then we have

of

the

<f>.

The

proof of these assertions

be the four vectors defining following relations

r*

r,+v*

we

Then we

these

equations the

first,

r*

;/*,

have

the

rt,

and

the fourth from

the

by

v,,

n,,

obtain

multiplying these equations by r*

we

Let

as follows.

?+? * r.+rfr,=0

from

subtract the second third

:

:

we multiply

If

is <*.

relations

following

between the components of the two plane

.

;/*-,or

by r*

.

?/*

.

obtain

.$,1

<i>*:

+

,,=0 and

**,

<*

y

^,,

+^?. ^..,=0

from which we have

In a corresponding

way we have

in whon the subscript (//) denotes the component of the plane contained by the lines other than (; ). Therefore </>

the theorem

We

have

is

proved.

^^*=^,

<+..,

XOTKS

The *"

way

the following

p and p*

denoting

perpend ic

ihir

f*

Vector"

the

contents of the pieces of mutually

may

it

p

We

|

"conjugate

and

//*

+

$*

of

is

/*)

4,

,,

can verify that

=

/*,

/

The

.

have,

f* = ,,*

|

/

be called the comi)lement

obtained by interchanging

We

composed from the vectors

is

:

composing

planes (or

/

vector

sii\-

^eneral

in

177

and (J f*}

vectors,

for

may

etc.

/.i

be said

to

bo invariants of the six

values are independent of

their

the choice of

the system of co-ordinates.

[M. N.

Note

Ligkt-tcfocity H*

12.

Page

and

-2-J,

8.]

maximum.

ti

Electro-dynamics

of

Moving

Bodies,

p. 17.

Flitting

+

1

r

=c

and

.,

/r

=c

A, \ve

c- -(-r-

(f-.r) (r-A)/c-

get

-(.r-fA)^-f

=, 2<-

,

I

^-(.r + A) + A) +J -A/c

-(.<

Thus

f<r,

Thus

the

a velocity

Let

velocity

We

velocity.

A

W

so lorig as

sh.il!

.rA |

of

i:n\v

>(). |

light

is

the absolute

maximum

see the consequences of admitting

>r.

and

B

be

velocity of a -Signal

"

separated in the

by distance

system S be

W>r.

/,

and

let

Let the

178

PRINCIPLE OF 11ELATIVITY

system

(observing)

S have

-H with 1

velocity

respect to

the system S.

Then

1T

Kivenbv Thus

"

Now

B

from A to

r is

if

less

W

W=? +

is

and

/A

than

then

e,

=c

Now

we can always choose than if

is,

is

c-,

+

fji

n)(c

A)

W

;.

given

0)

:.:

c

W>v.

+ X)c

/zA.

v in such a

way

is

(/*

>c-

if (/*

that

+ A)c

.which can always be

\>

i3

being greater than

v, i.e.,

= c- +

Wr

since

,

A.

Wr = (c

{-

S

;_;

greater than v

in

...

Then

that

measured

as

zggD -,

(by hypothesis)

greater

respect to system S

with

signal

!**., "

time

^ nw-

Let

of

velocity

Wv

yuA is

is

>0.

satisfied

by

a suitable choice of A.

Thus

way \V

for

as to /

i>;

W><?

make

we can always choose A

W?->e

always

2 ,

i.e., I

Wr/c

But

negative.

Hence with

positive.

such a

in

2

W>c,

we can

the time from A lo B in equation (1) always make f the signal starting from A will reach That is, negative." ,

"

B

(as observed in

system

S

)

in less

than no time. Thus the

perceived before the cause the future will precede the past.

effect will be i.e.,

Hence we conclude that

W>c

is

commences

Which

is

to act,

absurd.

an impossibility,

there

can be no velocity greater than that of light. It

than

is

conceptually possible to

that

of

light,

but such

imagine velocities greater cannot occur in

velocities

uot produce greater than c, will effect of any physical type can never Causal any travel with a velocity greater than that of light. reality.

Velocities

effect.

[P. C.

M.]

NOTES

Notes 13 and

We w I

<0

i

14.

have denoted

W3 w

2

179

the

It- is

I

four-vector

then

the matrix

by

t

at once seen that

&>

denotes

the reciprocal matrix

!

>4

It

is

now

The vector-product

[<..]

may

evident that while

w

the four-vector w and

of

-t

be represented by the combination

[w] It

now easy

is

Supposing

to

= w

Sta

the

verify

= [ A~ = A-

Now

a

w,sA

A

1

[(^-,va>JA

a>,

<v,

/

!

./

=A

M

represents

/A. the

we have

"

*w

=A

remembering that generally

Where

p,

p* are scalar

mutually perpendicular unit

seeming that

Note

formula

for the sake of simplicity that

vector-product of two four-vectors

in

= A~

w

=w\,

<>

15.

<*

(|iiantities,

planes,

<^,

there

is

are

two

no difficulty

/ = A-VA. The rector product

(trf).

(L\

3(>).

four- vector and This represents the vector product of a six-vector. Now as combinations of this type are of ;

180

i;l\(

l

ll

OF KEI.ATIVITY

I.K

frequent occurrence in this paper,

better to form

be

will

it

The following an idea of their geometrical meaning. is taken from the above mentioned paper of Sommerfeld. "

We

can also form a vectorial combination of a

four-

vector

and a six-vector, giving us a vector of the third

type.

If the six-vector be of a special

of

then

plane,

type,

parallelepiped formed of this four-vector and

ment

of

this

of

piece

a piece

i.e.,

of the third type denotes the

vector

this

plane.

In

the

comple

general case, the

the

product will be the geometric sum of two parallelepipeds, but it can always bo represented by a four-vector of the

For two pieces of 3-space volumes can always

1st type.

be

together by the vectorial addition of their com So by the addition of two 3-space volumes,

added

ponents.

we do not (loc, cit.

in

the

be

second type first

tvpe

vector

(/.

.,

is

B = (B,,

type.

From

first

.

,

denote

B., ,)

,

=(B,,, ,

is

veetor-

a vector of the

scheme of

this

.B,,,

B,

s

zero,

of

the

first

special vectors of

n

B,,, B y .V-(B l4 1}

of

-

= -B tl

.

B,,=0).

., B..)J B,

is

perpendicular

to the

vector-product of to the scalar product of A and U,, equivalent

The /-component

the

a vector of the second

form three

this lust, let us

B, = (B,.,

,

unnal

Hie same as

a vector of the

A.) denote a vector

A,,,

B -.,,

B,=(B rr

Since H

and

we obtain I

is

by

taken from the three-dimensional ease.

kind, namely

B..

first,

Tl;c

vector).

A=(A,.

type,

where

calculus,

a polar vector),

(axial

by a four-vector

represented

state of affairs here

of a vector of the

multiplication

Let

The

75!)).

p.

ordinary

mu Implication

the

a vector of a more general type, but

obtain

which can always

one

the

^A, B,,+A

B,,

+A.

B,,.

A

/-axis.

and

i.e.,

B

is

MHKS

We

that

see easily

for the vector-product,

= A, Correspondingly the

case

fory

B,,-A.. let us

define

The /-component (P/, )

We of

is

.

in

the

four-dimciibional

=

-r,

//,

:,

P and

or

is

/)

= p,/v + ?,/, + p,,/, + p./;, ,

Eac-h one of these

and

.

four- vector

(j ,

obtained from

can also find

vectors of

the six-

given by

,

,

components is obtained as the scalar f which is perpendicular to

product of P, and the vector j-axis,

=

B..,,

product (P/) of any

vector./".

with the usual rule

coincides

this c. y.,

181

,

/

bv the ruley ,

= [(f}1 J ,

,

v

>

out here the geometrical significance

the third type,

when

,/

=<,

/.<?.,./

represents

only one plane.

We

by the parallelogram defined by the two U, V, and let us pass over to the conjugate which is formed by the perpendicular four-vectors

replace

</>

four-vectors

plane

</>*,

U*, V.* 1

Tiie

three-rowed

components of

(P</>)

under-determiuants

are then eijual to

D D

1),

v

;

Di

of

matrix

Leavin

P,

P.

P.

U,*

U,*

U,*

\

V

aside the

tirst

which coincides with (P

* v

V

P/

* :

I

\ ,*

column we obtain

M

according to our

di-linition.

the the

182

Examples of ibe

this

* = /t-F,

page 36,

vectors

be found ou

will

and ^ = * I) =

electrical-rest-force,

The

rest-ray

same type (page

the

to

of

type

the

magnetic- rest-force.

belong

OF RELATIVITY

llJNCll LE

I

i>o/,*

also

?o>

[<I>i/T]

It is easy to

89).

show

that

When (w n w 2 o),)=o, dimensional vector

=/.

,

^

we have

=

(Q)

l

reduces to the

F 14 =r,

Since in this case, *!=oj 4

=

il

(the electric force)

}J\^r=m

1

i<

thi-ee-

J,

(the magnetic force)

e,

analogous

,

to

the

Poynting-vector.

m,

[M. N. S.]

Note

16.

The the

27*e eleclric-rcd force.

four- rector

^>

= o-F

clectric-rest-force

which

(Page 37.) is

called

p ,=:

\Vu have

o* 4

X.J>. J>.

to

we have, when e=l,

Nn\v since

(

^

</

V

l

(

:

),

2

=p[d.+ j

\\\c have liavc

/,, dy,

Minkowaki

Rub-Kraft) Ponderomotive

is

(elektrische

a closely connected to Lorentz the force acting on a moving charge.

of charge,

b\ r

j)ul j)ut

i? i>

/ .<".,

the

or

density

for free space

V2

(* .-*)]

c component- of e equivalent components of in equivalent to

the

and the

//.= ],

If

very

force,

18i

NOTES

//

// ,

s

with

accordance

in

//.),

a

Lorent/

the

notation

used

in

on

the

of Electrons,

Theory

\Ye have therefore

i. /., /j

(

(<,,

,

electron.

</>,

fourth

component the

/-times

represents

force

.=i p

<

4

-%/

.-

_

by /j done by

multiplied

work

which

/>,,

>

Pn</

acting

of Electrons, page 14.

when

<,

at

rate

moving electron, for i+ r y Po^2 + r Po^s-

r

the

represents

)

Compare Lorent/, Theory

The the

</>.,

,<!* [>

is

+ ?v7 +

()

r.c?.]

=

times the power pos represents the fourth

l

sessed

by the electron therefore component, or the time component, of the force-fourThis component was first introduced by Poineare vector. in

1900.

The

^ = /wK*

four-vector

the force acting on a

Note

has a similar

moving magnetic Lor"

Operator

mechanics a

four-dimensional

o

/

sional

I

Q

/

fC-,

p\

g

~,

>+

^

plays in

j

similar

o

+j o

which

, (

role

cs

11).

1:2, p.

( \

-5,

^-

operation

the operator

to

"

17.

\

The

relation

pole.

to

that

of

\

"aT=

V)m

three-dimen

geometry has been called by ^TinUowski

Lorent /-

or shortly in honour of H. A. Lorent/, lor Operation Later writers the discoverer of the theorem of relativity. have sometimes used the symbol Q to denote this

operation.

Physik,

p.

In

the above-mentioned

(ill),

Bd.

paper (Annalen der

Sommerfeld has introduced the

33)

terms, Div (divergence), as

Hot ( Rotation), (I rad (gradient) four-dimensional extensions of the corresponding three-

dimensional lor.

operations

The ph\

sical

in

si^ni

place

ii-iiu-e

of of

the

general

svmho!

thes- operations

will

184-

I

become

we

also

where 8

S,

()!

UKLATIVITY

the

study

met hod of

s

method

geometrical

Minkowski begins here with the

Sommerfeld. lor

I.K

when along with Minkowski

clear

treatment

lUNCIl

is

a six- vector

(^pace-time

vector

of

case of of

the

2nd kind). being a complicated case, we take

This catffe

of lor

where

A-

is

the simpler

*,

= =

a four- vector

and

.s

-fj,

j

*,

*

A-O

4

|

*,

i

The following geometrical method

is

from Som-

taken

m erf eld. volume

space-time

of

A2

Let

Scalar Divergence sional

in

any shape

of the

neighbourhood the

denote

//S

Q,

point

denote a small four-dimen the

three-dimensional

be the outer normal to ifS. bounding surface of A2, Let S be anv four-vector, P its normal component. Then /

N

Div S

Now

if

for

A2

we choose the four-dimensional paral ^.r n (?.r 4 } we have then

lelepiped with sides

O

is

(//>-,,

O

i

,

//.>,

2

C/

3

}

O

i

f

denotes a space-time vector of the second kind, lor equivalent to a space-time vector of the first kind. The

If /

= Lim

We have

geometrical significance can be thus brought out. seen fiat the operator

a four-vector. six-vector

is

lor

beirives in every

The vector-product again

a

four-vector.

respect

of a four-vector

Therefore

it

is

like

and a easy

XOTKS see

tn

thr l.i

that

lor

S

will

of

component

Q

(./,, .r,, .r.,,

us

find

any direction

in

*.

its

to this

/

.r

4 )

and

is

perpendicular to

AS

,v,

a

Q, tfn- is an element two-dimensional surface. Let the perpendicular surface lying in the space be denoted by //, and

very small part of

let

Let

four-vector.

fon r- vector

tliis

S denote the three-space whieh passes through the

t

point

of

a

lie

185

denote

.

which

it

the

in the region of

component of

f

the

in

evidently conjugate to the plane

is

v-component of tho

vector

operator lor multiplies

/

plane of

of

divergence

(*//)

Then

tlv.

the

because the

/

vectorially)

= Divf.=Lim Ih^?. A *=0 AS Where

the integration in

is

</<r

to

be

extended

over

the whole surface.

now

If

x

is

selected

a three-dimensional i//,

as

the .r-direction,

parallelopiped with

A*

then

is

the sides

t1i/ }

<h,

we have

then

and generallv

=

+

+-

H.-nce the four-components or Div. pag.-

/

is

+ of the

a four-vector with the

four- vector

lor

components given

S on

1-2.

According to the formulae of space geometry, D, the (//->0 space, formed

denote-; a parallelopiped laid in

out of the vectors (P, P. ?,), ( IT

*

r* U*)

(v*

vt

V*,).

18H

I

lilXCII LK

01

UKLATIVITY

D, is therefore the projection on the y-:-l space of 1he peralielopiped formed out of these three four- vectors (P, U*, V*), and could as well be denoted by Dyxl.

We

see directly that the four-vector of the kind represent

ed by (D,,

D

y

,

D., D,)

piped formed by (P

U*

is

perpendicular to

the

parallele

V*).

Generally we have

(P/)=PD + P*D*. The vector of the third type represented by (P/) .. given by the geometrical sum of the two four-vectors of the first type PD and P*D*. is

[M. N.

S.]

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Einstein, Albert The principle of relat