408 DOC.

53

DEVELOPMENT OF

RELATIVITY

February

17,

1921]

NATURE

783

Now in order that the

special principle

of rela-

tivity may hold,

it is

necessary

that all the equa-

tions of

physics

do

not

alter their form

in

the

transition

from

one

inertial

system

to another,

when

we

make

use

of the

Lorentz

transformation

for the

calculation of this

change.

In the lan-

guage of

mathematics,

all

systems

of

equations

that

express physical

laws

must

be co-variant with

respect to

the Lorentz

transformation. Thus,

from the

point

of view of

method,

the

special prin-

ciple of

relativity

is

comparable

to

Carnot’s prin-

ciple of

the

impossibility

of

perpetual

motion

of

the second kind,

for,

like

the

latter,

it

supplies us

with

a general

condition which

all

natural laws

must

satisfy.

Later,

H. Minkowski found

a particularly

elegant and suggestive

expression

for this

condition of co-variance,

one

which reveals

a

formal

relationship

between Euclidean

geometry

of three dimensions

and the

space-time

continuum

of

physics.

Euclidean

Geometry of

Three

Dimensions.

Corresponding to two

neighbouring

points

in

space,

there

exists

a

numerical

measure (dis-

tance ds)

which

conforms

to

the equation

ds*=dx12+dx22+dx32.

It

is independent

of the

system

of

co-ordinates

chosen,

and

can

be

measured

with

the

unit

measuring-rod.

The

permissible trans-

formations are

of

such

a

character that

the expression

for ds2 is

invariant,

i.e.

the linear

orthogonal

transformations

are per-

missible.

With

respect to

these

transformations,

the

laws

of

Euclidean

geometry

are

invariant.

Special Theory

of

Relativity.

Corresponding to

two

neighbouring points

in

space-time (point events),

there exists

a

numerical

measure

(distance

ds)

which conforms

to

the

equation

ds2

=

dx12+dx22+dx32+dx42

It

is

independent

of

the

inertial

system

chosen,

and

can

be

measured

with

the unit measuring-

rod and

a

standard

clock.

x1, x2, x3

are

here

rectangular co-ordinates,

whilst

X4=V-1ct is the

time

multiplied by

the

imaginary unit

and

by

he

velocity

of

light.

The

permissible

trans-

formations are

of such

a

character that the expres-

sion for ds2

is invariant,

i.e.

those linear ortho-

gonal

substitutions

are

permissible

which

main-

tain the

semblance of

reality of

x1,

x2, x3, x4.

These substitutions

are

the

Lorentz transforma-

tions.

With

respect to these

transformations,

the

laws

of

physics

are

invariant.

From this it follows that,

in

respect

of

its rôle

in

the

equations

of physics, though not

with

regard

to

its

physical significance,

time is

equivalent

to

the

space

co-ordinates (apart

from the relations

of

reality).

From this

point

of

view,

physics

is,

as

it

were, a

Euclidean

geometry

of four dimen-

NO.

2677,

VOL.

106]

sions, or,

more correctly, a

statics in

a four-

dimensional Euclidean continuum.

The

development

of the

special theory

of rela-

tivity consists of

two

main

steps, namely,

the

adaptation

of the

space-time

"metrics"

to

Maxwell's

electro-dynamics,

and

an

adaptation

of

the

rest

of

physics to

that altered

space-time

"metrics." The first of these

processes yields

the

relativity

of

simultaneity,

the

influence of

motion

on measuring-rods

and

clocks,

a modifica-

tion of

kinematics,

and

in particular

a new

theorem

of addition of velocities. The second

process

supplies us

with

a

modification of Newton’s

law

of motion for

large velocities,

together with

information of fundamental

importance on

the

nature

of inertial

mass.

It

was

found that

inertia

is not

a

fundamental

property

of

matter, nor, indeed,

an

irreducible

magnitude,

but

a property

of

energy.

If

an

amount

of

energy

E be

given

to

a body,

the

inertial

mass

of the

body

increases

by

an

amount

E/c2,

where

c

is

the

velocity

of

light in vacuo.

On the other

hand,

a

body

of

mass m

is

to

be

regarded as a

store

of

energy

of

magnitude mc2.

Furthermore,

it

was soon

found

impossible

to

link

up

the science

of

gravitation

with the

special

theory

of

relativity in

a

natural

manner.

In

this

connection

I was

struck

by

the fact that the force

of gravitation

possesses a

fundamental

property,

which

distinguishes

it from

electro-magnetic

forces.

All

bodies fall

in

a

gravitational

field

with

the

same

acceleration,

or-what

is

only

another

formulation of the

same

fact-the

gravitational

and inertial

masses

of

a

body are numerically

equal

to

each other. This numerical

equality

suggests identity

in

character. Can

gravitation

and inertia be identical? This

question

leads

directly

to

the General

Theory

of

Relativity.

Is it

not possible

for

me

to regard

the earth

as

free

from

rotation, if I

conceive of the

centrifugal

force,

which

acts

on

all bodies

rest relatively

to

the

earth,

as being a

"real"atfield

of gravita-

tion, or

part of such

a

field? If this idea can

be

carried

out,

then

we

shall have

proved

in

very

truth

the

identity

of

gravitation

and inertia.

For

the

same

property

which is

regarded as

inertia

from the

point

of view

of

a system

not

taking

part in

the rotation

can

be

interpreted as gravita-

tion when considered with

respect to

a

system

that

shares the rotation.

According

to Newton,

this

interpretation

is

impossible,

because

by

Newton's

law

the

centrifugal

field cannot

be

regarded as

being

produced by

matter,

and because

in

Newton’s

theory

there is

no place

for

a

"real"

field

of the "Koriolis-field"

type.

But

perhaps

Newton’s law

of

field

could be

replaced

by

another

that fits

in

with

the

field

which holds with

respect

to a

"rotating"

system

of co-ordinates?

My

conviction

of the

identity

of

inertial and gravita-

tional mass

aroused

within

me

the

feeling

of abso-

lute confidence

in

the

correctness

of this interpre-

tation. In

this

connection I

gained encourage-

ment from

the

following

idea.

We

are

familiar

with

the

"apparent"

fields which

are

valid rela–