DOC.

53

DEVELOPMENT OF RELATIVITY 409

784

NATURE

[February

17,

1921

tively to systems of

co-ordinates

possessing arbi-

trary motion with

respect to

an

inertial

system.

With the aid

of

these

special

fields

we

should be

able

to

study the

law

which

is

satisfied

in general

by

gravitational

fields.

In this connection

we

shall

have

to

take account

of the fact that the ponder-

able masses

will

be the

determining

factor in pro-

ducing the

field, or,

according to

the fundamental

result of the

special theory

of

relativity,

the

energy

density-a

magnitude having

the transformational

character of

a

tensor.

On the other

hand,

considerations based

on

the

metrical results of the

special theory

of

relativity

led to

the result that Euclidean metrics

can no

longer

be

valid with

respect to

accelerated sys-

tems of co-ordinates.

Although

it retarded the

progress

of the

theory

several

years,

this enor-

mous difficulty was mitigated

by

our knowledge

that Euclidean metrics

holds for small domains.

As

a

consequence,

the

magnitude

ds,

which

was

physically

defined

in

the

special theory

of rela-

tivity hitherto,

retained its

significance

also in

the

general

theory

of

relativity.

But the co-ordinates

themselves lost their direct

significance,

and

degenerated simply

into numbers with

no

physical

meaning,

the sole

purpose

of which

was

the num-

bering of the

space-time points.

Thus

in

the

general theory of relativity

the co-ordinates per-

form the

same

function

as

the Gaussian co-ordi-

nates

in

the

theory

of

surfaces. A

necessary con-

sequence of the

preceding

is

that

in

such

general

co-ordinates the measurable

magnitude

ds must

be

capable

of

representation in

the form

dx2=3Rvrguvdx1dx2

where the

symbols

guv

are

functions

of

the space-

time co-ordinates.

From the above it also

follows

that the

nature

of the

space-time

variation

of

the

factors

guv

determines,

on one

hand the space-

time

metrics,

and

on

the other the gravita-

tional

field

which

governs

the mechanical

behaviour of material

points.

The

law

of the

gravitational

field

is determined

mainly

by

the

following

conditions:

First,

it shall

be

valid for

an arbitrary

choice of the

system

of

co-ordinates;

secondly,

it shall

be

determined

by

the

energy tensor

of

matter;

and

thirdly, it

shall

contain

no

higher

differential coefficients of

the

factors

guv

than the

second,

and

must

be

linear in

these. In this

way

a

law

was

obtained

which,

although fundamentally

different from Newton’s

law, corresponded

so

exactly to

the latter

in

the

deductions derivable from

it

that

only

very few

criteria

were

to

be found

on

which the

theory

could

be

decisively

tested

by

experiment.

The

following

are some

of the

important ques-

tions which

are

awaiting

solution at the

present

time. Are electrical and

gravitational

fields

really

so

different

in

character that there is

no

formal

unit

to

which

they can

be

reduced? Do

gravita-

tional fields

play a

part in

the constitution of

matter, and

is the continuum within

the

atomic

nucleus to

be

regarded as

appreciably non-

Euclidean?

A final question

has reference

to

the

cosmological

problem.

Is inertia to

be traced

to

mutual action with distant masses? And

con-

nected with the

latter

: Is the

spatial

extent

of the

universe finite? It is here that

my opinion

differs

from that of

Eddington.

With Mach,

I

feel that

an

affirmative

answer

is

imperative,

but for

the

time

being nothing can

be

proved.

Not until

a

dynamical investigation

of the

large systems

of

fixed stars

has been

performed

from the

point

of

view

of

the limits of

validity

of the Newtonian

law of

gravitation

for immense

regions

of

space

will it perhaps

be

possible

to

obtain

eventually an

exact

basis

for the solution of this

fascinating

question.

Relativity:

The Growth of

an

Idea.

By

E.

Cunningham.

SACCHERI,

in

his

"Logica Demonstrativa,"

published

in

1697,

ten

years

after

Newton's

"Principia Mathematica,"

lays

down

a

distinction

between

real

and

nominal definitions

which should

be

kept

in

mind

if

we are to

do

justice to

Newton.

Euclid defines

a

square

as

a

four-sided

figure

the

sides

of

which

are

all

equal,

and the

angles

of

which

are

all right-angles.

That

is

what he

means

by

the

name

"square." It

is

a

nominal definition.

It remains

to

be shown that such

a figure

exists.

This is done

in

Book

1., Prop.

46.

The definition

then becomes real. Euclid

is not

guilty

of

the

error

of

presupposing

the existence of the

figure.

Newton

prefixes to

his

principles

of natural

philosophy

certain definitions of

absolute

true,

and mathematical

space

and time. The former

remains

fixed

and immovable; the latter

flows

uniformly on,

without

regard to

material

bodies.

He strives here

against

the

imperfections

of lan-

NO. 2677, VOL.

106]

guage to give

words

to

the

thought in

the back

of

his

mind. The

philosopher

attacks

him

on

these

definitions; he

has

no

right

to

presuppose

that

these words

correspond

to any reality.

What

then?

Suppose

these

offending

definitions

re-

moved, or

recognised as purely

nominal. Then

the definitions of

velocity,

acceleration,

mass,

and

force

are

nominal, too,

and the whole of Newton’s

structure

of

dynamics

is

a paper

scheme of words

and relations

which

may or may

not

correspond

to

the world of

sense.

But

that

is exactly

what

it is.

That is

what

all

scientific

theory

is,

until

experiment

demonstrates

that the

correspondence

exists. The

justification

of

Newton’s

theory comes, not in

the

discovery

of

a

time that flows

uniformly on,

but

in

the fact

that the observed

phenomena

of the

tides,

of

planetary

motion,

and of mechanics

in

general

do

fit on to

his scheme.

But the fit does

not

consist