DOC.

28

NORDSTRÖM'S THEORY OF GRAVITATION

293

Doc.

28

Nordström's

Theory

of Gravitation from the Point of View

of the Absolute Differential Calculus

by

A.

Einstein and

A. D.

Fokker

[Annalen

der

Physik

44

(1914):

321-328]

[1]

In all

previous presentations

of

Nordström's

theory

of

gravitation1

Minkowski's

theory

of

covariants

was

the

only

invariant-theoretical tool

employed, i.e.,

all that

was

required

of

the

equations

of the

theory

was

that

they

be covariant with

respect

to

linear

orthogonal space-time

transformations. But when this condition

is

imposed a

priori

on

the

equations

it does

not

restrict the theoretical

possibilities

to

such

an

extent

that

one

could

arrive in

an

unforced

way

at

the fundamental

equations

of

the

theory,

without

recourse

to

special physical assumptions.

We will show in what

follows that it is

possible

to

arrive

at

a

perfectly

complete

and

satisfying representa-

tion

of

the

theory

if-as

had

already

been done in the Einstein-Grossmann

theory-one

uses

the

invariant-theoretical tool

given

to

us

in the absolute differential

calculus. Since

nature does

not

present

us

with reference

systems

to

which

we

could

refer

things,

we

at

first refer the

four-dimensional manifold

to

totally arbitrary

coordinates

(corresponding

to

Gaussian coordinates in the

theory

of

surfaces),

and

[3]

restrict the

choice

of

the

reference

system only

when the

problem being

treated itself

induces

us

to

do

so.

It

turns out

that

one

arrives

at

Nordström's

theory,

rather than

at

the Einstein-

Grossmann

theory,

if

one

makes the

single assumption

that it is

possible

to

choose

privileged

reference

systems

in

such

a

way

that the

principle

of

the

constancy

of

the

velocity

of

light

will be

preserved.

§1.

The Characteristics of

the

Gravitational Field.

The

Influence

of the

Gravitational Field

on

Physical

Processes

[4]

We assume2

that

a point moving

in

a

gravitational

field

obeys

a

law of motion

whose Hamiltonian form

is

1Cf. G.

Nordström,

Ann.

d.

Phys.

42

(1913): 533;

A.

Einstein,

Phys.

Zeitschr. 14

(1913):

1251.

2Cf.

A.

Einstein,

"Entwurf einer

verallgemeinerten

Relativitätstheorie

un

einer Theorie

der Gravitation"

["Outline

of

a

Generalized

Theory

of

Relativity

and

of

a Theory

of

Gravitation"].

Zeitschr.

f.

Math. und

Phys.

62

(1913):

6.

[2]

[5]