6

DOC. 2 COVARIANCE PROPERTIES

Doc. 2

Covariance

Properties

of the

Field

Equations

of the

Theory

of Gravitation Based

on

the

General

Theory

of

Relativity

by

Albert Einstein in Berlin and Marcel Grossmann in Zurich

[p. 215]

In

a paper1

published

in

1913

we

based

a

generalized theory

of

relativity upon

absolute differential

calculus in

a manner

such that it also embraces the

theory

of

[3]

gravitation.

Two

basically

different kinds of

systems

of

equations occur

in this

theory.

For

a

gravitational

field considered

as

given,

we

first established

systems

of

equations

for material

(e.g.,

mechanical,

electrical) processes.

These

equations are

covariant

under

arbitrary

substitutions

of

the

space-time

variables

("coordinates")

and

can

be

considered

as

generalizations

of the

corresponding equations

of

the

original theory

of

relativity.

Second,

we

established

a system

of

equations

that determines the

gravitational

field insofar

as

the

quantities

that determine the material

processes are

given;

and this

system can

be considered

a

generalization

of

the Poisson

equation

of

Newton's

theory

of

gravitation.

In the

original theory

of

relativity,

there is

no

corresponding system

of

equations

for this. In contrast

to

the

equations

mentioned

above,

we

could not demonstrate

general

covariance for those

"gravitational

equations."

The

reason

is that their derivation

was

based

(besides

the conservation

theorems) only upon

the covariance with

respect

to

linear

transformations,

and thus

left it

an

open question

as

to

whether

or

not

there exist other substitutions that would

transform

the

equations

into themselves.

There

are

two

reasons why

the resolution

of

this

question

is of

particular

importance

to

the

theory.

The

answer

to

this

question

gives,

first,

information

on

how

far the basic idea

of

relativity theory

can

be

developed;

and this is

of

great import

to

[p. 216]

the

philosophy

of

space

and time. And

second,

the

judgment

about the value of the

theory

from the

point

of view of

physics depends

to

a

high degree upon

the

answer

to

this

question,

as

is shown

by

the

following

consideration.

The entire

theory

evolved from the conviction that all

physical processes

in

a

gravitational

field

occur

just

in

the

same way as

they

would without

it,

if

an

appropriately

accelerated

(three-dimensional)

coordinate

system

would be introduced

("hypothesis

of

equivalence").

This

hypothesis,

which is based

upon

the

experimental

fact

of

the

equality

between

gravitational

and inertial

mass,

gains

additional

convincing

force if the

"apparent" gravitational

field-which

exists relative

to

the

[1]

[2]

1"Entwurf

einer

verallgemeinerten

Relativitätstheorie

und einer Theorie der Gravitation"

(Leipzig:

B. G. Teubner,

1913).

[In

the

following

it

is

abbreviated

as

"Outline." The

paper

is

printed

in this

journal, Zeitschrift

für

Mathematik

und

Physik,

vol.

62, pp.

225-261.]