DOC. 26 COMMENTS ON DOC.

13

289

Doc. 26

"Comments"

on

"Outline of

a

Generalized

Theory

of

Relativity

and of

a

Theory

of Gravitation"

by

A. Einstein

[Zeitschrift

für

Mathematik und

Physik

62

(1914):

260-261]

Comments

Regarding §5

and

§6.

When

we

were writing

this

paper

we

felt it

a

deficiency

of the

theory

that

we

did

not succeed

in

constructing equations

for the

gravitational

field

that

are

generally covariant, i.e.,

covariant with

respect

to

arbitrary

substitutions.

Subsequently

I found

out, however,

that

equations

that determine the

yuv univocally

from the

0uv

and

are generally

covariant

cannot

exist

at

all;

the

proof

of

this

is

obtained

as

follows.

Suppose

that

our

four-dimensional manifold contains

a

part

L

in which

no

"material

process"

takes

place,

in

which, therefore,

the

0uv

vanish.

According

to

our

assumption,

the

yuv

are

completely

determined

everywhere,

and thus also in the

interior of

L,

by

the

0uv

given

outside of

L.

Let

us

now imagine

that

new

coordinates

x'v

of

the

following

kind

are

introduced instead

of

the

original

coordinates

xv.

Outside

of

L,

let

us

have

everywhere

xv

=

xv';

but inside L let

us

have

xv

=

xv'

at

least for

a

part

of

L

and

at

least for

one

index

v.

It is

clear that

by

means

of

such

a

substitution

one can

obtain

y'uv

#

yuv

for

at

least

a part

of

L.

On the other

hand,

one

will

have

O'uv

=

0uv

everywhere, namely,

outside

L

because for this

region

x'v

=

xv,

but inside

L because for this

region

Ouv =

0

= 0'uv.

From this it follows

that in the

case

under

consideration,

where

all

substitution

are

admitted

as

justified,

more

than

one

system

of the

yuv

is associated with the

same

system

of

the

0uv.

Thus,

if

one

sticks

to

the

demand-as

has been done in

our

paper-that the

yuv

should be

completely

determined

by

the

0uv,

then

one

is forced to restrict the choice

of the reference

system.

In

our paper we

realized this restriction

by postulating

the

validity

of

the conservation

laws, i.e.,

the

validity

of

four

equations

of

the form

of

equations (19),

for the material

process

and the

gravitational

field taken

together.

This

is

in fact the

postulate

from which

we

derived

equations

(18)

for the

gravitational

field in

§5.

Equations (19)

are

covariant

only

with

respect

to

linear

transformations,

so

that

in the

theory developed

in

our

paper only

linear

transformations

are

to

be considered

justified. Hence,

we can

designate

the

axes

of such

systems

as

"straight

lines,"

and

the coordinate surfaces

as "planes."

It is

very noteworthy

that the conservation laws

[1]

[2]

[3]

[4]