284 DOC. 25 FOUNDATIONS OF GENERAL THEORY

[8]

to

arbitrary

transformations. It is

owing

to

this fact that the construction of the

theory

can go on

without

arbitrariness

despite

the

occurrence

of the

guv.

6.

But the

new

theory

of

relativity

also has

to solve

a

problem

to

which

none

corresponds

in the

original theory

of

relativity.

That is

to

say,

it also has

to

yield

equations

that the

gravitational

field itself

satisfies,

i.e., equations

from which the

quantities

guv

are

to

be calculated when the

quantities referring

to

the material

processes

are

known. The

energetic

behavior of

a

system

is

characterized

by

the

$

energy

tensor

Iov/v-g

(mixed tensor).

Since

energy

and inertia

on

the

one

hand,

and

f-g

inertia and

gravitation on

the other

hand,

determine

one

another,

we

must

demand

that the

gravitational

field

be determined

by

the

quantities

$ov.

Thus,

we are

to

seek

differential

equations

that

are

to

be considered

a generalization

of

Poisson's

equation

and

which, therefore,

permit

the

calculation

of

the guv

from the $ov; these

equations

must

be

generally

covariant.

7.

We have

not

succeeded in

setting up

this relation between the

guv

and Xov

[9]

in

a generally

covariant

form;

and it is this circumstance that

my colleagues

believe

[10] they

can use

to set

up a

lethal

trap

for

our theory.

In what

follows,

I shall

explain

why they

are,

in

my opinion,

wrong.-

If

we are

given equations connecting any quantities

whatsoever2 that

are

valid

only

for

a special

choice of

the coordinate

system,

then

one

has

to

distinguish

between

two

cases:

1.

To these

equations

there

correspond generally

covariant

equations, i.e.,

equations

valid with

respect

to

arbitrary

reference

systems.

2.

There

are no

generally

covariant

equations

that

can

be

deduced

from the

equations given

for

the

specially

chosen reference

system.

In

case 2,

the

equations say

nothing

about the

things

described

by

the

quantities

in

question; they only

restrict the choice of the reference

system.

If the

equations say

anything

at

all about the

things represented by

the

quantities,

then

we are

dealing

with

case

1,

i.e.,

in that

case,

there

always

exist

generally

covariant

equations

between the

quantities.

Thus,

if,

without

knowing

the

generally

covariant

equations

of the

gravitational

field,

we

specialize

the reference

system

and

set

up

the field

equations

of

gravitation

for the

special

reference

system only,

then the sole

objection

that

can

be raised

against

the

theory

is that the

equations

we

have

set

up might, perhaps,

be void

of

any

physical

content.

But

no

one

is

likely

to

think

in earnest

that this

objection

is

justified

in

the

present

case.

2Of

course,

the

transformation

properties

of

the

quantities

themselves must be

considered

here

as being

given

for

arbitrary

transformations.