DOC. 2 COVARIANCE PROPERTIES

7

accelerated three-dimensional coordinate system-can be viewed

as a

"real"

gravitational

field;

in other

words,

if acceleration-transformations

(i.e.,

nonlinear

transformations)

become

permissible

transformations

in

the

theory.

At first

glance

it

appears

desirable

to

look for

gravitational equations

that

are

covariant toward

arbitrary

transformations.

However,

in

§2

of the

present

paper2 we

will

show

by a simple

consideration that the

quantities

guv

which characterize the

gravitational

field cannot

completely

be

determined

by generally-covariant equations.

In

the

following

we

shall demonstrate

that

the

gravitational equations

established

by us are

generally

covariant

just

to

the

degree imaginable

under the condition that

the

fundamental

tensor

guv

must be

completely

determined.

It

follows

in particular

that the

gravitational

equations

are

covariant with

respect

to

quite

varied

accelera-

tion

transformations (i.e.,

nonlinear

transformations).

[5]

§1.

The Basic

Equations of the

Theory

We characterized the

energetic response

of

a

physical process by means

of

a

covariant tensor

Tuv

or

its

reciprocal

contravariant tensor

©uv,

respectively.

This

tensor satisfies

equations (10)

of the

"Outline," viz.,

[6]

or respectively

.9

•

___

4uv

`Lv,

_

I

c~z

4uv/

S

a

dx~

go

_

I-

/4

2

Ftp

09pv

die0

S,sv,

and

they represent

the

energy-momentum equations

of the material

process.

All [p.

217]

equations

of the

theory

take

a particularly comprehensive

form

if

one

introduces the

quantities

(1)

Lav=-gyauTuv=-ggauQuv,4"

which differ from the

components

of

a

mixed

tensor3

only by

a

factor of \pg.

Conceptually

we

call them the

complex

of

energy-density

of

the

physical process.

Our

equations

above

can now

be rewritten

as

2Compare

also the remark in the

appendix

of

the

reprint

in Zeitschr.

f.

Math.

u. Phys., [4]

vol. 62.

3Compare

§1

of

part

II

of

the "Outline."

[7]