DOC.

21

GENERAL RELATIVITY

103

serve as a

counterargument

because

the first

term

on

its

right-hand

side

can

be

brought

into the form

EvT{avT}TvT.

Therefore,

from

now on we

shall call the

quantities

Tmv

=

-{mv}

=

-Egav

)U,V

a

=

-^s

**

n

dgfja

dgv d8,TiV

dx" dx"

dx

(15)

the

components

of

the

gravitational

field.

Kv

vanishes when

Tva

denotes the

energy

tensor

of

all "material"

processes,

and the conservation theorem

(14)

takes the form

E

aTav/SXa

=

-E

TaaB

TBa.

(14a)

We note

that

the

equations

of motion

(23b)

l.c. of

a

material

point

in

a

gravitational

field take

the

form

d2x

,__._

dxfl dx

T

_

V

t-iT

lb2 =

(15)

{2}

2.

The considerations in paragraphs

10

and

11

of the quoted paper remain

unchanged, except that the structures which were there called

V-scalars

and

V-tensors

are now ordinary scalars and tensors, respectively.

§3.

The Field

Equations

of Gravitation

From what has been

said,

it

seems

appropriate

to

write the field

equations

of

gravitation

in

the

form

Ruv=

-kTmv

(16)

since

we already

know that these

equations

are

covariant under

any

transformation

of

a

determinant

equal

to

1. Indeed,

these

equations satisfy

all conditions

we can

demand. Written out in

more

detail,

and

according

to

(13a)

and

(15),

they are

£

If

*

E

(16a)

a

oxa

"ß.

We wish to show

now

that these field

equations can

be

brought

into the

Hamilto-

nian

form

[p.

784]