DOC.

41

HAMILTON'S PRINCIPLE 245

J-

($;

+

o

=

o.

(21)

dxv

Equation (21) expresses

the conservation

of

the momentum and the

energy.

We

[p.

116]

call

zvo

the

components

of

the

energy

of

matter,

tvo

the

components

of the

energy

of

the

gravitational

field.

From the field

equations

(7)

of

gravitation

follows

(after

multiplication by

gouv,

summation

over

u.

and

v,

and

on

account

of

(20))

+

1

rMv

aa»

_

0,

dxv

280

agMv

or,

taking

(19)

and

(21)

into

account,

dzvo/dxv+1/2gouvZuv

=

0, (22)

[12]

{8}

where

Zuv

denotes the

quantities

gvoZou.

These

are

four

equations

that the

energy

components

of

matter have to

satisfy.

It is to be

emphasized

that the

(generally covariant)

conservation theorems

(21)

and

(22)

have been

derived-using

also the

postulate

of

general

covariance

(relativity)-from

the field

equations

(7)

of

gravitation

alone,

without

use

of

the

field

equations

(8)

for material

processes. [13]

Additional

notes

by

translator

In his footnote

1),

just

prior

to

equations (4)

and

(5),

Einstein introduces into tensor

calculus,

in

a

formal

manner,

the rule

of

abbreviated

writing

of

summations,

which

is

now generally

known

as

the Einstein summation convention. It

was

introduced in

Doc.

30,

p.

296.

{1}

The

"q"

with

2

subscripts

and

2

superscripts

has been corrected here to

"g";

the

2nd derivative

of

"q"

inside the

following parenthesis

has

also been corrected

to

"g,"

both with indices

as

indicated. Editorial notes

[6]

and

[7]

relate to

similar

typesetting

errors.

{2}

"quvor"

has

been corrected here to

"guvor."

{3}

"(4a)"

has

been corrected to

"(5)."

{4}

"q"

has been corrected here

to "guv".

{5}

"(13)"

and

"(14)"

have been corrected here to

"(11)"

and

"(12)."