DOC.

41

HAMILTON'S PRINCIPLE 241

depend only upon

the

guv.

The variational

principle (1) can

then

be

replaced by one

more

convenient for

us.

With suitable

partial

integration

one

gets

Jbdr

=

Jbdr

+

F,

(2)

where F is

an

integral

extended

over

the boundaries

of

the domain under

consider-

ation,

while the

quantity

b*

depends only upon

the

guv,

gouv,

q(p), q(p)a

but

no

longer upon

guvor.

For the variation

of

interest

to

us one gets

from

(2)

s{fbdr}

= d{fbd*r},

(3)

whereupon

we

can replace

the variational

principle

(1)

with the

more

convenient

one

d{fbdr}

=

0.

(1a)

By executing

the variation after the

guv

and the

q(p)

one

obtains for the field

equations

of

gravitation

and matter the

equations3

a

db

db*

dx

dgauv dguv

db*

db*

dxa

dq(p)

=

0

(4)

=

0.

(5)

§2.

Separate

Existence

of the Gravitational Field

The

energy components

cannot be

split

into two

separate parts

such that

one

belongs

to the

gravitational

field and the other to

matter,

unless

one

makes

special assump-

tions in which

manner

b

should

depend upon

the

guv,

goruv, q(p),

q(p)a.

In

order

to

bring

about this

property

of

the

theory

we

assume

b

=

R

+

M,

where

R

depends only upon

guv,

gouv, gorv

and

M

only upon guv,

q(p),

q(p)a.

Equations (4), (5)

then take the form

(6)

[p. 1113]

{3}

3As

an

abbreviation,

the summation

signs are

omitted in the formulas. A summation

has to

be

carried out

over

the indices that

occur

twice in

a

term.

For

example,

in

(4)

d

dx

db

denotes

the

term

Z

d

a

dxa

db*

dguva