242

DOC.

41

HAMILTON'S PRINCIPLE

a

(dR) dR

dm

dx

druva

duv

dguv

a

d

m

dm

dx

dq(p)

dq(p)

=

0.

(7)

(8)

R

is

here

in

the

same

relation

to

R

as

b

is

to b.

It

must

be noted that

equations (8)

or (5) respectively

would

have

to

be

replaced

by

others

if

we

would

assume

that

m

or b

resp.

would

depend upon higher

than the

first

derivatives

of

q(p)

Similarly,

one

could

imagine

the

q(p)

not

as

mutually

independent

but rather

as

connected

to

each other

by further,

conditional

equations.

All this is irrelevant for the

following development,

since it is

solely

based

upon

[7] {4}

equation

(7),

which is obtained

by varying our integral

after the

guv.

§3.

Properties

of

the Field

Equations

of Gravitation

Based

on

the

Theory

of Invariants

We

now

introduce the

assumption

that

ds2

=

guv dxudxv

(9)

is

an

invariant. This fixes the transformational character of the

guv.

We make

no

presuppositions

about the transformational character

of

the

q(p)

which describe

matter.

However,

the functions H

=

b/-g

and

G

= R/-g

and

M

= m/-g

shall

be

invariants under

arbitrary

substitutions

of

the

space-time

coordinates.

From these

suppositions

follows the

general

covariance of

equations

(7)

and

(8),

which have been

derived from

(1).

It follows furthermore that

(up

to

a

constant

factor)

G

is

equal

to

the scalar of the

Riemann

tensor

of

curvature,

because

there

is

no

other invariant

with the

properties

demanded for

G.4

With

this,

R,

and

hence

the

left-hand side

of

the field

equation (7)

is

completely

determined.5

[p. 1114]

The

postulate

of

general relativity

entails certain

properties

of

the function

R*

4This is the

reason why

the

requirements

of

general relativity

led to

a quite

distinct

theory

of

gravitation.

5Execution

of

the

partial

integration yields

R

=

-gg

uv

ua

ub

ab

b

a a

B.