240 DOC.

41

HAMILTON'S PRINCIPLE

Doc.

41

Hamilton's

Principle

and the General

Theory

of

Relativity

by

A. Einstein

[p.

1111]

[1]

H. A.

Lorentz

and

D.

Hilbert

have

recently

succeeded1

in

presenting

the

theory

of

general relativity

in

a particularly comprehensive

form

by deriving

its

equations

from

a

single

variational

principle.

The

same

shall be done in this

paper. My

aim

here is

to

present

the fundamental connections

as transparently

and

comprehensively

[3]

as

the

principle

of

general relativity

allows. In contrast

to

Hilbert's

presentation,

I

shall make

as

few

assumptions

about the constitution

of

matter

as

possible.

On the

other

hand,

and in contrast to

my own very

recent treatment

of

the

subject

matter,

the

[4]

choice

of

a

system

of coordinates shall remain

completely

free.

§1.

The Variational

Principle

and the Field

Equations

of

Gravitation

and Matter

The

gravitational

field shall be described

as

usual

by

the

tensor2

of

the

guv

(or

guv

resp.);

matter

(inclusive

of

the

electromagnetic

field)

by

an

arbitrary

number

of

space-

time functions

q(p)

whose invariance-theoretical character

we ignore.

Furthermore,

let

b

be

a

function

of

the

{1} guv,

guvo

=

dg

uv

dx

and

gor

uv

=

the

q(p)

and

q(p)a

dq(p).

dxa

The variational

principle

S{/bdr} =

0

(1)

[p. 1112]

then

provides

as

many

differential

equations as

there

are

functions

guv

and

q(p),

[5]

[6]

{2}

which

are

to

be

determined,

provided we

agree

that the

guv

and

q(p)

are

to

be varied

independently

of

each other such that

at

the boundaries of

integration

the

dguv,

and

ddguv/dxo

vanish.all

We

shall

now

assume

b

to be

linear

in

the

guvor

such that

the

coefficients

of

the

guvor

1Four

papers by

H.

a. lorentz in

volumes 1915

and 1916

of

the

Publikationer

d.

Koninkl. Akad.

van

Wetensch.

te Amsterdam;

D.

Hilbert, Gött. Nachr.

1915,

Heft.

3.

2For

the time

being,

the tensorial character

of

the

guv

is not used.

[2]