DOC. 9 FORMAL FOUNDATION OF RELATIVITY 69

a2

va

aav

dxa

dxa

8

(65a)

\

F

-/*£

=

a

8 va

dH^g

_

a

8

vo-

a//^

Ax (65b) {17}

OCTV/l

a^v

a*_ a*.

a&r

\

F

can

be transformed

into

a

surface

integral.

It vanishes when

AxU

and

5Ax

dAxu

dxo

vanish at the

boundary.

Adapted

coordinate

systems.

We consider

again

our

continuum and its domain

[37]

£,

which

we assume

finite in all of its

coordinates,

and referred to the coordinate

system

K.

Starting

from K

we

imagine successively

introduced coordinate

systems

K', K"

etc.,

all

infinitely

close to each

other such that the

Axu

and

the

dAxu/dAxa

vanish

at the boundaries.

We call all these

systems

"coordinate

systems

with

coinciding

boundary

coordinates." For

every

infinitesimal coordinate transformation between

neighboring

coordinate

systems

of

the

totality

K, K',

K"...

we

have

F=

0,

so

that instead

of

(65) we

have the

equation

JAJ=EfdrA*A- (66)

[38]

L

M

Among

all

systems

with

coinciding boundary

coordinates

will be

some

for which

J

attains

an

extremum

as compared

with the J-values

of

neighboring

systems

with

likewise

coinciding boundary

values.

We call these coordinate

systems

"coordinate

systems adapted

to the

gravitational

field."

The

equations

Bu =

0

(67)

hold

for these

adapted systems according

to

(66)

because the

Axu

can

be chosen

freely

inside

of

E.

Inversely, (67)

is

a

sufficient condition

that the coordinate

system

is

adapted

to

[p. 1071]

the

gravitational

field.

We avoid the

difficulty

mentioned in

§13

by

henceforth

writing only

differential

equations

of

the

gravitational

field that claim

validity merely

for

adapted

coordinate

systems.

Under the limitation to

adapted

coordinate

systems,

it is indeed

no longer

allowed to extend

a

coordinate

system

that is

given

for the exterior

of

E

into the

interior

of

E

in

an arbitrary manner.