40 DOC.

9

FORMAL FOUNDATION OF RELATIVITY

£(*wr)

i

«

I«;

i

=

i.

a

From this follows

\8r»

I

'

\SßV

I

=

I-

(11)

The

Invariant of

Volume.

For the immediate

neighborhood

of

a point

of

our

continuum

we can always put,

according

to

(2b),

fc1

=

£

8^

dxß

dxv

=

Y,dX° (12)

/xv a

provided we

allow

imaginary

values

for the

dXa.

For

the choice

of

the

system

of

the

dXa

there

are

still

infinitely

many possibilities; however,

all

these

systems

are

connected

[p.

1041]

by

linear

orthogonal

substitutions. From this follows that the

integral

dr*

=

fdX1dX2dX3dX

over a

volume element

is

an invariant,

i.e.,

completely independent

of the choice

of

the coordinates.

We

want to

find

a

second

expression

for this invariant. There

are

at

any

rate

relations

of the form

dXa

=

£

(13)

from which it follows that

dTo =

(14)

if

dr and

dr*0

resp.

denote the

integrals

fdx1...dx4

and

JdX1dX2dX3dX4

resp.

extended

over

the

same elementary

domain.

Furthermore,

according

to

(12)

and

(13),

=

Ev«v

(15)

a

and

consequently, according

to the

multiplication

theorem

of

determinants,

{6}

M

=

lEKAnJI

=

Kvl2-

(16)

a

Considering (14), one now

gets

fgdr

=

dr;,

(17)

where

we put

\guv\

=

g

for short. With this

we

have found the invariant

we were

looking

for.