DOC. 9 FORMAL FOUNDATION OF RELATIVITY 57

nw

=

(46)

[27]

2

"

3*a

i ;

l±

are

of crucial

importance,

and for this

reason we

want to

call them the

"components

of the

gravitational

field."

§10.

Equations

of Motion of

Continuously

Distributed Masses

Naturally

measured

quantities.

It has

already

been

emphasized

that it is

not

possible

in

a generalized theory

of

relativity

to

select

a

coordinate

system

such that

spatial

and

temporal

coordinate differences

are

in

a

direct connection with the

results obtained

from

measuring

rods and

clocks,

as

has been the

case

in the

original theory

of

relativity.

Such

preferred

choice of coordinates is

only possible

in the

infinitesimally

small

by setting

ds2

=

£

g^dxtidxv =

-dl\ -d\\ -d\\

-dl\.

(46)

{11}

/IV

The

d\

are (see

§2)

just

as

measurable

as

the coordinates in the

original theory

of

relativity,

but

they

are

not

complete

differentials. It is

possible

in the

infinitesimally

small to refer all

quantities upon

the coordinate

system

of the

d\.

When this is

done,

we

call them

"naturally

measured"

quantities.

And

we

call the coordinate

system

the

"normal

system."

According

to

(17a) we

have for

infinitesimally

small four-dimensional volumina

\pg

J

dx1dx2dx3dx4

=

J

d'iyd^2d^id^i. (47)

Let the volume under consideration consist

of

an

infinitesimally

short and

infinitesimally

thin four-dimensional

thread.

And let dv

be the

integral

J

dx1

dx2 dx3

extended

over

it.

We

now

choose the

dE,

such that the

dE4-axis

coincides with the axis of the

thread,

which effects

dE4

=

ds and the

integral

JdE1

dE2 dE3

to

be the

naturally

measured volume

dv0

of the thread

at rest.

According

to

(47) we

then find

sf^gdvdx^

=

dv0ds

(47a)

Unit of

mass.

The

comparison

of

the

masses

of

two

mass points can

be done

by

[p.

1059]

the usual methods. We

merely

need

a

unit

mass

to

measure masses.

Let

it

be defined

as

the

quantity

of

water that fits into

a naturally

measured unit volume

1

that is at

rest.

The

mass

of

a

material

point

is

by

definition

an

invariant under all transforma-

tions.

The

scalar of

density. By

scalar

density

of

continuously

distributed matter

we