80 DOC.

9

FORMAL FOUNDATION OF RELATIVITY

£

=

iK%av

a

dX:

(84)

rv

_

_

1 dA,

aß

~

2

dx'

(84a)

We

now

introduce

one

further

postulate

of

approximation by considering

in

Zvo

only

those

terms

that

correspond

to

ponderable matter,

while

terms

from

surface

forces

are

ignored.

Under these

postulates,

(48) represents

the

energy

tensor.

Since

the Zvo

remain finite according

to

(48),

one

already

obtains

a far-reaching approxima-

tion if

one

neglects

infinitesimals of first order in

(48).

In this

manner

one gets

[p. 1082]

T

=

«Po

dxa dxv

ds0 ds0

(84b)

Substituting

into

(84)

and

writing

the

left-hand side

Ohav,

one

obtains

dx

dx

l

=

kpo-^-

(IjCq

ds0

av

(85)

x1, x2, x3,

are spatial

coordinates in this

equation

and

x4

=

it

is

the

(imaginary)

{20}

time

coordinate,

while

dso =

dt

(.2

dx21

dx22

dx23

I

__

+

1~.

dt2 dt2

dt2

is the element of Min-

kowski's

"eigentime."

After

we

have

replaced equations

(81)

with

approximation equations,

their

similarity

with the poisson

equation

of

the

Newtonian

theory

of

gravitation

hits the

eye.

We shall

now

replace

the

equations

of

a

material

point, (50b)

and

(51), by

approximation equations.

One obtains

the

coarsest

approximation

if

one replaces

(51)

with

l°

=

~mH7~

dsQ

(86)

Introducing

the three-dimensional

velocity

vector

q

with

magnitude

q,

this

means

the

equations