194

DOC.

16 FOUNDATIONS OF

GRAVITATION

ds2

=

Y,

ik

as

the

absolute invariant

(scalar),

then

one

sees

at

once

how

one

attains

a generaliza-

tion of the

theory

of

relativity

that

encompasses gravitation

on

the basis of the

equivalence hypothesis.

While in the

original theory

of

relativity

the

independence

of

the

physical equations

from the

special

choice

of

the

reference

system

is based

on

the

postulation

of

the fundamental invariant ds2

=

e dxi2,

we are

concerned with

i

constructing a theory

in which the

most

general

line element

of

the form

ds2 =

£

gikdxidxk

ik

plays

the role of the fundamental

invariant. The

concepts

of

vector analysis

needed

for that

purpose

are

provided by

the method of the absolute differential

calculus,

which will be

explained

in the lecture

by

Grossmann which

is to

come

next.

It follows from the idea

outlined above that the

ten

quantities

gik

characterize

the

gravitational

field;

they replace

the scalar

gravitational potential

(p

of Newtonian

gravitation

theory,

and form the

second-rank fundamental covariant

tensor

of the

gravitational

field. The fundamental

physical significance

of

these

quantities

gik

consists, i.a.,

in the fact that

they

determine the

behavior of

measuring

rods and

clocks.

The method of

the absolute differential calculus allows

us

to

generalize

the

systems

of

equations

of

any physical process,

as

they occur

in the

original theory

of

relativity,

in such

a

way

that

they

fit into the scheme of the

new

theory.

The

components

gik

of

the

gravitational

field

always appear

in these

equations.

The

physical meaning

of

this

is that

the

equations

provide

information about the influence

of the

gravitational

field

on

processes

in the

region

under

study.

The

previously

indicated law of

motion of the material

point may serve as

the

simplest example

of

this kind.

Otherwise,

we

shall confine

ourselves

to

the formulation

of

the

most

general

law known

to

physics,

namely,

the law that

corresponds

to

the

momentum

and

energy

conservation law in the

original theory

of

relativity.

As

is

well

known,

one

has there

a

symmetric

tensor

Tuv,

the

components

of

which,

the

stress

components, yield

the

components

of

the

momentum,

and the

components

of

energy

flux

density

and

energy density.

These

quantities

can

be

specified

for

phenomena

in

any

domain. The laws

of

momentum and

energy

conservation

are

contained in the

equations

r)

T

(1)

£

=

0,

(v,o=1,2,3,4)

since

by integrating

with

respect

to

the

spatial

coordinates

over

the whole

system,

one

can

obtain from these

equations

the conservation

equations

[7]