DOC. 25 FOUNDATIONS OF GENERAL THEORY 283

substitutions. The

equation

of

the

freely moving

material

point

in its Hamiltonian

form

is

S{/ds}

=

0.

(2)

In

the

case

where

we

drop

the

postulate

of the

constancy

of the

velocity

of

light,

there

exist,

a

priori,

no

privileged

coordinate

systems.

The coordinates

xv can

therefore be

replaced

by

as

yet

totally arbitrary

functions of these

quantities.

Then,

if there

are

also four-dimensional

regions

in

which, given

a

suitable choice of the

coordinates

xv,

the material

point

moves according

to

(2)

and

(1),

then the latter

can

no

longer

be viewed

as

the

general

law

of

motion

of

the

point moving

in the absence

of force. What

follows,

instead of

these,

is

equation (2)

in

conjunction

with

ds2

=

Euv

guvdxudxv,

(1a)

where the

quantities

guv

are

functions of the

xv.

3.

This law

of

motion

[(2),

(1a)]

is derived

first of

all

only

for the

case

where the

point

moves

totally

force-free,

thus where

no

gravitational

field affects the

point

either

(as

judged from

an

appropriate

reference

system).

But since

we

know from

experience

that the law of motion of

a

material

point

in

a gravitational

field

does not

depend

on

the material of the

body,

and

since, in

any case,

it

ought

to

be

possible

to

reduce this law

to

the

Hamiltonian

form, it

seems

natural

to

consider

[(2),

(1a)]

in

general

as

the law of motion of

a

point

acted

upon by

forces other than

gravitational

forces. This is the

gist

of the

"equivalence hypothesis."

4.

According

to

what has

been said

above,

we

have

to

conceive of the functions

guv

as

of the

components

of the

gravitational

field with

respect

to

the

completely

arbitrarily

chosen

reference

system.

Since the Hamiltonian

equation

of

motion

must

determine the motion of

the

point wholly independently

of

the

choice of the reference

system,

guvdxudxv

is to

be

conceived

as an

invariant with

respect

to

all

/XV

substitutions. We call the

positive square

root

of this

quantity

the

(four-dimensional)

line element of the

temporo-spatial

manifold.

5.

Minkowski based

a

four-dimensional covariant

theory

on

the invariant

(1),

which

theory yields

the

equations

of

the

original theory

of

relativity.

In

an

analogous

way,

it is

possible,

with the aid of the "absolute differential

calculus,"

to

ground on

the invariant

(1a) a

covariant

theory yielding

the

corresponding equations

of the

new

theory

of

relativity.

Since the

quantities

guv

enter

into these

equations,

the latter

easily

allow

one

to find the

influence that the

gravitational

field

exerts

on

any

physical processes.

The

equations

of the

new theory

of

relativity

reduce to those

of

the

original theory

in

the

special

case

where the

guv can

be considered

constant

(given

a proper

choice of the

xv);

this

is

the

special

case

in which the

gravitational

field

can

be

neglected.

It

is essential for the

theory

that

(1a)

be invariant with

respect

[6]

[7]