DOC.

9

FORMAL FOUNDATION OF RELATIVITY

83

For

light rays

(ds

=

0),

one

has

dx2

+ dy2 +

dz2

=

1

+

o

dt

The

speed

of

light

is, therefore,

independent

in

its direction but varies with the

gravitational potential,

and

consequently one

has

a

curved

progression

of

light rays

in

a gravitational

field.

Finally,

we

calculate momentum and

energy

of

a

material

point

in

a

Newtonian

field for which

we

do not

use

the

equations

(86a)

but rather the

rigorous equations

(51).

If

we

substitute

into these for the

gou

the

values

-1

0

0

0

0 -1

0 0

0

0 -1

0

0 0

0 -1

+

h44

and for the

dxv

the

quantities

dx

dy dz

idt,

furthermore

limiting

ourselves for

h44

to

quantities

of

first

order,

and

replac-

ing

(h44/2)

with

o,

and neglecting terms of higher than second order in the velocities,

one

obtains

"Ii

=

m(

1

- ß)q;

[p. 1085]

il4

=

m

(W)

|(1

- f)q2

Since

(-I1)

is the

X-component

of the

momentum

and

(iI4)

is

the

energy

of

the

material

point,

one comes

to

the conclusion

that

the inertial

mass

increases

with

diminishing gravitational potential.

This is

very

well in

agreement

with the

spirit

of

the

interpretation

taken here. As there

are

no

independent physical qualities

of

space

in

our

theory,

the

inertia

of

mass

is

a consequence

of

the

mutual action between each

mass

and all the other

masses.

This interaction

must,

therefore,

increase when other

masses

are brought

closer

to

the

mass

that is under

consideration, i.e.,

if 0 is

decreased.