224 DOC.

18

DISCUSSION OF

DOC. 16

set

up a theory

of

gravitation.

It is

true, however,

that

my theory

is tucked

away

in

a

comprehensive

work

on

the

theory

of

matter

in

general,

and for that

reason my

investigations probably escaped

Mr. Einstein's notice.

(Einstein: No, no.)

Then he

probably

has

not

read

it,

otherwise he would have mentioned

it. I

believe that

my

theory

has the

advantage

of

being very perspicuous, so

that it

is

very easy,

for

example,

to

make

an

exact

calculation of the force exerted

on a particle,

which does

not

seem

to

me

to

be

quite successfully

done in Einstein's work. In

addition,

my

formulation is

very general

and allows for

many special

cases

that

seem

equally

justified,

because I do

not

identify

the

quantities

(g,

i

•

u)

and

(k, i

•

w); they only

become

equal

to

one

another in the ideal

vacuum, i.e.,

in

a space

that is

extremely

far

away

from

any

matter, and,

by contrast, I

do not have

to

make

any assumptions

about the

mutual

dependence

of the

two

quantities

inside

matter. I

have

not

yet

made

a

more

detailed

comparison

of Nordström's

theory

with mine: in the latter

(g,

iu)

is

equated

with

(k, iw),

otherwise the

two

theories

may

well be

nearly

identical.

I

took

for

p

the

quantity

that is

usually

called the

density

of the

rest

mass.

This is

a

four-dimensional scalar that is

indistinguishable

from the

density

of the inertial

mass

under normal

circumstances.

So,

just

as

the inertial

mass

of

a

body

is

identical with

its

energy

in the

theory

of

relativity, so

is the

rest

mass

identical with the Hamilto-

nian. The

density

of the

rest

mass

is

thus the

density

of

the

Hamiltonian,

and I

therefore denote it in

my study by

H and call

it

briefly

"the Hamiltonian function."

Einstein's

theory

is

by

no means

much

more

complicated

than this

theory.

Its

fundamental

equations are very

similar

to

those

I

have

just

written down. Einstein

has

already

pointed

out

that his

theory

is

characterized

by

the fact that the

gravitational potential

is not

a

four-dimensional scalar but

a

four-dimensional

tensor.

I

will

denote the

components

of this

potential by

wuv,

where

U

and

v are

numbers

running

from

1

through 4,

and

wuv

=

wuv. Accordingly,

the

gravitational

field is

not

to

be described

by

a simple

four-vector

but

by

a

space-time quantity

of

the fourth

rank,

which is formed from

10

four-vectors,

so to

speak,

each

one

of which

can

be

associated with

a

tensor

component

(u, v). I

will denote these

quantities

of

the

third

rank

by

guvx, guvy, guvz,

i

•

Uuv.

Just like

my theory,

Einstein's also contains

a

second

quantity by

means

of

which

one

could describe the field

equally

well,

which

only

becomes identical

with

(guv, i

•

uuv)

in the ideal

vacuum.

I

will denote it

by

kuvx,

Kuvy, kuvZ,

i•wuv.

The fundamental

equations

of

Einstein's

theory

of

gravitation

can

then be written in the

following

way:

A

_

3t0|.v

A

Ä

9,1

v*= ~äT'

9"v'=

~äT'

9"v*=

~bF'

=

dk^x

+ + +

_

_vY

.h

dx

By

dz

dt ~

v-

I

wrote the

equations

in

a way

that makes the

analogy

with the scalar

potential theory

[4]

[5]