DOC.

619

SEPTEMBER

1918 643

by a

complete

penetration

into the mathematical

foundations of

this

theory.

I

am

publishing

an

article

about

it entitled “Reine

Infinitesimalgeometrie” [“Pure

Infinitesimal

Geometry”]

in

the

Mathem.

Zeitschrift,

which

is appearing soon.[2]

There the natural

development

of

geometry

in

the three

stages

of

topology [reinen

“analysis situs"] (1),

affine

(2),

and metric

geometry

(3)

is followed through. I

say, now:

God

could have stopped

at

any

of

these

3

stages, namely,

created either

an

empty world,

or a

world

just

with

a

gravitational

field,[3]

or

the

one

familiar

to

us;

but

having

once

decided

on

the

metric

one,

the

most

richly

endowed,

he

had

to

adopt

mine,

the

pure one.[4]

I

can

perhaps

best

refute

the

objection

you

raised[5]

with the

following

remark: if

R

is

some

invariant of

weight -1,[6]

e.g.,

the

scalar

of

the

curvature,[7]

then the

quadratic

form

R(gikdxidxk)

is

an

absolute

invariant;[8] by multiplying by

this

factor,

an

absolute

normalization,

so

to

speak,

of

the unit

of

length

thus

is achieved.-[9]

When

I

sent

you my

note in

the

spring[10]

I

was already

convinced

that

your

gravitational

equations

follow

in

the

1st

approximation

from

my

theory

for static

fields,

that

is,

with

the

cosmological

A

term; yet

in such

a

way

that

from

the

laws

of nature it

only

follows

that

A

is

a

constant,

without

any specific

value

being prescribed

for this constant

(this

value

rather

appears

to

be

determined

by

the

world’s

total

mass).-[11]

In

my

case

also,

a

conservation

law for

the

total

energy

and momentum

is

valid,

dxk

=

0,

which

results

much

more

naturally

than

in

your

theory.[12]

For

the

matter in

the

more

restricted

sense,

tik,

the

equation[13]

dtf

-*_

_

pr

ts +

F•«*Sk

g

ZS r

+

=

0

(s

current)

applies;

whereas for

the

current,

the

postulate

si

=

pui

(u

four

velocity

dxi/ds,

p

charge

density)

is

possible,

tki

=

uuiuk

contradicts the

required

invariance

properties.[14]

In

consequence

of

this,

the

mechanical

equations

are

still

not

clear

to

me;

and

as long as

this

is not

settled,

it

is difficult

to

draw

any consequences

to

be

tested

against

experience

(the

influence of

the

electromagnetic

potential

on a

uncharged point-mass,

which

you

infer from

the

equation

for

the

geodesic

line,[15]

is

up

in

the

air

as

long

as

the

significance

the

geodesic

line has in mechanics

can-

not

be

ascertained).

This

will

be

the

next

task,

for

me

to arrive at

some

clarity

here.[16]

In

addition, I

have formulated the strict

equations

for

a

static,

rotation-

ally

symmetric

field;

they

are so complicated, though,

that

I

do not know

what

to do

with them

at

present;

the constitution

of

the

electron

and

the

possibility

of

its existence

ought

to result from

them.[17]