DOC.

470

FEBRUARY

1918

485

cases (for

both

coordinate

choices);

only

the

boundary

conditions

(for

the

spatial

boundaries

of

the

considered

system) are

different.

Now

you

will

say:

In

one

case (Galilean

coordinate

choice),

guv

=

-1

0

0 0

0 1

0

0

0 0

1

0

0 0 0

1

at

the

boundary,

in

the

other

case,

they

are

certain

functions in which not

even

the

equivalence

of

all

the

directions in

space

is

observed

(a mere

axial

symmetry);

there

the first

system

does

indeed

appear preferred

in

principle.

I

say,

though:

I

do

not

believe

that

the

boundary

conditions

(guu=

1,

guv =

0)

apply

in

principle.

If this

were

the

case, my

entire

theory

would have to be

rejected.

Since

a

theory, generally

invari-

ant

with

regard

to

the

differential

equations (regarding

arbitrary subst.)

but

not

generally

invariant with

regard

to the

boundary conditions, is

a

monstrosity.[5]

The

usefulness

of

the

boundary

conditions

(guv

=

1

or

0)

is

only

based

on

the

circumstance

that the

parts

of

the

world

we are

considering

are

sufficiently

small.

A sufficiently

small

part

of

a

continuously

curved

manifold

can

always be

treated

as

flat.

The

gravitating

individual

masses,

the

fields of

which

we are

examining,

then

appear as

local

sources

of

perturbation

curvature

in the

otherwise flat

manifold. Belief in

the

Euclidean nature

of

the

world

corresponds fully

with

the

belief of

antiquity

in

the

basically

flat

shape

of the Earth’s

surface;

in order to

be able to adhere to

this

belief, autonomous,

nonrelativistic

boundary

conditions

for

infinity

must be

introduced,

and

on

top of

this,

this

infinite world

must

be

considered

essentially

as

empty

so

that

it

is not

curved

by

its

own

matter!

If

the

boundary

conditions

(guv =

1

or

0)

fall

away,

then

all essential

grounds

for

the

preference

of

one

specific

choice

regarding

the

coordinate

system’s

rotational

state also fall

away.

This becomes

completely

clear when

the

world

is

viewed

as

(spatially) closed;[6]

for then

spatial

boundary

conditions

are

eliminated,

so

that

for

any

choice

of

coordinates the events

of

the

world

as a

whole

are

determined

entirely by

the differential

equations alone.

To

your

remark about

the electron

I

just comment

that the transformation

to

an

electron

constantly at

rest would indeed be

a

formal

simplification

for

some

problems.

The

simplification

is less

great, though,

than

in

the

case

of

pure

translation

(in

the absence

of

a

gravitational

field)

because

of

the

nonsta-

tionary gravitational fields

which

are

generally present.

As

an

example

to

the

contrary,

however,

I

mention the

Earth-Moon

system,

the

mechanics

of

which

is

represented

naturally

only

when the non-Galilean

system

is

introduced,

where

the

origin

constantly

occurs

in

the

center

of

gravity

of

both

masses.

The

excep-

tionality

of

the

ordinary

th.

of rel. lies

in

that

the Euclidean nature

is

retained