50

DOC.

43 JANUARY

1915

the condition

that

at

the transition

from

one

system

to

another

one

that

differs

from

it

to

an

infinitely

small

degree,

the

Axu

and

dAxu/dxa

terms vanish

at

the

boundary.[4]

If

it

is

assumed

that the

second differential

gradients

of Ax

with

respect

to

x

differ from

zero

at

the

boundary,

then,

as

formula

(63a)

shows,

this

also

applies

to

Aguv.

The

latter

do not vanish

on

the

inner side of

the

boundary,

and

since

they

are zero

at

the

outer side where

nothing

has been

changed,

an

inconsistency

thus

arises for these values.

Therefore,

if

to

begin

with

you

relate

a given

gravitational field

to

a

coordinate

system,

and

thereafter

to

a

second

system

adapted to

the

field

and

differing

from the

first

only

in

the

interior

by E,

you

are

then

introducing

inconsistencies at

the

boundary.

A

description

of the

phenomena

involving

this

can

hardly

be called

satisfactory,

however.

The

difficulty

remains

Thus

it

is

not

impossible

that

at

that

place

higher

differential

quotients

of

Ax

with

respect to

x

are

other than

zero.

If this

is

the

case,

then

as can

be

derived from formula

(63),

the variation

A,

with

respect

to

the coordinates

at

the

boundary,

for

certain

differential

quotients

of

the

guv’s

will

also differ from

zero.

Now,

since

nothing

has been

changed

on

the

outer side of

the

boundary,

an

inconsistency

would

necessarily

occur

in

the

values

of

those differential

quo-

tients,

if

you

introduce

a

coordinate

system

that

differs

only

in

the interior

of

the

delimited

area

from

the

one

first used. The

introduction

of such

an

inconsistency

in

the

description

of

the

gravitational

field is

scarcely gratifying, though.

Were

the

second differential

quotients

of

the

Ax’s with

respect

to

x

other

than

zero

at

the

boundary,

then

an

inconsistency

would result

already

in the first

differential

quotients of

the

guv's.

Such

an

inconsistency

can

exist

only

when

a

finite amount

of

the attractive

“agent” (thus

here

the

energy,

etc.)

is

distributed

throughout

a

surface.

It

is

clear, however,

that

when such

a

surface

distribution

does not exist in

one

description

of

the

gravitational

field,

it also cannot exist in

the

new

description.

Similar observations also

necessarily

follow

when

you imagine

that material

processes

take

place

in

the

interior of

E.

Only,

in this

case

attention

must

be

directed

to all

the

equations, therefore,

not

only

to

the

gravitation

eqs.

but

also

to those

that

determine

the material

processes. By

the

way,

it

seems

to

me

that

the

difficulty always

exists

even

when,

for

ex.,

the

only

restriction consists in

that

the

processes

are

also

analyzed

from

a

particular moment

onward,

so

that

region

E

is denoted,

shall

we

say,

by

the

inequality

t

F(x,

y,

z).

As

I

see

it,

it

is

permissible

to

require

that the

description

of

the

phenomena

after the

moment

t0

=

F(x,

y,

z)

conform with

the

description

of

the

processes

preceding

this

time,

and

that

therefore

no

discontinuities

re

introduced

at

the

boundary

t0.