174

DOC.

180 JANUARY

1916

at

the

Dutch

border,

mission

unaccomplished,

in

spite

of

regulation

passports.[4]

So I

am

going

to try

to

answer

your question

in

writing.

I

shall

make

an

effort

not

to

leave

a

word of

your

letter

out

of

consideration

but

shall address

everything.

First

of all

I

note

that

your equation (IV)

is

satisfied

identically,

as

you

will

have

gathered

from

my

last

postcard.[5]

Should

the

proof

not have convinced

you,

I

shall

provide you

with

another

one

that

is less

bumpy

mathematically

but

which

cannot be linked

so

conveniently

to

the

papers.

I

cannot hold

it

against you

that

you

have

not

yet

understood the

admissibility

of

generally

covariant

equations,

because

I

myself

needed

so long

to arrive at

total

clarity

on

this

point.[6]

The

root

of

your difficulty

lies

in

that

you instinctively

treat

the

reference

system as something

“real.” Your somewhat

simplified

ex-

ample:

You

examine two solutions

with

the

same

boundary

conditions

at

infinity,

in

which

the

coordinates for

the

star,

the

mate-

rial

point

at

the

aperture,

and

the

plate

are

the

same.

You

ask whether “the

direction

of

the

wave

normal” at

the

aperture

always

comes

out

the

same.

As

soon as

you

speak

of “the direction of

the

wave

normal

at the

aperture,” you

are

treating

this

space

with

regard

to

the

guv

functions

as an

infinitesimal

space.

This

and the

definiteness of

the coordinates

for

the

aperture points

have

as a

consequence

that

for

all solutions

the direction

of

the

average

waves

at

the

aperture

is

the

same.

star

aperture

plate

This

is

my

contention. For

a

finer illustration, the

following:

In the

above

spe-

cial

case

you

obtain all

the

solutions

that

are

a

consequence

of

general

covariance

in

the

following

way.

Trace

the

above

little

diagram

on

to

a

completely

flexible

piece

of

tracing

paper.

Then deform the

tracing

paper randomly along

the

paper

plane.

Then

make

another

tracing

on

the letter

paper.

You

then

obtain,

e.g.,

the

diagram

star

aperture

plate

If

you

now

again

relate

the

diagram

to

orthogonal

letter-paper coordinates,

the

solution is mathematically

a

different

one

to

beforehand,

naturally

also

with

regard

to

the

guv's.

But

physically

it

is exactly

the

same, simply

because

the