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
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