DOC.
495
MARCH
1918 517
Both
representations
are
merged
in the
transformation
i+~=~i+2~1-Yz
Both
are
Cartesian
coordinate
systems;
but
only
one can give
the best
possible
link
with
experience.
And
this then
is
the
justified one.
It
becomes
apparent
that this
is
the
case
for
(b)
and
that
here
g
=
-c2
is
satisfied;
the
equivalency
hypothesis,
on
the
contrary, yields
an
illegitimate
metric.
Coordinate
systems
other than Cartesian
ones
have
no new
physical
signifi-
cance.
Rather,
it
must
never
be
disregarded
that
these
are
merely
mathematical
operations
without
geometrical
content.
Likewise, a
surface does
not
change
its
shape
if I
represent
it
using
other
coordinates, except
in
the
case
of
pliability.
It
is
precisely
this
case
which would
occur
in
the
above
homogeneous
field
example
if
one
of
the
systems
had
not been excluded.
Thus the
uniqueness
of
the
designated
shape
of
our
curved
four-dimensional manifold
is
established. And
the
geometric
shape
is identical here to
physical
nature.
My
condition for these non-Cartesian coordinate
systems
now
undergoes
the
corresponding
alteration
g
=
(g)k=o,
e.g.,
for
polar
coordinates
r, d, p
g
=
-c2
.
r4
sin2d etc.
That
is
why
it
is
by no means
possible
to confuse it with
the
constraint
g
=
-1
you
have been
using
now
and then.
I
warned
against
this
in
a
separate
footnote
to
prevent
such
misunderstandings.
In
returning
again
to
the
case
where
g
-
-c2
in
Cartesian
coordinates,
or
generally, g
= (g)k=0
is not
satisfied,
it must be mentioned
that then the initial
conditions
(structure
of the
field,
material
tension
tensor,
boundary
conditions,
etc.)
ought
to be checked.
(Comp. §26
the
interior
of
homogeneous
spheres.)[8]
2)
I
have
read
your paper
on
gravitational
waves[9]
but
must
confess to
being
guilty
of
an
omission in
my
paper. Namely,
I
apply
as
gravitational
tensors
[Gravitationsspannungen]
Gin
=
\Kgik
-
Kik
=
pik
Y
^{rq,
sq}
-
" "
r,s,q
q
for which
Gik
+
KTik
=
0