552
DOC.
532
MAY 1918
this
solution
is
supposed
to be
stricter than the
one
in
which,
let’s
say, a
=
0
is
set,
hence
R
=
r,
or
any
other
value, is
not
at
all
clear, though. Any
other
value for
a
simply
means a
different
coordinate
system.
But
one
sees
from
this
yet again
how convinced Schwarzschild
very
obviously
was
that
one
preferred
coordinate
system
had to exist.
It
would indeed be
a
quite
intolerable situation,
of
course,
if such
arbitrary
quantities
always
had
to be
dragged along
within
the
solutions,
without
knowing
what
it
actually
means
if
a
certain
value
is
inserted
for these
arbitrary
variables,
if
one
does not
even
know whether
some
nonsensical
consequences,
such
as
apparent
waves
or
the
like, are
thereby
introduced
into
the
solution. Therefore
I believe
it
can
be said
that,
as
beautiful
as
the
general
transformability
of the
fundamental
equations
is
seen
to be from
the mathematical
point of
view,
physically
it
does
signify a
weakness in
the
theory
which must first
be
eliminated,
and
I
see
from
a
statement in
a
letter
of
yours
that
you
also
are
in
complete
agreement
with
me
about the
legitimacy
of
the
recipe
offered
in
my
Göttingen
lectures for
how, by means
of
a
general rule,
one can
always
obtain
a
coordinate
system
that
is
unobjectionable
and
that,
as
far
as
I
can see,
contains
the minimum of
arbitrary
elements.[17] I
am quite
convinced
that
this coordinate
system
indicated
by
me
is
the
natural
coordinate
system
and
that
one
would be
hard
pressed
to
find
another
as
good,
let alone
a
better
one.[18]
Well
now,
I have
attempted
to
examine
what
my suggested
coordinate
system
yields
in
the
case
of
the
gravitating sphere.
This
problem
is
soluble
without the
least
difficulty
and
one
arrives
exactly
at Schwarzschild’s
solution,
but
with
p
=
0.
Outside
of
the
sphere,
that
which Schwarzschild denotes
as
R
(where
R
=
(r3+a3)1/3),
measured
in
the
“natural” coordinate
system,
is
the
radius
vector
itself;
within
the interior
of
the
sphere
filled
uniformly
with
mass,
the radius
vector
R
=
n1/3
=
/-k.p0
•
sin
x
must
be
inserted.[19]
It
goes
without
saying
that, in the natural coordinate
system,
the
solution
is
fully unique.
In
addition,
it
is
evident
that the coordinate
system
chosen
by Schwarzschild,
in which R
=
(r3
+
a3)1/3
is inserted,
depends
on
a, i.e.,
on
the
gravitational
mass
of
the central
body.
Hence,
for
each
central
body
a
differently
defined coordinate
system
is
taken,
a
procedure
which
would,
extremely probably,
lead
occasionally
to
all sorts of inconsistencies if
one
wanted
to
move
from
the
sphere’s
solution to
some
other
more
general
solutions.
This
is
avoided
with the
“natural” coordinate
system.
It
is
especially interesting
that
in
my
“natural” coordinate
system,
/-g
=
1
does in fact result in
the
field
outside
of
the
sphere,
though
not in
the
interior
of the
sphere.
Whether
some general
principle
lies behind
this
or
whether it
is just
a
coincidence
I
do
not
know,
but
one can see clearly
from this
exactly
why, as
you
say,
“there
is
profound physical
justification
for
a
coordinate choice
according
to
the
condition
/-g
=
1.”
This