DOC.
355
JUNE
1917
345
Lately,
I
have been
working on
the
problem
of
computing
the
field
of
a
small
sphere
in
the three
systems.[3]
This has
long
since been done for
system
C,
of
course (Schwarzschild,
etc.).[4]
In
system A,
I
thus
imagine
the
world matter
condensed into
a
sun
at
one
point only
(x
=
y
=
z
=
0, or
r
=
0),
not
anywhere
else.
Then
I
am
surely
permitted
to
interpret
the
problem
such
that
I set
p
=
P0
+
p1,
where
p0
is
the
density
of
the
cosmic matter and
a
constant:
kp0
=
2A;
and
p1
the
density
of
the
“normal
matter,”
hence
p1
=
0
for
r
r,
if
r
is
a
small
positive
number. Then
p0
=
0 throughout
system
B.
In
A,
A
=
1/R,
in
B,
A
=
3/R2,
and
in
C,
A
=
0
(and
p0
=
0).
I
now imagine a
stationary state,
thus
the normal
matter and
the
cosmic matter
are
at
rest. If
g0uv
stands
for
the
guv's
without
normal
matter, thus[5]
A: ds2
=
-dr2
-
R2
sin2
-
(dfi2
+
sin2
'ipd'd2)
+
dt2
B: ds2
=
-dr2
-
R2 sin2
-^(d^2
+
sin2
'ipdd2)
+
cos2
jgdt2
C: ds2
=
-dr2
-
r2(dip2
+ sin2
ifidd2)
+
dt2,
then the
guv
- g0uv’s
(=
ruv)
must
be
small. The
Tuv’s
are:
T0uv
=
0,
apart
from
T440
=
g44p;
Tuv
=
T0uv
+
T’uv
and
T'uv
is of
the
order
ruv
x
p.
I
did not take
the
T'uv’s
into
account.
They may
be
disregarded,
though,
[in
an
approximate
solution
up
to
the
first
order in
ruv
(or
k)],[6]
only
when
kp0
is
of
the
same
order
as kp1,
that
is,
when
A
is of the
same
order
as kp1,
within
systems
B and
C,
p
=
0
outside
of
the
“sun,”
and
a complete
solution
can
be
obtained with
Tuv =
T0uv.
In
system
A,
however,
one
cannot
get beyond
the
first
approximation
without
an
assumption
about the
T'uv's,
that
is,
about the
pull
and
pressure
effects in
the
“world
matter”
as
a
consequence
of
the
presence
of
the
“sun.”
But
even
the
first
approximation
leads
to
interesting
results.[7]
For
I
find
in
system A:[8]
g44 =
1
-
a
cot
-
a
=
/
kR^pA sin2
-dr.
For
a
spherical world,
at
the
antipodal point of
the
sun,
we
would have
(r
=
nR)g44
=
oo.
For
r
=
1/2nR,
g44
becomes
=
1. Hence,
it would not be
permissible
to
assume
a
spherical world,
but
the world must be
imagined
as “elliptical,”[9]
that
is,
the
greatest
possible
distance between
two
points
is
1/2nR,
two
(straight)
lines intersect each
other
only
at
one
point,
not
two,
etc. Schwarzschild
already
concluded
the
same
in
1900
(Vierteljahrsch.
der Astron.
Gesellsch.,
p.
337)[10]