164 DOC.
169
DECEMBER
1915
perihelion[2]
and
solved
to
1st-order
approximation.[3] Initially,
one
factor
made
me
very
confused.
I
found for
the
coefficients
gnv
in
first-order
approximation,
in
addition
to
their
solution,
also
the
following
second
one:[4]
9
pa
-
ßxpx"
T
&
pa
ß_
3
r3
g44=1.
According
to
this
there
would be
a
second
a
in
addition
to
yours
and
the
problem
would be
physically
ambiguous.
Thereupon,
I
took
my
chances and made
an
attempt at
a
complete
solution.[5] A not
overly
lengthy
calculation
yielded
the
following
result: There
is
only
one
line element
that
satisifies
your
conditions
(1)
to
(4),[6]
aside from
the
field
and
determinant
eqs.,[7]
and
is
singular
at
the
origin
and
only
at
the
origin.
Let
x1
=
rcospcosd
x2
=
rsinpcosd x3
=
rsind
R=(r3+a3)1/3=r(1+1/3ar,)
then the
line
element
reads:[8]
ds2
=
(1
-
-r/R)dt2
-
R2(dd2
+ sin2
ddp2).
R, d, p
are
not
“admissible” coordinates with which
the
field
equations
could
be
formed,
because
they
do not have
the determinant
1,
but the
line element
is
written
most
neatly
in them.
The
equation
for
the orbit
remains
exactly
the
one
obtained
by you
in
first–
order
approximation
(11),[9]
except
under
x
not
1/r,
but
1/R
must
be
understood,
which
is
a
difference
of
the
order
of
10-12,
thus
practically absolutely
irrelevant.
The
problem
of
the
two
arbitrary
constants
a
and
ß,
which
the
first-order
approximation
had
yielded,
is
solved in that ß
must
have
a
specific
value of
the
order of
a4,[10]
the
way a
is
given,
otherwise in
continuing
the
approximations
the
solution
would be
divergent.
Thus
the
uniqueness
of
your problem
is
also in
the best
of
order.[11]
It
is
a
wonderful
thing
that the
explanation
for
the
Mercury anomaly emerges
so
convincingly
from such
an
abstract
idea.
As
you
see,
the
war
is
kindly disposed
toward
me,
allowing
me, despite
fierce
gunfire
at
a
decidedly
terrestrial
distance,
to
take this walk into
this
your
land
of
ideas.
[5]Draft version:
“In
order
to become versed in
your
gravitation theory,
I
set
myself
the task
of
solving completely,
if
possible,
the
problem
you posed
in
the
paper
on