DOC.
463
FEBRUARY
1918 473
force of
an
infinite stellar
system
and find
that
the
physical objection
to
the
in-
finity
of
the
universe and
the
mass
contained within
it,
based
on
Newton’s law
implying an
infinite
force,
is
invalid. On the other
hand,
I have also not been able
to convince
myself
of
the
necessity
of
assuming a
finite world
on
mathematical
grounds.[5]
You
think that the
boundary
conditions
cause
problems
at
infinity.
In
the
general
theory
of
relativity,
however,
since
it
interprets
the
world
with
differ-
ential
equations,
thus with
function
theory, infinity
(x1
=
oo,
x2
=
oo,
x3
=
oo,
x4
=
oo)
is
not
a limiting case
for
a large
sphere, perhaps,
but for
a
point
just
like
any
other
point
and
can
be transformed into
any
other
point.- Thereupon
I
reflected
on
which
boundary
conditions would
then
be at all suitable
to
define
clearly
a
solution to
the
system
of field
equations.
For
this
purpose
[:in
the
case
of
geodesic
coordinates[6]
gi4 = 0;
g44
=
1,
which
is not
a
limitation
of the
generality,
in connection with
the
equally
admissible
assumption
that the
coordinate
system
is
“quasi-orthogonal"
in
the
vicinity
of
the
observed
point
xi
=
xi0
(i
=
1, 2,
3,
4)
(:i.e.:
the
gii's
differ from
1
just
by
infinitesimal amounts of
the
2nd
order,
the
other
gik’s
are
themselves in
the
2nd order
:):],[7]
I
tried to
apply
the
system
of
field
equations
to Tresse’s canonical form
of
a “passive system” (Encyclop. d.
math.
Wiss.
II
A
5,
No.
2),[8]
and in
doing so
found
that,
in order to
single
out
a
solu-
tion,
the
following can
be
prescribed arbitrarily
at the
origin: (a)
All derivatives
of
g12, g13,
g23
with
the
exception
of
each
of
the
following
four
fourth
derivatives:
d4/1144, d4/2244, d4/3344,
d4/4444
and all
of
their
derivatives,
in
addition,
(b)
the
deriva-
tives of
g33
with the
exception
of six
specified ones (of
2nd-4th
order)
and
their
derivatives,
in
addition,
(c)
those
of
g22
with
the
exception
of
5
determined
ones
(of
2nd and 3rd
order) along
with
their
der[ivations],
and
finally,
(d)
those
of
g11
with
the
exception
of those of
the
2nd
order
according
to
22,
33,
44,
23, 24,
34
and
their
derivatives. From
this,
the
arbitrary
functions
appearing
in
the
general
solution
can
also be deduced
(:The
ranking
used here:
g12,
g13, g23, g33,
g22, g11
is
obviously
arbitrarily
chosen:).
Certainly
a
very complicated
boundary
value
problem.
With this the
matter
is
far from
finished, though.
The
system
of
your
field
equations
conceals
many
more
surprises.
I
would
just like to
point
to
one.
Through
adept
elimination,
20
equations
of
the 4th
order
can
be derived from
your general equations,
which
(except
for lower-order derivatives of all
the
guv's)
contain
4th-order
derivatives
just
for each
of
4
(or
3)
of
the
unknowns
(guv),
namely
for each
only a
second derivative
of
the
first differential
parameter.
In
the
above-mentioned
geodesical, quasi-orthogonal coordinates,
in
the
main,
6
of these
equations
remain of
interest,
for which the
highest-order
terms read:
d2/44
(d2gik/11
+
d2gik/22
+
d2gik/33
+
d2gik/44)
(i,
k
= 1,
2, 3)
(These
terms
thus
contain
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