426
DOC.
43
COSMOLOGICAL CONSIDERATIONS
an appropriate
choice of
the
system
of
coordinates the
stellar
velocities
are
very
small
in
comparison
with
that
of
light.
We
may,
therefore,
substitute
/g44
dx4
for
ds.
This
shows
us
that
all
components of
Tuv
must
be
very
small in
comparison
with the last
component
T44.
But it
was
quite
impossible
to
reconcile
this
condition with
the
chosen
boundary
conditions.
In the
retrospect
this result
does not
appear
astonishing.
The
fact
of
the
small
velocities
of
the
stars
allows
the
conclusion
that wherever there
are
fixed
stars,
the
gravitational potential (in
our case
/B)
can
never
be
much
greater
than here
on
earth. This
follows
from statistical
reasoning,
exactly
as
in the
case
of
the Newtonian
theory.
At
any rate,
our
calculations have
convinced
me
that such
conditions
of
degeneration
for
the
guv
in
spatial
infinity may
not
be
postulated.
After
the
failure
of
this
attempt,
two
possibilities
next
present
themselves.
(a)
We
may require,
as
in the
problem
of the
planets,
that,
with
a
suitable
choice of
the
system of reference,
the
guv
in
spatial infinity approximate
to
the
values

1 0
0
0
0

1
0 0
0 0

1
0
0 0 0
1
(b)
We
may
refrain
entirely
from
laying
down
boundary
conditions
for
spatial infinity
claiming
general validity;
but
at
the
spatial
limit
of the domain under
consideration
we
have
to
give
the
guv
separately
in
each
individual
case,
as
hitherto
we were
accustomed to
give
the
initial conditions
for time
separately.
[10]
[9]
The
possibility (b)
holds
out
no
hope
of
solving
the
prob
lem,
but
amounts to
giving
it
up.
This
is
an
incontestable
position,
which
is
taken
up
at the
present
time
by
de
Sitter.*
But I must
confess
that
such
a
complete
resignation
in this
fundamental
question
is for
me a
difficult
thing.
I
should
not
make
up my
mind
to
it until
every
effort
to make
head
way
toward
a
satisfactory
view
had
proved
to be vain.
Possibility (a)
is
unsatisfactory
in
more
respects
than
one.
*de
Sitter, Akad.
van
Wetensch. te
Amsterdam, 8 Nov., 1916.