356
EINSTEIN-DE
SITTER-WEYL-KLEIN DEBATE
in which the time
coordinate
is
introduced
(see
also
Klein,
F.
1918b, 1919).
The main
point
of
the
letter
is
something
different:
Klein wanted to retract his
earlier
objection
(in
Doc.
518)
that Einstein’s
cosmological
model is not time-orientable
if
antipodal points are
iden-
tified. Klein had
come
to realize that he had
been
conflating
the
cosmological
models
of
Einstein and De
Sitter
(see
Doc.
552,
note
3,
for
more
details)
and that his
objection applied
to De Sitter’s model
rather
than Einstein’s.
Partly
because the
result
concerning
the
singu-
larity
in the De
Sitter
solution
was
not
emphasized
in the
letter
and
partly
perhaps
because
the letter
relied
heavily on
notions from
projective geometry,
Einstein
failed
to
appreciate
that
Klein’s
analysis
of
the De Sitter solution showed
that
the
singularity
at
the
equator
can
be transformed
away
and does not
indicate
the
presence
of
matter
after
all. In his
response
(Doc. 556),
Einstein
simply
reiterated the
argument
of
his
critical note
on
the De Sitter
so-
lution,
for which
Weyl,
he
thought,
had
just
provided
new support.
In his next
letter
(Doc. 566),
Klein
was more
direct. As in his
earlier
letter,
he wrote the
transformation from the
pseudo-Cartesian
coordinates
of
the De
Sitter
hyper-hyperboloid
in 4+1-dimensional Minkowski
space-time
to the coordinates
used
to write the solution in
static form. This
shows,
Klein
explicitly pointed
out,
that the
singularity
at the
equator
has
to be
an
artifact
of
the static coordinates.
The
point
can
be made
more generally
than Klein
did. Since the De Sitter solution
can
be
represented
geometrically as a
fully regular hyper-
surface in
a higher-dimensional embedding
space,
any
singularity
in
a
coordinate
represen-
tation
of
the solution must be
an
artifact
of
the
coordinates.
Given Klein’s transformation from
the
pseudo-Cartesian coordinates
of
the
embedding
space
to static
coordinates,
the
properties
of
such static coordinate
systems
become
fully
perspicuous
(see
Doc.
566,
note
7,
for discussion and
a diagram).
Static
coordinates,
it turns
out,
only cover a double-wedge-shaped region
of
the
hyper-hyperboloid,
and the
hypersur-
faces
of
simultaneity
all intersect
on
the
edge
of
this
wedge,
the
region
Einstein and
Weyl
called the
equator.
The
singular
behavior of
the
temporal component
of
the
metric,
which
vanishes
on
the
equator,
reflects the fact that in the immediate
vicinity
of
the
equator, points
infinitesimally
close in
proper
time will be
infinitely
far removed
from
one
another in
co-
ordinate time. The
double-wedge-shaped
region
covered
by
static coordinates lies
fully
out-
side the
light
cones
of
points
on
the
equator.
This
explains why
De Sitter
concluded that the
equator
can never
be
reached.
This time Klein’s
point
was
not lost
on
Einstein. He
accepted
(in
Doc.
567)
that the De
Sitter solution
is
matter-free,
fully regular,
and
homogeneous.
This does not
mean,
howev-
er,
that Einstein
now accepted
the De Sitter solution
as a possible cosmological
model. He
still held that
any
acceptable
cosmological
model would have to be static. Klein had shown
that
in the static form
of
the De Sitter
solution,
the time coordinate breaks down
on
the
equator.
In
Weyl’s
hybrid
static
solution,
on
the
other
hand,
which coincides with the De
Sitter solution
outside
a zone
of
matter
around
the
equator,
the time
coordinate
is well de-
fined
everywhere. Only
this
hybrid
solution, therefore,
provides an
acceptable
static
cos-
mological
model.
Einstein thus had to
accept
that the De
Sitter
solution forms
a
counterexample
to Mach’s
principle as
he had
formulated
it in
March
1918
and that
his critical note
on
the De Sitter