EINSTEIN-DE
SITTER-WEYL-KLEIN DEBATE
355
ment
outlined above for
why
the De Sitter solution is not
a
counterexample
to this
principle.
De
Sitter’s
defense
of
his
solution
(in
Doc. 501
and in De
Sitter
1918)
was
that the
singular
equator
cannot
be
reached from
any
point
in the
space-time, despite
the fact that the
proper
distance
from
any point
to the
equator
is finite.
Only
a
few months
later,
it became
clear
that
both
this
peculiar property
and the
singularity
itself
are
artifacts
of
the static
coordinates
used (see below).
III
The
discussion
of
the
cosmological
models
of
Einstein and De Sitter continued in Einstein’s
correspondence
with Hermann
Weyl
and Felix Klein.
The
central issue in both
exchanges
was
whether
or
not
antipodal
points
should be identified in
spherically symmetric
solutions
of
Einstein’s
field
equations
with
cosmological
term
(see,
for
instance,
Docs. 511 and
518).
This issue
was
also touched
upon
in the
correspondence
with De Sitter
(in
Docs. 355 and
359).
The
most
important
results
coming
out
of
the discussion
with
Weyl
and
Klein,
how-
ever,
concern
the
interpretation
of
the
singular
equator
in the static form
of
the De Sitter
solution.
In
the
section
on cosmology
of
Weyl
1918c,
which Einstein read in
proof
(see
Doc.
476),
the author discussed various static
spherically symmetric
solutions
of
the field
equations
with
cosmological
term. Although
not identified
as
such,
one
of
these solutions
is,
in
fact,
the static form
of
the De Sitter solution. To
turn
this into
a
solution that is
regular every-
where,
Weyl pieced
together
a complete
solution out
of
the static form
of
the De Sitter
so-
lution
and the solution for
an incompressible
fluid. The
resulting
solution has
a zone
of
mat-
ter
around the
equator (see
Doc.
511, note
5,
for
more details).
Einstein
pointed
out
(in
Docs.
511 and
513)
that such
hybrid
solutions need not be
symmetric
around the
equator,
a symmetry Weyl
had
emphasized
and
used
as an argument
for
identifying antipodal points.
As
Weyl was revising
the
page
proofs
in
response
to Einstein’s
criticism,
he
considered
what would
happen
if
he let the thickness
of
the
zone
of
matter around the
equator
go
to
zero
in his combination
of
the
De Sitter solution and the solution for
an incompressible
fluid
(see
Doc.
544).
He found
that the
resulting
surface
layer
of
matter has
a
finite
mass.
This
result,
published
in
Weyl
1919b and contested
by
several later authors
(see
Doc.
544,
note
9,
for
more details), initially
seemed to vindicate Einstein’s hunch that the De
Sitter
solu-
tion describes
a
universe
very
much like his
own cylindrical
universe,
the
only
difference
being
that all
mass
is
concentrated
on
the
equator.
In his last
letter
to De Sitter
(Doc. 506)
before this
exchange
with
Weyl,
Einstein
had,
in
fact,
already
announced that
Weyl
had
found
a
proof
for this
conjecture.
The
correspondence
with
Weyl
contains
only
allusions to
this
implication
of
Weyl’s
work.
Weyl
wrote to
Einstein
that the result
of
the
calculation
mentioned above
“might
meet with
your approval” (Doc. 544).
Einstein wrote
back that
he
was
happy
that
Weyl
had
finally
resolved the “zone
issue,”
and added: “Now the
result of
your calculation is
just
what
one
had to
expect” (Doc. 551).
Einstein’s satisfaction
was
short-lived. On
the
same day
that he wrote these lines to
Weyl,
Felix
Klein
sent
him
a
letter
(Doc. 552),
which
states,
among
other
things,
that
the
singularity
at
the
equator
in
the
static form
of
the De
Sitter
solution
is
an
artifact
of
the
way
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