354 EINSTEIN-DE SITTER-WEYL-KLEIN DEBATE
hypersurface)
turns out
to
be
an
artifact
of the
stereographic
coordinates
(see
Doc.
356,
note
8).
Meanwhile,
De Sitter
began
to take his
model
more
seriously
as a
possible
description
of
the actual universe.
Originally (see
Doc.
321),
he had looked
upon
the
discussion
of
the
two
new
models
merely
as
idle
speculation
about how to
extrapolate
the
approximately
Minkowskian values
of
the metric field
beyond
the visible
part
of the
universe.
By
mid-
1917, however,
he
was busy exploring
the
physical
consequences
of
three different
options
available
for
the
global
structure
of
space-time
(see
Doc. 355 and De
Sitter
1917b, 1917c):
(A)
Einstein’s
cylindrical space-time,
(B)
his
own hyperboloid space-time,
and
(C)
Minkowski
space-time.
De
Sitter,
like Mie
a year
later
(in
Doc.
465),
distinguished
between
a purely
inertial and
a gravitational part
of
the
metric field. The three alternatives above
per-
tain to the
purely
inertial field.
For
these three
cases,
De Sitter calculated the
gravitational
field, i.e.,
the deviations from the inertial
field, produced by a
massive
spherical body
such
as
the
sun.
Einstein
was
pleased
that De Sitter
was now
willing
to
engage
in the sort
of
spec-
ulation that he had rejected earlier
(see
Doc.
356).
But
he took
exception
to De Sitter’s
de-
composition
of
the metric field into
an
inertial and
a gravitational
part, and,
in
particular,
to
the distinction it
implied
in
model
(A)
between
“world matter”
generating
the
inertial field
and
“ordinary
matter”
generating
the
gravitational
field
(see
Docs.
356, 359).
Ordinary
mat-
ter,
Einstein
explained,
should not be
thought
of
as existing
in addition to the uniform
mass
distribution
of
his
cosmological
model,
but in terms
of
local condensations in this uniform
mass
distribution.
The
spatial geometry
of
a more
realistic version
of
his model with local
inhomogeneities,
he
added,
might
resemble
that of
the “surface
of
a potato”
(Doc. 356)
more
than that
of
a
perfect
sphere.
Einstein did not
develop any
such models. De Sitter’s
calculations,
and Mie’s
a
year
later
(see
Docs.
465,
488),
made it doubtful that static inho-
mogeneous
variants
of
Einstein’s model
could
be constructed.
To
facilitate
comparison
with
options (A)
and
(C),
De Sitter had written the line element
for
his
own
model
(B)
in
new
coordinates.
In these
coordinates,
the
components
of
the
met-
ric tensor
are time-independent.
In Doc.
363,
Einstein wrote that he found this
new
form
“very
instructive,”
and
the remainder
of
the
exchange
between Einstein and De Sitter
during
1917-18
focused
on
this so-called static form
of
the
De
Sitter solution. In this
form,
the
spa-
tial
geometry
of
the De Sitter model is the
same as
that
of
Einstein’s
model,
namely
that
of
a hypersphere
in
a
4-dimensional Euclidean
space. Contrary
to Einstein’s
model, however,
the
temporal component
of
the
static De Sitter metric is variable and vanishes
on
the
“equa-
tor”
of
this
hyperspherical
space.
Einstein
argued
(in
Docs.
363, 366,
and
370)
that such
singular
behavior
of
the metric
was unacceptable,
and
suggested
that
it
indicated
the
pres-
ence
of
matter
on
the
equator.
Careful not to make
the
same
mistakes that
he
had made in
his
analysis
of
De
Sitter’s solution in
stereographic
coordinates,
Einstein convinced
himself
that the
singularity
of
the metric in static coordinates occurred at
a
finite
proper
distance
from
an arbitrarily
chosen
point
of
the
space-time
and that
it
could not be transformed
away.
The threat that De Sitter’s solution
posed
to Mach’s
principle
thus
finally
seemed to be
re-
moved. In
early
March
1918,
Einstein
completed
two
papers,
Einstein
1918f
(Vol.
7,
Doc.
4) introducing
Mach’s
principle,
and Einstein 1918c
(Vol.
7,
Doc.
5) presenting
the
argu—