DOCUMENT 566 JUNE 1918 807
1918c
(Vol.
7,
Doc.
5),
as
well
as
De
Sitter
1917a and 1917c
(see
Jahresbericht der
Deutschen Mathe
matikerVereinigung
27
(1919), part 2, pp.
4244).
[5]Two
weeks
earlier,
Klein had
already
written that
contrary
to
what is
claimed
in Einstein 1918c
(Vol.
7,
Doc.
5),
the
singularity
in
the static
form of
the De Sitter solution
at
what Einstein called the
equator
is
nothing
but
an
artifact
of the
coordinates used
(see
Doc.
552).
Einstein
had
not, however,
acknowledged
this
result in
his
response (see
Doc.
556).
[6]See
Doc. 552
(in
the
equations
below,
the coordinate labels
u
and
co
have been switched
com
pared
to Doc.
552).
[7]For
the De Sitter line
element
in static
coordinates, see,
e.g.,
Doc. 556. The
figure
below illus
trates
some
of
the
properties
of
these
co
ordinates
(see Eddington
1923,
pp.
164166,
and
Schrödinger 1956,
pp.
121,
for
further
discussion).
The
figure
shows
a hyperboloid
representing a
1+1dimensional De Sitter
spacetime
embedded in
a
2+1dimensional
Minkowski
spacetime
with
(following
Klein’s
notation) pseudoCartesian
coordinates
E,
u,
and
to
(the
coordinates
T
and
Ç
are suppressed).
Planes
through
the
origin
rotated around
any
axis in the £uplane
over
any angle
45°
a
45°
cut out
ellipses on
the
hyper
boloid
which
are spacelike geodesics
of
the
spacetime.
(If
a
=
±45°,
such
planes
cut out
pairs
of
straight
lines,
generators
of
the
hyper
boloid,
which
are
null
geodesics;
if
a
45°,
they
cut out
pairs
of
hyperbolae,
which
are
timelike
geodesics.)
The
figure
shows
a num
ber of
such
ellipses
for
different values
of
a,
with the
£axis as
the axis
of
rotation. In
terms
of
the metric induced
on
the
hyperboloid
by
the metric
of
the
embedding
Minkowski
spacetime,
all
these
ellipses are equivalent.
The
set
of
all
of
them
covers
the
doublewedgeshaped region on
the
hyperboloid
indicated
by
the
dark
shading
in the
figure.
In
the static form
of
the
(1+1dimensional)
De Sitter
solution,
these
ellipses (which
would be
3dimensional
spherical spaces
in the
3+1dimen
sional
case)
are
taken to
represent
simultaneous events. This
means
that the time coordinate will be
a
function
only
of
the
angle
a, or,
equivalently,
the ratio
co/t).
The actual function
chosen,
which is
t

t0
=
R/carctanh
w/v
=
R/2clnf11
+
01,
has the further
property
that for
w/v

±1
(a

±45°)
t+O.
This time coordinate
t
is defined
only
in
the
region
between
the
planes u +
co
=
0 and
v

w
=
0,
where the
argument
of
the
logarithm
is
positive.
This is just the
doublewedgeshaped
region
in the
figure.
The time coordinate is
not
defined
on
the
edge
where the
two
wedges
meet
and
where
v
=
w
=
0.
This is reflected
by
the fact that the static form
of
the 1+1dimensional De Sitter
solution
is
singular
in the
two
points (i;
=
±R
,
v
=
co
=
0). The static form
of
the 3+1dimension
al
De Sitter solution will likewise be
singular
on
the surface
(Ç2
+12
+
Ç2 =
R2
,
o
=
co
=
0
),
the
socalled
equator.
Depending on
whether
one
adds
spherical or
Cartesian
spatial
coordinates
to
the time coordinate
t, one recovers
either
the static form
of
the De Sitter line
element
used
by
De Sitter and Einstein
or
the static
form
used
by Weyl
(see
Doc.
511, note 5).
One arrives
at
the latter form
by introducing co
ordinates
(xi,
z,
t)
(i
=
1,
2,
3)
related to the
pseudoCartesian
coordinates used
by
Klein via:
Í;
=
x1,
T
=
x2,
=
x3, v
=
z
sinhct/R,
and
0 =
z
coshct/R
(see
Weyl
1921a,
p.
256
[Weyl
1922,
pp.
281282],
and
Weyl
1923a, p. 293).