572 DOCS.
552 MAY 1918
the
projective
metric with
4 variables
is
obtained,
for which
x2
+
y2
+
z2
-
u2
+
R2
=
0
is
the absolute.-
From
the
elliptic
variables
one
returns
to
the
spherical
ones
by
“adjugating”
the
irrationality
O =
x2
+
y2
+
z2
-
u2
+ R2
and
setting
Rx
Ry
Rz
í
=
ir
c
=
tt-
v
= Ru/O
LO
=
IP
n
'
The
question
now
is
how to
introduce the time t.
a)
I
find
that
the
Schwarzschild-de
Sitter[6] ds2 is
obtained
by
inserting[7]
E =
Rsind
cosp,
r) =
Rsind
sinp
cosp,
E =
Rsind sinp sinp
v
=
RcosdGint,
w
=
Rcosd
.
Cost
Sin,
Cos
=
hyperbolic
functions.
Hence
inserting[8]
t
=
arctan
v/w.
And here arises
the
inconsistency
with
our
tacit
assumption
about
the
time
con-
cept,
which
you
point
out in
your
note of
7
March,[9]
namely,
that
t
is
undefined
for those
spatial
points
for which
v
=
0,
u
=
0.[10]
This
inconsistency
is
not
eliminated
even
if
we
move
over
to
an
elliptic concept
of
space:
at
M2,
located
in
the
space
and
which
is
of itself not
special
in
any
way,
t
has
a singular
region.[11]
b) We
shall avoid this defect and generally have a
natural point
of
departure
if we
set t in
the
spherical case equal
to the perpendicular distance
of some
diametrical
plane
of
the
sphere,
e.g.,
the plane v
=
0:
t
=
R•arcGinv/R.