DOC.
566
JUNE
1918 593
may
be
occupied
almost
exclusively
with
the
practical
tasks
attached
to
the 20th
anniversary
of
the
foundation of the
Göttingen
Association for
Applied Physics
and
Mathematics.[3]
But last
Tuesday
I
did
report
on
your
note
re.
de Sitter’s
ds2.[4]
I
came
to
the
result
that
the
singularity you
noticed
can
indeed
simply
be transformed
away.[5]
The formulas I
prepared I
had
actually already
written
down
recently.[6]
I
set
£2
+
if
+
C2
+
v2

u1
=
R2
d£2
+
dn2
+
d(2
+ dv2

du2
=
ds2
(1)
and
arrive at de Sitter’s line
element[7]
through
the substitution:
£
=
Rsin

cos
ifj,
r¡

Äsin

sin
•¡/'•cosí?,
£
=
Äsin

sin
íp
sint?
D
r
rf
(C(t
¿0)
r.
r
(c(t
~
¿o)
v
=
Reo
s 
•
(Eos

u =
R
cos
 •
6m
 R
[
R R \
R
(2)
where
(Eos.
and
6in.
mean
hyperbolic
functions and
t0
is arbitrary.
I do
not
need
to
write down
the
inverse
of
these
formulas,
which
is
selfevident;
let
me
just
note
the
value for
t:
gq34
4tq34
34t3qt 4q3t
(3)
Through
this
inverse, however,
de Sitter’s
ds2 is
changed
into
d£2

dn2

d(2

dv2
+
dto2,
which
on
the
“quasisphere”
£2
+
r)2
+
Ç2

v2 +
w2
=
R2
certainly
has
no
singularity,
q.e.d.[8]
In the last
analysis,
there
are
oo6
many
legitimate
ways
of
moving
from
equa
tion
system
(1)
to
a
de
Sitter
ds2.
This
is
because
the
“hyperplanes”
(v
+
w)
=
0
and
(v

w)[9]
are
ultimately
only
two
arbitrary tangential
planes
of
the
quasi
sphere’s “asymptotic
cone”
:
£2
+
n2
+
C2
+
v2

w2
=
0.
(4)
It
is
amusing
to
picture
how
two
observers
living
on
the
quasisphere
and
equipped
with
differing
de
Sitter
clocks
would
squabble
with each other. Each of
them would
assign
finite
time
ordinates
to
some
of
the
events
that
for
the
other
would be
lying
within
infinity
or
that
would
even
show
imaginary
time
values.[10]
In order to
give a
physical
turn to
my
letter after
all,
I
note
that
de Sitter’s
ds2
appears implicitly already
in Schwarzschild’s
paper
of 24
Feb.
1916.[11]
One
just