5 6 D O C . 6 7 J U L Y 1 9 1 9
67. From Jakob Grommer[1]
Berlin, 1 July 1919
Dear Professor,
The proof of the validity of the conservation laws for a pseudospherically closed
world is easily
completed.[2]
Following your suggestion, think of this world
mapped onto a sphere and the sphere mapped by stereographic projection on a hy-
persurface. In the hypersurface’s cartesian coordinates the only singular point is the
hypersurface’s spatial
infinity.[3]
It suffices to show that
(1) ,
where denotes the tensor density of the matter and gravitation, r =
, [with] , , the coordinates of the hypersurface.
Now, if (1) is satisfied, then[4]
will more surely vanish (where the integral is taken over a sphere’s surface at the
origin).
To prove (1), first imagine the vicinity of the singular point, hence at the north
pole, projected normally onto the hypersurface so that each point corresponds to
the perpendicular coordinates , of the point of projection. In these
primed coordinates, the north pole is a regular point, and all quantities , ,
including derivatives, are finite and regular. We perform the transformation
,
R = radius of the sphere, and express by the primed quantities.
, will have the limiting value zero. Since
(2) .
Now,

ν
r=∞
lim
r2
0=

v
x1 2 x2 2 x3 2 + + x1 x2 x3
Uσ----
1
x1
r
- Uσ----
2
x2
r
- Uσ----⎠
3
x3⎞
r
- + +

do
r=∞
lim⎝
x1′ x2′ x3′
g′ik g′
ik
xi
2R
R
R2 r′2
––
--------------------------------xi′, = r′2 x1′2 x2′2 x3′2 + + =

v

v

v

ν
+=
v
r2

v
gTσ
v
g′
D′(x′)
D x) (
---------------
β′
∂xv-----------
∂xβ′
---------
∂xα′
∂x
σ
= =
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