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,

Uσ

ν

r=∞

lim

r2

0=

Uσ

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 + + =

Uσ

v

Uσ

v

Tσ

v

tσ

ν

+= Tσ

v

r2

Tσ

v

gTσ

v

g′

D′(x′)

D x) (

--------------- Tα

β′

∂xv-----------

∂xβ′

---------

∂xα′

∂x

σ

= =