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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