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The Flatter Regions of Newman, Unti, and Tamburino's Generalized Schwarzschild Space

### Abstract

The ``generalized Schwarzschild'' metric discovered by Newman, Unti, and Tamburino, which is stationary and spherically symmetric, is investigated. We find that the orbit of a point under the group of time translations is a circle, rather than a line as in the Schwarzschild case. The time‐like hypersurfaces *r* = const which are left invariant by the group of motions are topologically three‐spheres *S* ^{3}, in contrast to the topology*S* ^{2} × *R* (or *S* ^{2} × *S* ^{1}) for the *r* = const surfaces in the Schwarzschild case. In the Schwarzschild case, the intersection of a spacelike surface *t* = const and an *r* = const surface is a sphere *S* ^{2}. If σ is any spacelike hypersurface in the generalized metric, then its (two‐dimensional) intersection with an *r* = const surface is not any closed two‐dimensional manifold, that is, the generalized metric admits no reasonable spacelike surfaces. Thus, even though all curvature invariants vanish as *r* → ∞, in fact *R* _{μναβ} = *O*(1/*r* ^{3}) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which *g* _{μν} − η_{μν} = *O*(1/*r*). An apparent singularity in the metric at small values of *r*, which appears to be similar to the spurious Schwarzschild singularity at *r* = 2*m*, has not been studied. If this singularity should again be spurious, then the ``generalized Schwarzschild'' space would represent a terminal phase in the evolution of an entirely nonsingular cosmological model which, in other phases, contains closed spacelike hypersurfaces but no matter.

© 1963 The American Institute of Physics

Received Mon Dec 17 00:00:00 UTC 1962
Published online Wed Dec 22 10:27:59 UTC 2004

/content/aip/journal/jmp/4/7/10.1063/1.1704019

http://aip.metastore.ingenta.com/content/aip/journal/jmp/4/7/10.1063/1.17040

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