On the differentiability conditions at spacelike infinity

Details On the differentiability conditions at spacelike infinity On the differentiability conditions at spacelike infinity

1997-12-12

arxiv.org/abs/gr-qc/9712058v1

Class.Quant.Grav. 15 (1998) 3873-3889

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

24 pages, Latex

Scientific paper

10.1088/0264-9381/15/12/016

We consider space-times which are asymptotically flat at spacelike infinity, i^0. It is well known that, in general, one cannot have a smooth differentiable structure at i^0, but have to use direction dependent structures. Instead of the oftenly used C^{>1}-differentiabel structure, we suggest a weaker differential structure, a C^{1^+} structure. The reason for this is that we have not seen any completions of the Schwarzschild space-time which is C^{>1} in both spacelike and null directions at {i^0}. In a C^{1^+} structure all directions can be treated equal, at the expense of logarithmic singularities at {i^0}. We show that, in general, the relevant part of the curvature tensor, the Weyl part, is free from these singularities, and that the (rescaled) Weyl tensor has a certain symmetry.

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