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Register a new userFurther results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
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Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R times S$ in a canonical way. Even more, accura
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Given a globally hyperbolic spacetime $M$, we show the existence of a {em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $R times S$.
This paper has been withdrawn because the new one gr-qc/0512095 includes all its results (as well as those in gr-qc/0507018), in a clearer way.
We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance $d_S$ from a spacelike hypersurface $S$ is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.
This paper has been withdrawn because the new one gr-qc/0512095 includes all its results (as well as those in gr-qc/0511016) in a clearer way.
This paper gives a detailed pedagogic presentation of the central concepts underlying a new algorithm for the numerical solution of Einsteins equations for gravitation. This approach incorporates the best features of the two leading approaches to computational gravitation, carving up spacetime via Cauchy hypersurfaces within a central worldtube, and using characteristic hypersurfaces in its exterior to connect this region with null infinity and study gravitational radiation. It has worked well in simplified test problems, and is currently being used to build computer codes to simulate black hole collisions in 3-D.
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