Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions

By A. Bernal, Miguel Sánchez Caja. 2005

203 citations recorded by the source corpus.

AbstractRecently, 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 $$\mathcal{T}$$ (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with $$S= \mathcal{T}^{-1}(0)$$ is constructed – thus, the spacetime splits orthogonally as $$\mathbb{R} \times S$$ in a canonical way. Even more, accurate versions of this last result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/3b101248b39cec23ab9af3e2b6aab274b79b328a

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In Applied Mathematics

1. Ettore Minguzzi
2. Miguel Sánchez Caja
3. Valter Moretti

In Astronomy and Astrophysics

1. Ettore Minguzzi
2. Igor Khavkine
3. Miguel Sánchez Caja
4. Valter Moretti

Citations in the corpus

• Future stability of the Einstein-non-linear scalar field system (2008, unknown): “[4]-[6].”
• The 1965 Penrose singularity theorem (2014, unknown): “morphic Cauchy hypersurfaces [32, 148, 279].”
• The causal hierarchy of spacetimes (2006, methodology): “The solution to the problems on smoothability in [4, 6, 7] involves technical procedures very different to the expected approaches in previous attempts.”
• Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction (2014, background): “Such an adapted partition of unity always exists if O is globally hyperbolic in its own right and, if not, since M is globally hyperbolic, any open neighborhood of Σ will contain a possibly smaller neighborhood of Σ that is also globally hyperbolic [6, 5].”