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Constraint equations for general hypersurfaces and applications to shells

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Abstract

Hypersurfaces of arbitrary causal character embedded in a spacetime are studied with the aim of extracting necessary and sufficient free data on the submanifold suitable for reconstructing the spacetime metric and its first derivative along the hypersurface. The constraint equations for hypersurfaces of arbitrary causal character are then computed explicitly in terms of this hypersurface data, thus providing a framework capable of unifying, and extending, the standard constraint equations in the spacelike and in the characteristic cases to the general situation. This may have interesting applications in well-posedness problems more general than those already treated in the literature. As a simple application of the constraint equations for general hypersurfaces, we derive the field equations for shells of matter when no restriction whatsoever on the causal character of the shell is imposed.

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Notes

  1. In general, no such global basis exists, and we would need to work with bases defined on each element of a suitable open cover of \(\Sigma \). Since all the expressions below will be tensorial (unless explicitly stated), there is no loss of generality in working as if the global basis did exist. This difficulty is general to the use of index notation and it is both well-understood and harmless. An alternative is to view indices in the sense of the abstract index notation of Penrose.

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Acknowledgments

I wish to thank José M. M. Senovilla and Alberto Soria for useful comments on a previous version of the manuscript and José Luis Jaramillo and an anonymous referee for suggesting interesting references related to this work. Financial support under the projects FIS2012-30926 (Spanish MICINN) and P09-FQM-4496 (Junta de Andalucía and FEDER funds).

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  1. Instituto de Física Fundamental y Matemáticas (IUFFyM), Universidad de Salamanca, Plaza de la Merced s/n, 37008 , Salamanca, Spain

    Marc Mars

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Correspondence to Marc Mars.

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This article belongs to the Topical Collection: Progress in Mathematical Relativity with Applications to Astrophysics and Cosmology.

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Mars, M. Constraint equations for general hypersurfaces and applications to shells. Gen Relativ Gravit 45, 2175–2221 (2013). https://doi.org/10.1007/s10714-013-1579-9

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