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Geometry of General Hypersurfaces, Constraint Equations and Applications to Shells

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Progress in Mathematical Relativity, Gravitation and Cosmology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 60))

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Abstract

The constraint equations are well-understood for hypersurfaces which are either everywhere non-null or null everywhere. The corresponding form of the equations is very different in both cases. In this paper, I discuss a general framework capable of analyzing the intrinsic and extrinsic geometry of general hypersurfaces of a spacetime. This framework is then applied to derive the form of the constraint equations in this general context. As an application, I will generalize the Israel equations for spacetime shells to the case when the shell is allowed to have varying causal character.

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Notes

  1. 1.

    In fact, with only minor changes, everything below would apply equally well to Riemannian manifolds of arbitrary (non-degenerate) signature. We work with Lorentzian signature for definiteness and because it is the most interesting case in the context of gravitation.

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Acknowledgements

Financial support under the projects FIS2009-07238, FIS2012-30926 (Spanish MICINN) and P09-FQM-4496 (Junta de Andalucía and FEDER funds).

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Authors and Affiliations

  1. Dept. Física Fundamental, Universidad de Salamanca, Plaza de la Merzed s/n, 37008, Salamanca, Spain

    Marc Mars

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  1. Marc Mars
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Correspondence to Marc Mars .

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Editors and Affiliations

  1. Dept. of Mathematics, University of Minho, Braga, Portugal

    Alfonso García-Parrado

  2. Dept. of Mathematics, University of Minho, Braga, Portugal

    Filipe C. Mena

  3. Dept. of Mathematics, University of Minho, Braga, Portugal

    Filipe Moura

  4. Dept. of Mathematics, University of Minho, Braga, Portugal

    Estelita Vaz

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Mars, M. (2014). Geometry of General Hypersurfaces, Constraint Equations and Applications to Shells. In: García-Parrado, A., Mena, F., Moura, F., Vaz, E. (eds) Progress in Mathematical Relativity, Gravitation and Cosmology. Springer Proceedings in Mathematics & Statistics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40157-2_5

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