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.
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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Financial support under the projects FIS2009-07238, FIS2012-30926 (Spanish MICINN) and P09-FQM-4496 (Junta de AndalucĂa and FEDER funds).
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Mars, M. (2014). Geometry of General Hypersurfaces, Constraint Equations and Applications to Shells. In: GarcĂa-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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