Constraint equations for general hypersurfaces and applications to shells

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.

34 pages, no figures

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Cited by in corpus (17)

• Equations for general shells
• A Fundamental Theorem for Hypersurfaces in Semi-Riemannian Warped Products
• Hypersurface data: General properties and Birkhoff theorem in spherical symmetry
• Double Null Data and the Characteristic Problem in General Relativity
• Covariant definition of Double Null Data and geometric uniqueness of the characteristic initial value problem
• Null shells: general matching across null boundaries and connection with cut-and-paste formalism
• The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces
• General matching across Killing horizons of zero order
• On the Penrose inequality along null hypersurfaces
• Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons
• Supertranslations: redundancies of horizon data, and global symmetries at null infinity
• Transverse expansion of the metric at null hypersurfaces II. Existence results and application to Killing horizons
• First order perturbations of hypersurfaces of arbitrary causal character
• Variational formalism for generic shells in general relativity
• Killing and homothetic initial data for general hypersurfaces
• Junction conditions in a general field theory
• General Shells and Generalized Functions