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Hypersurface data: General properties and Birkhoff theorem in spherical symmetry

Marc Mars

Abstract

The notions of (metric) hypersurface data were introduced in [Mars,2013] as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-riemannian manifolds. In this paper, general geometric properties of these notions are studied. In particular, the properties of the gauge group inherent to the geometric construction are analyzed and the metric hypersurface connection and its corresponding curvature tensor are studied. The results set up the stage for various potential applications. The particular but relevant case of spherical symmetry is considered in detail. In particular, a collection of gauge invariant quantities and a radial covariant derivative is introduced, such that the constraint equations of the Einstein field equations with matter can be written in a very compact form. The general solution of these equations in the vacuum case and Lorentzian ambient signature is obtained, and a generalization of the Birkhoff theorem to this abstract hypersurface setting is derived.

Hypersurface data: General properties and Birkhoff theorem in spherical symmetry

Abstract

The notions of (metric) hypersurface data were introduced in [Mars,2013] as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-riemannian manifolds. In this paper, general geometric properties of these notions are studied. In particular, the properties of the gauge group inherent to the geometric construction are analyzed and the metric hypersurface connection and its corresponding curvature tensor are studied. The results set up the stage for various potential applications. The particular but relevant case of spherical symmetry is considered in detail. In particular, a collection of gauge invariant quantities and a radial covariant derivative is introduced, such that the constraint equations of the Einstein field equations with matter can be written in a very compact form. The general solution of these equations in the vacuum case and Lorentzian ambient signature is obtained, and a generalization of the Birkhoff theorem to this abstract hypersurface setting is derived.
Paper Structure (8 sections, 33 theorems, 206 equations)

This paper contains 8 sections, 33 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Abstract data and embedded data
  3. Gauge structure
  4. Metric hypersurface connection
  5. Metric hypersurface curvature
  6. Matter-hypersurface data and constraint equations
  7. Spherically symmetric hypersurface data
  8. Birkhoff theorem at the abstract hypersurface data level

Key Result

Lemma 2.2

Let $\{{\mathcal{N}}^m,\gamma,\underaccent{\check}{\bm{\ell}},\ell^{\hbox{\tiny $(2)$}}\}$ be metric hypersurface data and $p \in {\mathcal{N}}$. Then the radical of $\gamma$, with the usual definition $\hbox{Rad}|_p := \{ X \in T_p {\mathcal{N}} \, ; \, \gamma(X,\cdot)=0 \} \subset T_p {\mathcal{N

Theorems & Definitions (84)

  • Definition 2.1: Metric hypersurface data and hypersurface data
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 74 more