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Initial data sets with dominant energy condition admitting no smooth dec spacetime extension

Jonathan Glöckle

Abstract

There are two versions of the dominant energy condition (=dec): The original one for Lorentzian manifolds and an associated one for initial data sets. If a Lorentzian manifold satisfies dec, then so does the induced initial set on any embedded spacelike hypersurface. In this article, we discuss the question of a potential converse of this: Is every dec initial data set the induced one on a spacelike hypersurface within a suitably chosen dec Lorentzian manifold? We provide an example showing that in general the answer is no if we require all structures to be smooth.

Initial data sets with dominant energy condition admitting no smooth dec spacetime extension

Abstract

There are two versions of the dominant energy condition (=dec): The original one for Lorentzian manifolds and an associated one for initial data sets. If a Lorentzian manifold satisfies dec, then so does the induced initial set on any embedded spacelike hypersurface. In this article, we discuss the question of a potential converse of this: Is every dec initial data set the induced one on a spacelike hypersurface within a suitably chosen dec Lorentzian manifold? We provide an example showing that in general the answer is no if we require all structures to be smooth.
Paper Structure (2 sections, 3 theorems, 16 equations)

This paper contains 2 sections, 3 theorems, 16 equations.

Table of Contents

  1. Acknowledgements
  2. Proofs of the results

Key Result

Proposition 2

If an initial data set $(g,k)$ on a manifold $M$ satisfies $\rho > |j|_g$, then there is a dec spacetime $(\overline{M}, \overline{g})$ containing $M$ as spacelike hypersurface with induced initial data set $(g,k)$.

Theorems & Definitions (8)

  • Proposition 2: Gloeckle:2019
  • Lemma 3
  • Theorem 4: Main Theorem
  • proof : Proof of \ref{['Lem:Rigid']}
  • Remark 5
  • Remark 6
  • proof : Proof of \ref{['Thm:Main']}
  • Remark 7