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Conformally isometric embeddings and Hawking temperature

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We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in 3+1 dimensions as a hypersurface in 4+1 dimensional Minkowski space. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the r=0 singularity. We discuss the asymptotic properties of the embedding in the context of Penrose’s theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metric agrees with the Unruh temperature measured by an observer moving along hyperbolae in 4+1 dimensions. The talk is based on a joint work with Paul Tod arXiv:1812.05468

This talk is part of the DAMTP Friday GR Seminar series.

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