Conformally isometric embeddings and Hawking temperature

Abstract

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 $R^{4,1}$. 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 $R^{4,1}$.

arXiv commentFinal version, to appear in Classical and Quantum Gravity

References in corpus (4)

• Conformal Invariance of Black Hole Temperature
• Mapping Hawking into Unruh Thermal Properties
• Embeddings for Schwarzschild metric: classification and new results
• Some examples of the behaviour of conformal geodesics

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• Bridging Dimensions: General Embedding Algorithm and Field-Theory Reconstruction in 5D Braneworld Models
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• On Schwarzschild causality in higher dimensions
• Conformal and isometric embeddings of gravitational instantons
• Explicit isometric embeddings of collapsing dust ball