# Conformally isometric embeddings and Hawking temperature

## TL;DR

This paper establishes conditions for embedding vacuum spacetimes into conformally-flat 5-spaces, explicitly constructs such embeddings for spherically symmetric metrics, and relates Hawking and Unruh temperatures through these embeddings.

## Contribution

It provides necessary and sufficient conditions for conformally isometric embeddings and explicitly constructs global embeddings for Schwarzschild spacetime.

## Key findings

- Embedding exists for spherically symmetric vacuum spacetimes.
- The Schwarzschild embedding extends through the horizon to the singularity.
- Hawking temperature matches Unruh temperature in the embedding space.

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