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We find necessary and sufficient conditions for existence of a locallyisometric embedding of a vacuum space-time into a conformally-flat 5-space. Weexplicitly construct such embeddings for any spherically symmetric Lorentzianmetric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For theSchwarzschild metric the embedding is global, and extends through the horizonall the way to the $r=0$ singularity. We discuss the asymptotic properties ofthe embedding in the context of Penrose's theorem on Schwarzschild causality.We finally show that the Hawking temperature of the Schwarzschild metric agreeswith the Unruh temperature measured by an observer moving along hyperbolae in$R^{4, 1}$.

Conformally isometric embeddings and Hawking temperature