A tale of two horizons

Abstract

I revisit the fate of coinciding horizons and the volume between them in the extremal limit of spherically symmetric black holes in four space–time dimensions, focusing on the Schwarzschild – de Sitter black hole for concreteness. The two Killing horizons in the limit space–time that are traditionally identified with the limiting event horizons of the non-extremal black hole are shown to instead be generated by an enhanced symmetry of the near horizon geometry (NHG). This dismantles the interpretation of the four-volume between the horizons remaining finite in the extremal limit. The NHG is reinterpreted as a tangent space–time to the degenerate black hole horizon, and geometrical objects, such as Killing vectors and Killing horizons, are carefully mapped between the bulk and the NHG. The implications for extremal black hole entropy are then discussed.