Extremal limits and black hole entropy

TL;DR

This paper investigates the extremal limit of Reissner-Nordstr"om black holes, revealing a discontinuous transition where the near-horizon geometry approaches an $AdS_2 imes S^2$ space, offering insights into extremal black hole entropy.

Contribution

It demonstrates that the extremal limit of non-extremal black holes results in a split spacetime, providing a new perspective on the entropy of extremal black holes.

Findings

The extremal limit leads to a disconnected $AdS_2 imes S^2$ region.

The transition to extremality is discontinuous.

This behavior may shed light on extremal black hole entropy.

Abstract

Taking the extremal limit of a non-extremal Reissner-Nordstr\"om black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate -- while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of $AdS_2\times S^2$. In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordstr\"om solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.