Hamiltonian evolution and quantization for extremal black holes

TL;DR

This paper explores two Hamiltonian quantization approaches for extremal black holes in spherically symmetric Einstein-Maxwell theory, analyzing their implications for black hole entropy and the classical limit.

Contribution

It introduces two distinct Hamiltonian quantization methods for extremal black holes, contrasting their effects on entropy and classical behavior.

Findings

Classical Hamiltonian with extremal boundary conditions yields zero entropy.

Quantization as a limit of nonextremal black holes supports Bekenstein-Hawking entropy.

Wave packet analysis shows entropy continuity in the extremal limit.

Abstract

We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein-Maxwell theory. First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only. The Hamiltonian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yields a vanishing black hole entropy. Second, we give a Hamiltonian quantization that incorporates extremal black holes as a limiting case of nonextremal ones, and examine the classical limit in terms of wave packets. The spreading of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus with the Bekenstein-Hawking entropy even for the extremal holes. The discussion takes place throughout within Lorentz-signature spacetimes.