Notes on the Area Theorem

TL;DR

This paper investigates the behavior of inner and outer horizons of various black holes under quasi-stationary processes, revealing that while outer horizons' areas never decrease, inner horizons can shrink, leading to a generalized area theorem.

Contribution

It extends the understanding of black hole horizon dynamics by analyzing both inner and outer horizons across different black hole solutions, including exotic and higher derivative gravities.

Findings

Outer horizon area never decreases during quasi-stationary processes.

Inner horizon area can decrease, contrary to traditional expectations.

A generalized area theorem combining inner and outer horizon behaviors.

Abstract

Hawking's area theorem can be understood from a quasi-stationary process in which a black hole accretes $positive$ energy matter, ``independent of the details of the gravity action''. I use this process to study the dynamics of the ``inner'' as well as the outer horizons for various black holes which include the recently discovered exotic black holes and three-dimensional black holes in higher derivative gravities as well as the usual BTZ black hole and the Kerr black hole in four dimensions. I find that the area for the inner horizon ``can decrease'', rather than increase, with the quasi-stationary process. However, I find that the area for the outer horizon ``never decrease'' such as the usual area theorem still works in our examples, though this is quite non-trivial in general. There exist the instability of the inner horizons and the connected effects of ``mass inflation'' but, according to more detailed analysis, it seems that the instability is not important in my analysis. I also find a ``generalized'' area theorem by combining those of the outer and inner horizons.