Some Comments on Gravitational Entropy and the Inverse Mean Curvature Flow

TL;DR

This paper extends the positive energy theorem approach to derive bounds on the mass and entropy of black holes in various spacetime topologies, highlighting the significance of the function |Z(Q,P)| and potential higher-dimensional generalizations.

Contribution

It introduces new lower bounds for mass and entropy in asymptotically Anti-de-Sitter spacetimes and black hole initial data, extending previous methods beyond static spherically symmetric cases.

Findings

Lower bounds for mass in exotic topologies with cosmological constant

Lower bounds for black hole horizon area and entropy

Extension of attractor behavior beyond static spherically symmetric black holes

Abstract

The Geroch-Wald-Jang-Huisken-Ilmanen approach to the positive energy problem to may be extended to give a negative lower bound for the mass of asymptotically Anti-de-Sitter spacetimes containing horizons with exotic topologies having ends or infinities of the form $\Sigma_g \times {\Bbb R}$, in terms of the cosmological constant. We also show how the method gives a lower bound for for the mass of time-symmetric initial data sets for black holes with vectors and scalars in terms of the mass, $|Z(Q,P)|$ of the double extreme black hole with the same charges. I also give a lower bound for the area of an apparent horizon, and hence a lower bound for the entropy in terms of the same function $|Z(Q,P)|$. This shows that the so-called attractor behaviour extends beyond the static spherically symmetric case. and underscores the general importance of the function $|Z(Q,P)|$. There are hints that higher dimensional generalizations may involve the Yamabe conjectures.