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Universal properties and the first law of black hole inner mechanics

Alejandra Castro, Maria J. Rodriguez

TL;DR

This work establishes two universal properties for black holes with non-spherical horizons in five dimensions: (1) the product of outer and inner horizon areas, $A_+A_-$, is independent of the mass across neutral, charged, and string-like solutions, with explicit expressions for each class, suggesting a mass-invariant structural character linked to charges; (2) a universal inner-horizon first law, $-dM = T_- dA_- / (4 G) - ( ext{potentials})$, together with a Smarr-type relation, holds for these solutions and mirrors the outer-horizon thermodynamics. These results motivate a microscopic entropy framework where $A_\± /(4 G_d) = S_R \,\pm\,S_L$ and $S_{R,L} = 2\pi\sqrt{N_{R,L}}$, pointing to a CFT_2-like decomposition even for black rings and strings. The findings extend holographic ideas to less symmetric geometries, offering a pathway to understanding ring entropy and inviting further exploration of inner-horizon thermodynamics in broader backgrounds such as de Sitter spacetimes.

Abstract

We show by explicit computations that the product of all the horizon areas is independent of the mass, regardless of the topology of the horizons. The universal character of this relation holds for all known five dimensional asymptotically flat black rings, and for black strings. This gives further evidence for the crucial role that the thermodynamic properties at each horizon play in understanding the entropy at the microscopic level. To this end we propose a "first law" for the inner Cauchy horizons of black holes. The validity of this formula, which seems to be universal, was explicitly checked in all cases.

Universal properties and the first law of black hole inner mechanics

TL;DR

This work establishes two universal properties for black holes with non-spherical horizons in five dimensions: (1) the product of outer and inner horizon areas, , is independent of the mass across neutral, charged, and string-like solutions, with explicit expressions for each class, suggesting a mass-invariant structural character linked to charges; (2) a universal inner-horizon first law, , together with a Smarr-type relation, holds for these solutions and mirrors the outer-horizon thermodynamics. These results motivate a microscopic entropy framework where and , pointing to a CFT_2-like decomposition even for black rings and strings. The findings extend holographic ideas to less symmetric geometries, offering a pathway to understanding ring entropy and inviting further exploration of inner-horizon thermodynamics in broader backgrounds such as de Sitter spacetimes.

Abstract

We show by explicit computations that the product of all the horizon areas is independent of the mass, regardless of the topology of the horizons. The universal character of this relation holds for all known five dimensional asymptotically flat black rings, and for black strings. This gives further evidence for the crucial role that the thermodynamic properties at each horizon play in understanding the entropy at the microscopic level. To this end we propose a "first law" for the inner Cauchy horizons of black holes. The validity of this formula, which seems to be universal, was explicitly checked in all cases.

Paper Structure

This paper contains 5 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Universality of $A_+A_-$
  4. First Law of black holes Inner mechanics
  5. Conclusions

Figures (1)

  • Figure 1: Phase diagram of black holes: $a_\pm$ vs. $j$ where $a_\pm=A_\pm/(16\pi M)$ and $j=J/M$. The solid curve shows the horizon area of the Kerr and $5d$ Myers-Perry (with $J_1=J_2=J$) black hole while the dashed line outlines the inner horizon area both as functions of angular momenta $j$ and fixed mass $M$. As $j$ increases the $a_-$ grows while $a_+$ decrease coinciding at the extremal value (points A, B).