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A Universality Theorem for the Quantum Thermodynamics of Near-Extremal Black Holes

Leopoldo A. Pando Zayas, Jingchao Zhang

Abstract

We prove that the one-loop contribution from tensor modes to the thermodynamic entropy of near-extremal black holes is universal. Our proof applies to asymptotically flat, Anti-de-Sitter and de-Sitter black holes; it also covers spherical, axial and planar symmetries. We consider black hole configurations with and without matter sectors and explicitly discuss Abelian gauge fields and neutral scalar fields with arbitrary potential. We demonstrate that under certain conditions, the thermodynamics of near-extremal black holes contains a one-loop contribution from the tensor modes that equals $\frac{3}{2}\log (T_{\rm Hawking}/T_q)$. The proof of this theorem also shows explicitly how the Schwarzian modes appear universally in near-extremal geometries in dimensions four, five and six. We apply this theorem to Kerr-de-Sitter black holes as an explicit example.

A Universality Theorem for the Quantum Thermodynamics of Near-Extremal Black Holes

Abstract

We prove that the one-loop contribution from tensor modes to the thermodynamic entropy of near-extremal black holes is universal. Our proof applies to asymptotically flat, Anti-de-Sitter and de-Sitter black holes; it also covers spherical, axial and planar symmetries. We consider black hole configurations with and without matter sectors and explicitly discuss Abelian gauge fields and neutral scalar fields with arbitrary potential. We demonstrate that under certain conditions, the thermodynamics of near-extremal black holes contains a one-loop contribution from the tensor modes that equals . The proof of this theorem also shows explicitly how the Schwarzian modes appear universally in near-extremal geometries in dimensions four, five and six. We apply this theorem to Kerr-de-Sitter black holes as an explicit example.
Paper Structure (24 sections, 5 theorems, 124 equations, 1 figure)

This paper contains 24 sections, 5 theorems, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of one-loop temperature corrections
  3. The near-horizon extremal configuration
  4. The background metric
  5. The matter sector
  6. The small temperature perturbation
  7. Tensor zero modes
  8. Existence of tensor zero modes
  9. Relation to the boundary curve reparameterization
  10. The cancellation and lifted eigenvalues
  11. The cancellation
  12. The lifted eigenvalues
  13. Planar symmetric black branes
  14. Kerr-dS$_4$ black holes as an example
  15. Near-cold black holes
  16. ...and 9 more sections

Key Result

Lemma 1

(Extended Kunduri-Lucietti-Reall Lemma 1) Assume that a black hole solution of the theory given by the action in Eq: Maxwell action is at quadratic extremality and satisfies Assumption ass: 1, the Euclidean near-horizon metric takes the form of where repeated $\{i,j\}$ follows the Einstein summation convention while $g_{mm}d\theta_m^2$ term only sums once over $m$.

Figures (1)

  • Figure 1: Phase space of Kerr-dS$_4$ black hole: We set $\ell=1$ in this plot. The arrows represent different ensemble choices, the red arrow is the choice used in Section \ref{['Sec: Near-cold']}.

Theorems & Definitions (21)

  • Definition 1
  • Remark 1
  • Lemma 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof
  • Remark 4
  • Remark 5
  • Definition 2
  • ...and 11 more