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An extremization principle for the entropy of rotating BPS black holes in AdS$_5$

Seyed Morteza Hosseini, Kiril Hristov, Alberto Zaffaroni

TL;DR

This work identifies an entropy-extremization principle for a class of 1/16 BPS rotating black holes in AdS$_5\times S^5$, showing that the Bekenstein-Hawking entropy can be obtained from the Legendre transform of $E = - i \pi N^2 \frac{\Delta_1\Delta_2\Delta_3}{\omega_1\omega_2}$ under the constraint $\Delta_1+\Delta_2+\Delta_3+\omega_1+\omega_2 = 1$. The extremization reproduces the black hole entropy and reveals a formal similarity to anomaly polynomials and the supersymmetric Casimir energy of ${\cal N}=4$ SYM, hinting at deep field-theory/gravity connections. In the special case of equal angular momenta, a dimensional reduction to four dimensions maps the problem to the well-understood attractor mechanism for static BPS black holes, providing a concrete link between five- and four-dimensional extremization. The results reinforce the idea that black hole entropy in AdS contexts can be captured by extremization principles tied to holographic anomalies and may illuminate the five-dimensional attractor structure for rotating solutions.

Abstract

We show that the Bekenstein-Hawking entropy of a class of BPS electrically charged rotating black holes in AdS$_5\times S^5$ can be obtained by a simple extremization principle. We expect that this extremization corresponds to the attractor mechanism for BPS rotating black holes in five-dimensional gauged supergravity, which is still unknown. The expression to be extremized has a suggestive resemblance to anomaly polynomials and the supersymmetric Casimir energy recently studied for $\mathcal{N}=4$ super Yang-Mills.

An extremization principle for the entropy of rotating BPS black holes in AdS$_5$

TL;DR

This work identifies an entropy-extremization principle for a class of 1/16 BPS rotating black holes in AdS, showing that the Bekenstein-Hawking entropy can be obtained from the Legendre transform of under the constraint . The extremization reproduces the black hole entropy and reveals a formal similarity to anomaly polynomials and the supersymmetric Casimir energy of SYM, hinting at deep field-theory/gravity connections. In the special case of equal angular momenta, a dimensional reduction to four dimensions maps the problem to the well-understood attractor mechanism for static BPS black holes, providing a concrete link between five- and four-dimensional extremization. The results reinforce the idea that black hole entropy in AdS contexts can be captured by extremization principles tied to holographic anomalies and may illuminate the five-dimensional attractor structure for rotating solutions.

Abstract

We show that the Bekenstein-Hawking entropy of a class of BPS electrically charged rotating black holes in AdS can be obtained by a simple extremization principle. We expect that this extremization corresponds to the attractor mechanism for BPS rotating black holes in five-dimensional gauged supergravity, which is still unknown. The expression to be extremized has a suggestive resemblance to anomaly polynomials and the supersymmetric Casimir energy recently studied for super Yang-Mills.

Paper Structure

This paper contains 14 sections, 121 equations.

Table of Contents

  1. Introduction
  2. Supersymmetric AdS$_5$ black holes in $\mathrm{U}(1)^3$ gauged supergravity
  3. The asymptotic AdS$_5$ vacuum
  4. Properties of the solution
  5. An extremization principle for the entropy
  6. Dimensional reduction in the limiting case: $J_\phi = J_\psi$
  7. The near-horizon geometry
  8. Dimensional reduction on the Hopf fibres of squashed $S^3$
  9. Attractor mechanism in four dimensions
  10. Comparison with five-dimensional extremization
  11. Discussion and future directions
  12. Five-dimensional $\mathcal{N} = 2$ gauged supergravity
  13. Four-dimensional $\mathcal{N} = 2$ gauged supergravity
  14. Generalities about the supersymmetric Casimir energy