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One entropy function to rule them all

Kevin Goldstein, Rudra P. Jena

TL;DR

The work presents a unified entropy-function framework for extremal black holes and black rings across four and five dimensions by dimensional reduction of a 5D theory with gravity, abelian gauge fields, neutral scalars, and a Chern-Simons term to a gauge-invariant 4D action. By exploiting near-horizon symmetries such as $AdS_2\times S^2\times U(1)$ and $AdS_2\times U(1)^2$, the authors derive a single entropy function $\mathcal{E}$ whose extremization yields an effective potential $V_{eff}$ and attractor values for radii and scalars, with entropies matching 4D and 5D Bekenstein-Hawking results. Specialization to very special geometry (5D ${\cal N}=2$) provides explicit BPS and non-BPS attractor equations for both black rings and static black holes, expressed in terms of intersection numbers $C_{IJK}$ and charges, and reveals the 4D-5D lift structure. A generalized entropy function for less symmetric near-horizon geometries is also developed, reproducing known non-SUSY ring entropy and underscoring the robustness and universality of the approach for extremal horizon data.

Abstract

We study the entropy of extremal four dimensional black holes and five dimensional black holes and black rings is a unified framework using Sen's entropy function and dimensional reduction. The five dimensional black holes and black rings we consider project down to either static or stationary black holes in four dimensions. The analysis is done in the context of two derivative gravity coupled to abelian gauge fields and neutral scalar fields. We apply this formalism to various examples including $U(1)^3$ minimal supergravity.

One entropy function to rule them all

TL;DR

The work presents a unified entropy-function framework for extremal black holes and black rings across four and five dimensions by dimensional reduction of a 5D theory with gravity, abelian gauge fields, neutral scalars, and a Chern-Simons term to a gauge-invariant 4D action. By exploiting near-horizon symmetries such as and , the authors derive a single entropy function whose extremization yields an effective potential and attractor values for radii and scalars, with entropies matching 4D and 5D Bekenstein-Hawking results. Specialization to very special geometry (5D ) provides explicit BPS and non-BPS attractor equations for both black rings and static black holes, expressed in terms of intersection numbers and charges, and reveals the 4D-5D lift structure. A generalized entropy function for less symmetric near-horizon geometries is also developed, reproducing known non-SUSY ring entropy and underscoring the robustness and universality of the approach for extremal horizon data.

Abstract

We study the entropy of extremal four dimensional black holes and five dimensional black holes and black rings is a unified framework using Sen's entropy function and dimensional reduction. The five dimensional black holes and black rings we consider project down to either static or stationary black holes in four dimensions. The analysis is done in the context of two derivative gravity coupled to abelian gauge fields and neutral scalar fields. We apply this formalism to various examples including minimal supergravity.

Paper Structure

This paper contains 19 sections, 148 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Black thing entropy function and dimensional reduction
  3. Algebraic entropy function analysis
  4. Set up
  5. Charges
  6. Preliminary analysis
  7. Black rings
  8. Static 5-d black holes
  9. Very Special Geometry
  10. Black rings and very special geometry
  11. Static black holes and very special geometry
  12. Non-supersymmetric solutions of very special geometry
  13. General Entropy function
  14. Dimensional reduction
  15. Notes on Very Special Geometry
  16. ...and 4 more sections

Figures (2)

  • Figure 1: A Black ring away from the centre of Taub-NUT is projected down to a black hole and naked Kaluza-Klein magnetic monopole in four dimensions. The angular momentum carried in the compact dimension will translate to electric charge in four dimensions. An $AdS^2\times S^2 \times U(1)$ near horizon geometry will project down to $AdS^2\times S^2$. On the other hand, an $AdS^2 \times U(1)^2$ will go to $AdS^2\times U(1)$.
  • Figure 2: A black hole at the centre of Taub-NUT caries $NUT$ charge. Using the Hopf fibration it can be projected down to black hole carrying magnetic charge. A spherically symmetric black hole with near horizon geometry of $AdS^2\times S^3$ will project down to an $AdS^2\times S^2$. On the other hand, a rotating black hole with a $AdS^2 \times U(1)^2$ geometry will go to $AdS^2\times U(1)$.