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Macroscopic and Microscopic Description of Black Diholes

Roberto Emparan, Edward Teo

TL;DR

This paper constructs and analyzes a class of four-dimensional dihole solutions: pairs of non-extremal black holes with equal mass and opposite charges, in Einstein–Maxwell theory and its dilatonic and U(1)^4 extensions. It provides explicit macroscopic descriptions in Weyl coordinates, computes simple closed-form expressions for areas, charges, temperatures, and interaction energies, and connects these configurations to string/M-theory via an effective string model. In the extremal and near-extremal regimes, the authors derive a microscopic entropy that matches the Bekenstein–Hawking result and exhibits the interaction between the antiparallel string/anti-string, offering insights into non-supersymmetric black hole microphysics. They also discuss limitations due to conical defects and outline future work on balanced configurations, rotation, and broader brane configurations.

Abstract

We study configurations consisting of a pair of non-extremal black holes in four dimensions, both with the same mass, and with charges of the same magnitude but opposite sign---diholes, for short. We present such exact solutions for Einstein-Maxwell theory with arbitrary dilaton coupling, and also solutions to the U(1)^4 theories that arise from compactified string/M-theory. Despite the fact that the solutions are very complicated, physical properties of these black holes, such as their area, charge, and interaction energy, admit simple expressions. We also succeed in providing a microscopic description of the entropy of these black holes using the `effective string' model, and taking into account the interaction between the effective string and anti-string.

Macroscopic and Microscopic Description of Black Diholes

TL;DR

This paper constructs and analyzes a class of four-dimensional dihole solutions: pairs of non-extremal black holes with equal mass and opposite charges, in Einstein–Maxwell theory and its dilatonic and U(1)^4 extensions. It provides explicit macroscopic descriptions in Weyl coordinates, computes simple closed-form expressions for areas, charges, temperatures, and interaction energies, and connects these configurations to string/M-theory via an effective string model. In the extremal and near-extremal regimes, the authors derive a microscopic entropy that matches the Bekenstein–Hawking result and exhibits the interaction between the antiparallel string/anti-string, offering insights into non-supersymmetric black hole microphysics. They also discuss limitations due to conical defects and outline future work on balanced configurations, rotation, and broader brane configurations.

Abstract

We study configurations consisting of a pair of non-extremal black holes in four dimensions, both with the same mass, and with charges of the same magnitude but opposite sign---diholes, for short. We present such exact solutions for Einstein-Maxwell theory with arbitrary dilaton coupling, and also solutions to the U(1)^4 theories that arise from compactified string/M-theory. Despite the fact that the solutions are very complicated, physical properties of these black holes, such as their area, charge, and interaction energy, admit simple expressions. We also succeed in providing a microscopic description of the entropy of these black holes using the `effective string' model, and taking into account the interaction between the effective string and anti-string.

Paper Structure

This paper contains 11 sections, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Black holes in Weyl coordinates
  3. The non-extremal dihole
  4. Physical properties of the dihole
  5. Dilatonic and string/M-theory diholes
  6. Dilatonic dihole solution
  7. $U(1)^4$ dihole solution
  8. Statistical mechanics of the dihole
  9. Energy and entropy of the extremal dihole
  10. Microscopic entropy of the near-extremal dihole
  11. Conclusions and outlook

Figures (3)

  • Figure 1: Structure of the axis, and interpretation of $R_\pm$, $r_\pm$, for the Weyl solutions corresponding to (a) Reissner-Nordström, (b) Israel-Khan neutral two-black hole solution. The black hole horizons correspond to the 'rods' marked by the bold lines.
  • Figure 2: The extremal dihole solution in Weyl coordinates. The extremal black hole horizons correspond now to the black dots.
  • Figure 3: Structure of the axis, and interpretation of $R_\pm$, $r_\pm$, (\ref{['rpm']}), and $\kappa_\pm$. The black hole horizons correspond to the 'rods' marked by the bold lines.