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Kerr-Bertotti-Robinson Spacetime and the Kerr/CFT Correspondence

Haryanto M. Siahaan

Abstract

We construct the Kerr/CFT correspondence for extremal Kerr--Bertotti--Robinson (Kerr--BR) black holes, which are exact stationary solutions of the Einstein--Maxwell equations describing a rotating black hole immersed in a uniform Bertotti--Robinson electromagnetic universe. After reviewing the geometry, horizon structure, and thermodynamics of the Kerr--BR family, we demonstrate that the external field non-trivially modifies both the horizon positions and the extremality condition. For extremal configurations, the near-horizon limit yields a warped $\mathrm{AdS}_3$ geometry with an associated Maxwell field aligned to the $U(1)$ fibration. Imposing standard Kerr/CFT boundary conditions, the asymptotic symmetry algebra gives rise to a Virasoro algebra with central charge $c_L$ and left-moving temperature $T_L$ that depend explicitly on the external field strength. The Cardy formula then reproduces exactly the Bekenstein--Hawking entropy of the extremal Kerr--BR black hole for any admissible value of the Bertotti--Robinson field, thereby establishing a consistent Kerr/CFT dual description. Comparisons with the magnetized Melvin--Kerr and Kaluza--Klein black holes are briefly discussed, highlighting qualitative differences in their curvature profiles and horizon geometries.

Kerr-Bertotti-Robinson Spacetime and the Kerr/CFT Correspondence

Abstract

We construct the Kerr/CFT correspondence for extremal Kerr--Bertotti--Robinson (Kerr--BR) black holes, which are exact stationary solutions of the Einstein--Maxwell equations describing a rotating black hole immersed in a uniform Bertotti--Robinson electromagnetic universe. After reviewing the geometry, horizon structure, and thermodynamics of the Kerr--BR family, we demonstrate that the external field non-trivially modifies both the horizon positions and the extremality condition. For extremal configurations, the near-horizon limit yields a warped geometry with an associated Maxwell field aligned to the fibration. Imposing standard Kerr/CFT boundary conditions, the asymptotic symmetry algebra gives rise to a Virasoro algebra with central charge and left-moving temperature that depend explicitly on the external field strength. The Cardy formula then reproduces exactly the Bekenstein--Hawking entropy of the extremal Kerr--BR black hole for any admissible value of the Bertotti--Robinson field, thereby establishing a consistent Kerr/CFT dual description. Comparisons with the magnetized Melvin--Kerr and Kaluza--Klein black holes are briefly discussed, highlighting qualitative differences in their curvature profiles and horizon geometries.

Paper Structure

This paper contains 5 sections, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kerr--Bertotti--Robinson geometry
  3. Near horizon geometry of extremal Kerr--BR black holes and Cardy entropy
  4. Conclusion
  5. Equatorial squared Riemann tensor

Figures (2)

  • Figure 2.1: Equatorial dimensionless squared Riemann tensor $K^\ast = M^4 R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ in the Kerr--BR spacetime for $a=0.5 M$. The dimensionless radius is $r^\ast = r/M$ and the outer horizon is located at $r_+^\ast \simeq 1.87$. The solid, dashed, and dash-dotted curves represent the cases $BM=0.1$, $BM=0.2$, and $BM=0.3$, respectively.
  • Figure 2.2: Equatorial dimensionless squared Riemann tensor $K^\ast$ for the Melvin--Kerr geometry with $a=0.5$. The three curves represent the same magnetizations as in Fig. \ref{['fig:KBR-Kplots']}, allowing a direct comparison between Kerr--BR and Melvin--Kerr backgrounds. The solid, dashed, and dash-dotted curves represent the cases $BM=0.1$, $BM=0.2$, and $BM=0.3$, respectively.