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

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.

💡 Research Summary

This paper establishes the Kerr/CFT correspondence for extremal Kerr-Bertotti-Robinson (Kerr-BR) black holes. The Kerr-BR spacetime is an exact, stationary solution to the Einstein-Maxwell equations describing a rotating black hole immersed in a uniform Bertotti-Robinson electromagnetic universe, which is a product space AdS2 x S2 supported by a homogeneous electromagnetic field.

The work begins by thoroughly reviewing the geometry, horizon structure, and thermodynamics of the Kerr-BR family. A key finding is that the external Bertotti-Robinson field parameter B non-trivially modifies both the positions of the event horizons and the condition for extremality (where the inner and outer horizons merge), unlike in the magnetized Melvin-Kerr case where the horizon structure of the naked Kerr solution is preserved. The authors compute the black hole entropy and Hawking temperature, which depend on B.

The core achievement lies in the analysis of extremal configurations. By taking the near-horizon limit of an extremal Kerr-BR black hole, the authors obtain a warped AdS3 geometry with an SL(2,R) x U(1) isometry group, generalizing the Near-Horizon Extreme Kerr (NHEK) geometry. They then impose the standard Kerr/CFT boundary conditions on this background. The analysis of the asymptotic symmetry group reveals a Virasoro algebra. The central charge c_L and the left-moving Frolov-Thorne temperature T_L of this algebra are computed explicitly and are shown to depend on the external field strength B.

Finally, the Cardy formula for the entropy of a two-dimensional conformal field theory, S = (π^2/3) c_L T_L, is applied. Strikingly, this formula reproduces exactly the Bekenstein-Hawking entropy of the extremal Kerr-BR black hole for any admissible value of B. This successful matching establishes a consistent holographic dual description for these black holes within the Kerr/CFT framework, demonstrating its robustness even in non-asymptotically flat, homogeneous electromagnetic backgrounds.

The paper also includes a comparative study of curvature profiles, calculating the Kretschmann scalar for Kerr-BR and Melvin-Kerr spacetimes. This highlights qualitative geometric differences induced by the distinct nature of the external fields, such as the appearance of regions with negative curvature invariant near the horizon in Kerr-BR, suggesting an “hour-glass” shape deformation.