Charged and rotating near-horizon geometries in five dimensions

Technical Summary: Charged and Rotating Near-Horizon Geometries in Five Dimensions

Problem Statement
The classification of extremal black hole horizons in five-dimensional Einstein-Maxwell theory remains incomplete. While the full spacetime solutions for charged, rotating black holes with two independent angular momenta are not known analytically (existing only numerically or perturbatively), the near-horizon geometry of such extremal horizons offers a more tractable problem due to enhanced $SO(2,1)$ symmetry. Previous analytic results were limited to special cases: vacuum Myers-Perry horizons, static charged horizons in background electric fields, or homogeneous horizons where angular momenta are equal. A general analytic solution for a charged, rotating near-horizon geometry with two independent angular momenta in pure Einstein-Maxwell theory (zero Chern-Simons coupling) was previously unknown. Furthermore, while numerical constructions by Horowitz and Santos suggested the existence of such solutions and hinted at specific entropy relations, closed analytic forms were lacking.

Methodology
The authors work within five-dimensional Einstein-Maxwell-Chern-Simons theory, defined by an action with a coupling constant $\lambda$. They focus on extremal horizons, utilizing the near-horizon formalism where the geometry is determined by data on a compact cross-section $H$.

1. Reduction of Equations: The authors first reduce the near-horizon equations of motion for $U(1) \times U(1)$ symmetric geometries to a system of three coupled second-order ordinary differential equations (ODEs) for the functions $|K|^2$, $\Gamma$, and the co-rotating electric field $\chi$, subject to a first-order constraint.
2. Parameter Counting: By analyzing the boundary conditions and the structure of the ODEs, they establish that any smooth family of solutions on a spherical cross-section can depend on at most five continuous parameters.
3. Ansatz: To find explicit solutions, they impose the condition that the co-rotating electric field $\chi$ is a non-zero constant. This assumption simplifies the system significantly, allowing for the integration of the ODEs.
4. Sasakian Structure: The authors identify an underlying Sasakian structure on the spherical cross-sections of solutions with constant non-zero co-rotating electric fields. They utilize this geometric structure to prove that such solutions must admit a second commuting Killing vector, thereby establishing a $U(1) \times U(1)$ isometry without assuming it a priori.
5. Generalization: The construction is extended to higher odd dimensions ($2n+3$) by starting from a $(2n+1)$-dimensional Sasaki-Einstein manifold.

Key Contributions and Results

• New Analytic Families: The paper presents new closed-form analytic solutions for charged and rotating near-horizon geometries in five dimensions.
  • Pure Einstein-Maxwell ($\lambda=0$): Two distinct three-parameter families are constructed.
    • Family 1: Connects to known static solutions (squashed horizons) and generalizes them to rotating cases.
    • Family 2: Connects to a specific two-parameter slice of vacuum solutions (Kaluza-Klein type) rather than the standard extremal Myers-Perry horizon.
  • Minimal Supergravity ($\lambda=1$): Two three-parameter families are identified. One has both static and vacuum limits; the other connects to static solutions.
  • General Coupling ($\lambda \neq 0, 1$): A three-parameter family of solutions is derived for arbitrary Chern-Simons coupling, consisting of multiple disjoint regions in parameter space.
• Entropy Relations: The new solutions satisfy simple entropy relations that match those predicted numerically for extremal charged Myers-Perry black holes. Specifically:
  • Family 1 satisfies $S \propto |Q|^{3/2} + |J_1 J_2|/|Q|^{3/2}$.
  • Family 2 satisfies $S \propto \sqrt{|J_1 J_2|}$.
  • Notably, Family 2 satisfies the entropy relation expected for charged Myers-Perry horizons despite not reducing to the Myers-Perry horizon in the vacuum limit.
• Classification Theorem: The authors prove Theorem 1: Any smooth five-dimensional rotating near-horizon geometry with a constant non-zero co-rotating electric field and compact horizon cross-sections is given by one of the solutions presented. This classification holds without assuming the existence of symmetries on the horizon cross-sections; the $U(1) \times U(1)$ symmetry is a derived consequence of the constant electric field condition and the Sasakian structure.
• Higher Dimensions: The construction is generalized to $(2n+3)$ dimensions, showing that any Sasaki-Einstein manifold generates a two-parameter family of charged and rotating horizons.

Significance and Claims
The paper claims to provide the first analytic description of charged, rotating near-horizon geometries with two independent angular momenta in pure Einstein-Maxwell theory. A central finding is the existence of solutions that satisfy the entropy relations of the conjectured extremal charged Myers-Perry black holes but are distinct from the Myers-Perry horizon in the vacuum limit. This suggests that the charged Myers-Perry horizon might belong to a more general family of solutions or that the entropy relation is a robust feature of a broader class of geometries.

The authors emphasize that their solutions are the most general rotating extremal horizons for which the co-rotating electric field is constant. They do not claim to have found the full analytic solution for the extremal charged Myers-Perry black hole itself, noting that their solutions do not reduce to it in the vacuum limit (except for specific subcases). Instead, they propose that their Family 1 may coincide with the numerical solutions found by Horowitz and Santos, while Family 2 might represent a different slice of a more general solution space.

The work also resolves a question regarding the existence of non-vacuum solutions with vanishing co-rotating electric field on spherical topologies, proving they do not exist. Finally, the identification of the Sasakian structure provides a powerful geometric tool for classifying these horizons and constructing higher-dimensional generalizations, extending previous vacuum constructions to the charged, rotating sector.