A classification of near-horizon geometries of extremal vacuum black holes (0806.2051v3)

Abstract: We consider the near-horizon geometries of extremal, rotating black hole solutions of the vacuum Einstein equations, including a negative cosmological constant, in four and five dimensions. We assume the existence of one rotational symmetry in 4d, two commuting rotational symmetries in 5d and in both cases non-toroidal horizon topology. In 4d we determine the most general near-horizon geometry of such a black hole, and prove it is the same as the near-horizon limit of the extremal Kerr-AdS(4) black hole. In 5d, without a cosmological constant, we determine all possible near-horizon geometries of such black holes. We prove that the only possibilities are one family with a topologically S^1 X S^2 horizon and two distinct families with topologically S^3 horizons. The S^1 X S^2 family contains the near-horizon limit of the boosted extremal Kerr string and the extremal vacuum black ring. The first topologically spherical case is identical to the near-horizon limit of two different black hole solutions: the extremal Myers-Perry black hole and the slowly rotating extremal Kaluza-Klein (KK) black hole. The second topologically spherical case contains the near-horizon limit of the fast rotating extremal KK black hole. Finally, in 5d with a negative cosmological constant, we reduce the problem to solving a sixth-order non-linear ODE of one function. This allows us to recover the near-horizon limit of the known, topologically S^3, extremal rotating AdS(5) black hole. Further, we construct an approximate solution corresponding to the near-horizon geometry of a small, extremal AdS(5) black ring.

Overview of Near-Horizon Geometries of Extremal Vacuum Black Holes

The paper by Kunduri and Lucietti provides a comprehensive classification of near-horizon geometries associated with extremal, rotating black hole solutions within the framework of vacuum Einstein equations, considering both asymptotically flat and anti-de Sitter (AdS) spacetime scenarios in four and five dimensions. Their analysis is conducted under the assumption of specific symmetries and horizon topologies.

Key Results

Four-Dimensional Geometries:

The authors derive the most general near-horizon geometry for axisymmetric extremal black holes with S^2 horizon topology in four dimensions. Remarkably, they show that the derived geometry is isometric to the near-horizon limit of the extremal Kerr-AdS black hole, when a negative cosmological constant is included. When the cosmological constant is zero, the near-horizon geometry aligns with the extremal Kerr solution. This extends previous work on the subject, affirming the uniqueness of these solutions under the stated symmetries and conditions.

Five-Dimensional Geometries:

The paper extends to five-dimensional scenarios, segregating cases based on the presence of a cosmological constant. Without a cosmological constant, three distinct families of solutions are identified based on horizon topology: one with S^2 × S^1 horizon topology and two with S^3 topology. These families cover known solutions such as the boosted extremal Kerr string, the extremal black ring, and the extremal Myers-Perry black hole. The work successfully delineates these solutions using rotational symmetries and integrates existing knowledge of these exotic black hole solutions.

When a cosmological constant is introduced in five dimensions, the problem is simplified to solving a sixth-order non-linear ordinary differential equation (ODE). This solution regime recovers the near-horizon limits of certain known AdS_5 black hole solutions and suggests potential configurations for hypothetical extremal AdS_5 black rings.

Implications and Future Directions

The classification addresses theoretical aspects of black hole uniqueness and symmetry properties. The implications of this framework help refine the characterizations of black holes in higher-dimensional theories, including those in the context of string theory and the AdS/CFT correspondence. For instance, the paper provides insights into the uniqueness and consistency of extremal black hole solutions beyond the standard four-dimensional Kerr family, emphasizing the structural stability across planar and hyper-surface ambiguities in near-horizon spacetimes.

Additionally, the analysis employing differential topology and geometrical techniques offers a methodology applicable to broader gravitational settings, possibly extending to include charged and electrically coupled black holes. This work is instrumental in identifying whether new black hole solutions, particularly those involving intricate horizon topologies like black rings, can be constructed within known theoretical frameworks, further informing on their thermodynamical and quantum mechanical properties.

The paper hints at future avenues, notably investigating the existence and uniqueness of supersymmetric versus non-supersymmetric solutions, and extending the near-horizon geometry analysis to cases with additional fields or in alternative gravity theories. This highlights an expansive field in general relativity and theoretical physics, setting the groundwork for more complex explorations into black hole energetics and the role of higher-dimensional physics in shaping fundamental gravitational phenomena.