Near-horizon symmetries of extremal black holes (0705.4214v3)

Abstract: Recent work has demonstrated an attractor mechanism for extremal rotating black holes subject to the assumption of a near-horizon SO(2,1) symmetry. We prove the existence of this symmetry for any extremal black hole with the same number of rotational symmetries as known four and five dimensional solutions (including black rings). The result is valid for a general two-derivative theory of gravity coupled to abelian vectors and uncharged scalars, allowing for a non-trivial scalar potential. We prove that it remains valid in the presence of higher-derivative corrections. We show that SO(2,1)-symmetric near-horizon solutions can be analytically continued to give SU(2)-symmetric black hole solutions. For example, the near-horizon limit of an extremal 5D Myers-Perry black hole is related by analytic continuation to a non-extremal cohomogeneity-1 Myers-Perry solution.

• The paper establishes that any extremal black hole with D-3 rotational symmetries exhibits a global SO(2,1) near-horizon symmetry under generic conditions.
• The paper demonstrates the stability of this symmetry by showing that it persists to all orders when higher-derivative corrections are included.
• The paper provides analytic examples with Myers-Perry and black ring solutions, highlighting practical implications for gravitational thermodynamics and microstate counting.

Overview of Near-Horizon Symmetries of Extremal Black Holes

The paper authored by Hari K. Kunduri, James Lucietti, and Harvey S. Reall provides a detailed examination of the near-horizon symmetries of extremal black holes. This paper is nestled in the broader investigation of the attractor mechanism within the framework of gravity theories that include both abelian vectors and uncharged scalars. The focus is distinctly on establishing the emergence of an SO(2,1) symmetry in the vicinity of the horizon of extremal black holes, a feature that extends established results traditionally limited to static, spherically symmetric cases.

Key Theorems and Results

1. Theorem 1: The authors prove that the near-horizon limit of any extremal black hole solution in a given gravitational theory with D-3 rotational symmetries invariably exhibits a global G_3× U(1)^D−3 symmetry, where G_3 is either SO(2,1) or the 2D Poincaré group. The latter possibility, however, is excluded under specific conditions related to the positivity of the scalar potential and toroidal horizon topology, reinforcing the pervasiveness of the SO(2,1) symmetry.
2. Theorem 2: A significant advancement is the stability of the SO(2,1) symmetry when higher-degree corrections to the theory are factored in. The authors contend that if an extremal black hole solution has a regular horizon with SO(2,1)×U(1)^D−3 symmetry at the zeroth-order in the coupling constant λ, this symmetry remains to all orders in λ, thus offering robustness against perturbative corrections.

Analytical Examples and Implications

The research further explores specific instances of five-dimensional black hole configurations, such as the Myers-Perry and black ring solutions, providing explicit calculations of their near-horizon geometries. These constructions are pivotal in showcasing the applicability and predictive power of the theoretical framework established. Notably, the developed methodology also offers pathways for the analytic continuation of near-horizon solutions, enriching the landscape with SU(2)-symmetric configurations such as in the case of Myers-Perry extensions.

Implications and Future Developments

The implications of this research are multi-faceted. Practically, understanding the symmetry structures near the horizons of extremal black holes has bearings on calculating entropy and other thermodynamic quantities. Theoretically, these findings fortify the linkages within string theory, especially concerning microstate descriptions and the interplay of higher-derivative corrections.

The paper also sparks speculation on various fronts for future exploration:

• Extension beyond the restriction of two rotational symmetries in five dimensions.
• Consideration of non-abelian vectors or charged scalars in similar models.
• A deeper exploration into 'small' black holes, where higher derivative terms play a crucial role in black hole formation.

Moreover, this paper resonates with ongoing debates surrounding the role of supersymmetry and the AdS/CFT correspondence, potentially catalyzing further developments in supergravity and higher-dimensional black hole configurations. The current findings generate a structured basis for addressing the stability and uniqueness of black hole solutions, with potential expansions into non-static or warping scenarios.

In conclusion, this paper furnishes essential insights into extremal black hole symmetries, systematically bridging gravitational theory with quantum computations of entropy, and challenges researchers to extend these boundaries in both conceptual and calculational paradigms.