Extended Symmetries at the Black Hole Horizon (1607.05703v2)

Abstract: We prove that non-extremal black holes in four-dimensional general relativity exhibit an infinite-dimensional symmetry in their near horizon region. By prescribing a physically sensible set of boundary conditions at the horizon, we derive the algebra of asymptotic Killing vectors, which is shown to be infinite-dimensional and includes, in particular, two sets of supertranslations and two mutually commuting copies of the Virasoro algebra. We define the surface charges associated to the asymptotic diffeomorphisms that preserve the boundary conditions and discuss the subtleties of this definition, such as the integrability conditions and the correct definition of the Dirac brackets. When evaluated on the stationary solutions, the only non-vanishing charges are the zero-modes. One of them reproduces the Bekenstein-Hawking entropy of Kerr black holes. We also study the extremal limit, recovering the NHEK geometry. In this singular case, where the algebra of charges and the integrability conditions get modified, we find that the computation of the zero-modes correctly reproduces the black hole entropy. Furthermore, we analyze the case of three spacetime dimensions, in which the integrability conditions notably simplify and the field equations can be solved analytically to produce a family of exact solutions that realize the boundary conditions explicitly. We examine other features, such as the form of the algebra in the extremal limit and the relation to other works in the literature.

Extended Symmetries at the Black Hole Horizon

This paper provides a thorough examination of the extended symmetries at the black hole horizon within the framework of four-dimensional general relativity. Authors Laura Donnay, Gaston Giribet, Hernán A. González, and Miguel Pino, present a detailed analysis of asymptotic symmetries and surface charges associated with non-extremal and extremal black holes, offering insights into infinite-dimensional symmetry groups at the horizon and their implications.

Overview of Main Contributions

The authors begin by establishing a physically reasonable set of boundary conditions at the black hole horizon. They demonstrate that non-extremal black holes possess an infinite-dimensional algebra of asymptotic Killing vectors, comprising two sets of supertranslations and two copies of the Virasoro algebra. Surface charges linked to these asymptotic symmetries are carefully defined, with particular attention given to integrability conditions and the proper formulation of Dirac brackets. The computations for stationary solutions reveal that zero modes are significant, encapsulating the black hole entropy.

Further analysis extends into the extremal limit, recovering the Near-Horizon-Extremal-Kerr (NHEK) geometry. Authors identify modifications to the algebra of charges and integrability conditions in this singular limit, ensuring that zero-mode calculations accurately represent black hole entropy once more. A comparative paper in three-dimensional spacetime illustrates a different simplification of integrability conditions, allowing the analytical solution of field equations to generate exact solutions aligning with specified boundary conditions.

Technical Highlights and Claims

The paper presents several strong numerical results and bold claims:

1. Infinite-Dimensional Symmetries:
   • In the vicinity of the horizon, non-extremal black holes exhibit an algebra that includes two Virasoro algebras and two sets of supertranslations.
2. Surface Charges and Black Hole Entropy:
   • Calculations demonstrate that zero modes of surface charges embody the Bekenstein-Hawking entropy in stationary black holes.
3. Extremal Limit Validity:
   • Despite altered algebra structures in the extremal limit, methods provide consistent results for entropy calculations.
4. Three-Dimensional Solutions:
   • Authors introduce a complete family of exact solutions in three-dimensional spacetime, fulfilling near-horizon boundary conditions.

Implications and Future Directions

The paper's implications impact both practical and theoretical realms, illuminating potential paths for resolving the information paradox and contributing to the discourse on black hole entropy in fundamental physics. From the practical standpoint, articulating the role of entropy and symmetry at horizons enhances understanding of black hole thermodynamics and quantum mechanics interaction.

Furthermore, the theoretical framework established here initiates possibilities for exploring new boundary conditions and asymptotic algebra configurations. Such explorations could foster deeper insights into the physics at null surfaces in various dimensional spacetimes. Future developments in AI and computational methods may afford advanced capabilities to simulate and test these theoretical models, enabling more profound and accurate predictions.

Finally, this research lays a foundational basis for further investigations into the symmetries of extremal configurations and their interplay within the Kerr/CFT correspondence, a prominent topic in contemporary gravitational physics research.

In essence, "Extended Symmetries at the Black Hole Horizon" presents an exhaustive paper of the profound structural symmetries surrounding black holes, paving the way for comprehensive understanding and novel inquiries into the intricate nature of spacetime and gravitational phenomena.