Black Hole Entropy and Soft Hair (1810.01847v4)

Abstract: A set of infinitesimal ${\rm Virasoro_{\,L}}\otimes{\rm Virasoro_{\,R}}$ diffeomorphisms are presented which act non-trivially on the horizon of a generic Kerr black hole with spin J. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Integrability and associativity of the charge algebra are shown to require the inclusion of `Wald-Zoupas' counterterms. A counterterm satisfying the known consistency requirement is constructed and yields central charges $c_L=c_R=12J$. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the central charges reproduce the macroscopic area-entropy law for generic Kerr black holes.

• The paper demonstrates that Virasoro diffeomorphisms acting on Kerr black hole horizons yield a central charge c = 12J, underpinning the entropy-area law.
• The paper employs a covariant phase space formalism with Wald-Zoupas counterterms to derive integrable charges on the black hole horizon.
• The paper highlights that non-Abelian soft hair may explain black hole microstate degeneracy, bridging classical thermodynamics and quantum gravity.

Analysis of "Black Hole Entropy and Soft Hair"

The paper "Black Hole Entropy and Soft Hair" by Sasha Haco, Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger presents a comprehensive paper on the connection between black hole entropy, represented by the Bekenstein-Hawking area formula, and the concept of 'soft hair,' interpreted as low-energy excitations on the black hole horizon due to large diffeomorphisms. This paper explores the applicability of Virasoro symmetries emerging from these large diffeomorphisms on the horizon of Kerr black holes with spin $J$.

Main Contributions

1. Virasoro Algebra and Black Hole Horizon: The authors present a set of infinitesimal Virasoro $\otimes \mathbb{R}$ diffeomorphisms which act on the horizon of a generic Kerr black hole. Using a covariant phase space formalism, the paper derives a formulation for the Virasoro charges via surface integrals on the horizon. The introduction of the Wald-Zoupas counterterms ensures integrability and associativity of the charge algebra, yielding central charges expressed by $c = 12J$.
2. Quantum Hilbert Space and Cardy Formula: Assuming the existence of a quantum Hilbert space upon which these charges generate symmetries, the authors extrapolate the applicability of the Cardy formula. This allows the computation of the central charges, correlating the results with the macroscopic area-entropy law for Kerr black holes. The usage of this formalism provides a potential bridge between classical and quantum understandings of black hole entropy.
3. Implications of Soft Hair: The paper highlights the necessity for a richer type of soft hair beyond supertranslations, pointing towards non-Abelian large diffeomorphisms as a plausible explanation for black hole microstate degeneracy.
4. Central Charge and Black Hole Configuration: By identifying and computing the Wald-Zoupas boundary counterterm, the authors effectively remove obstructions such as non-integrability and violations of associativity, deducing that the central charge $c = 12J$. This development supports the hypothesis that hidden conformal symmetry could account for the leading degeneracy of black hole microstates.

Implications and Future Research

The theoretical implications of these results provide an elegant framework for extending our understanding of black hole thermodynamics with a quantum perspective. The notion that large diffeomorphism symmetries can potentially unveil a complete understanding of black hole entropy is compelling. However, future investigations will need to address the uniqueness of the counterterm and establish the completeness of the diffeomorphisms acting on black hole horizons. Furthermore, the paper leaves open the task of establishing the integrability of the charges and verifying their symmetry-generating capacity via Dirac brackets.

Speculative Outlook

The research contributes to ongoing efforts to tie together dark horizons of general relativity with the discrete spectra of quantum gravity approaches. Extending these results could produce novel insights into the holographic principle and gravitational anomalies. The explicit realization of these concepts may provide substantial evidence towards understanding the unitarity of black hole evaporation, potentially aiding in resolving the information paradox. Future work could explore these ideas in non-Kerr black holes or further generalize the foundational framework into higher-dimensional gravitational theories.

In conclusion, this paper represents an important advance in the paper of black hole physics by elucidating the significance of large diffeomorphisms and their connection to black hole entropy through the lens of quantum field theory and general relativity. Such insights pave the way for new theoretical developments in understanding one of the most enigmatic objects in the universe.