Hidden Conformal Symmetry of the Kerr Black Hole (1004.0996v1)

Abstract: Extreme and very-near-extreme spin J Kerr black holes have been conjectured to be holographically dual to two-dimensional (2D) conformal field theories (CFTs) with left and right central charges c_L=c_R=12J. In this paper it is observed that the 2D conformal symmetry of the scalar wave equation at low frequencies persists for generic non-extreme values of the mass M. Interestingly, this conformal symmetry is not derived from a conformal symmetry of the spacetime geometry except in the extreme limit. The periodic identification of the azimuthal angle is shown to correspond to a spontaneous breaking of the conformal symmetry by left and right temperatures (T_L,T_R). The well-known low-frequency scalar-Kerr scattering amplitudes coincide with correlators of a 2D CFT at these temperatures. Moreover the CFT microstate degeneracy inferred from the Cardy formula agrees exactly with the Bekenstein-Hawking area law for all M and J. These observations provide evidence for the conjecture that the Kerr black hole is dual to a c_L=c_R=12J 2D CFT at temperatures (T_L,T_R) for every value of M and J.

• The paper uncovers that non-extreme Kerr black holes show hidden conformal symmetry in low-frequency scalar wave equations.
• It employs an analysis of SL(2,R) × SL(2,R) symmetries to tie black hole parameters with dual 2D CFT central charges, c_R = c_L = 12J.
• These findings refine the Kerr/CFT correspondence and provide actionable insights for gravitational wave research and quantum gravity theories.

An Overview of "Hidden Conformal Symmetry of the Kerr Black Hole"

The paper "Hidden Conformal Symmetry of the Kerr Black Hole" by Alejandra Castro, Alexander Maloney, and Andrew Strominger makes significant observations related to the double copy of the Kerr/CFT (Conformal Field Theory) correspondence for Kerr black holes. This research explores the notion that even for generic non-extreme Kerr black holes, outside the extreme limit, a hidden conformal symmetry is present in the solution space of scalar wave equations at low frequencies, despite not being a symmetry of the spacetime geometry.

Key Observations and Conjectures

1. Duality with 2D CFTs: The paper presents evidence that Kerr black holes, particularly near the extreme spin state, can be conjectured as holographically dual to two-dimensional CFTs. The key claim is that the central charges $c_R = c_L = 12J$, capturing a notion that aligns microstate degeneracy from the Cardy formula with the entropy expressions from the Bekenstein-Hawking area law.
2. Hidden Conformal Symmetry: The investigation reveals that the low-frequency persistence of conformal symmetry in scalar wave equations does not solely derive from the geometry in non-extreme cases. This unearths a potential for a match between known low-frequency Kerr scattering amplitudes and CFT correlators at specific temperatures, driven predominantly by the periodicity in the azimuthal angle and the spontaneous breaking of symmetry.
3. SL(2,R) × SL(2,R) Symmetry: The authors highlight an SL(2,R)×SL(2,R) symmetry, with vector fields satisfying the Casimir and Virasoro algebra, underpinning the conformal symmetry in the wave equation’s solution space. This symmetry is disrupted globally by the azimuthal angle’s periodic identification, suggesting that finite temperatures $T_L$ and $T_R$ are crucial elements in the dual CFT picture.

Implications and Future Directions

The paper proposes that understanding Kerr black holes' dynamics, even beyond extreme conditions, can benefit from CFT approaches, particularly emphasizing the need for further exploration of conformal symmetries not visible in spacetime geometry. The implications extend to refining the Kerr/CFT conjecture and elucidating microstate structure via two-dimensional conformal fields.

1. Practical Implications: The alignment of low-frequency scattering results with 2D CFT correlators suggests practical methodologies in gravitational wave research and potential implementations in black hole thermodynamics.
2. Theoretical Insights: By bridging the symmetry disparities via the solution space, the research may advance quantum gravity theories. It pushes towards a concrete CFT dual construct for non-extreme black chronicling, potentially enriching D-brane and string theory approaches in holography.
3. Speculation on AI's Role: As the understanding of complex systems grows, AI methodologies might become integral in modeling such high-dimensional dualities and optimizing computations involved in verifying the hidden symmetries and corresponding CFT properties.

Conclusion

This paper provides a robust speculative framework for further understanding the Kerr black holes through conformal symmetry, connecting to 2D CFTs beyond merely extreme conditions. While definitive systematic derivations are pending, the evidence and approach illuminate a cohesive conjecture deserving deeper investigation and may very well lay the groundwork for future breakthroughs in black hole physics and quantum field correspondence.