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Kerr-CFT and gravitational perturbations (0906.2380v1)

Published 12 Jun 2009 in hep-th

Abstract: Motivated by the Kerr-CFT conjecture, we investigate perturbations of the near-horizon extreme Kerr spacetime. The Teukolsky equation for a massless field of arbitrary spin is solved. Solutions fall into two classes: normal modes and traveling waves. Imposing suitable (outgoing) boundary conditions, we find that there are no unstable modes. The explicit form of metric perturbations is obtained using the Hertz potential formalism, and compared with the Kerr-CFT boundary conditions. The energy and angular momentum associated with scalar field and gravitational normal modes are calculated. The energy is positive in all cases. The behaviour of second order perturbations is discussed.

Citations (168)

Summary

Kerr-CFT and Gravitational Perturbations

This paper presents a comprehensive paper of perturbations in the near-horizon extreme Kerr (NHEK) spacetime, motivated by the Kerr-CFT conjecture. The authors investigate the Teukolsky equation for massless fields of arbitrary spin and categorize the solutions into normal modes and traveling waves. Imposing suitable outgoing boundary conditions, they find that the NHEK geometry is stable against linearized gravitational perturbations, as no unstable modes exist.

The authors utilize the Hertz potential formalism to obtain the explicit form of metric perturbations, comparing these with the Kerr-CFT boundary conditions. They calculate the energy and angular momentum associated with both scalar field and gravitational normal modes, demonstrating that the energy is positive in all cases studied. The work discusses the implications of second-order perturbations, suggesting that while NHEK is stable at the linearized level, nonlinear interactions may introduce complexities.

Key Findings

  1. Stability of NHEK: The paper concludes that NHEK is stable against linearized gravitational perturbations, meaning it does not exhibit instability due to ergoregion-induced negative energy phenomena.
  2. Boundary Conditions: The authors use outgoing boundary conditions at infinity, which converts some traveling wave modes into quasinormal modes. These signal the decay of perturbations via radiation to infinity.
  3. Energy Calculations: The energy, calculated using the Landau-Lifshitz pseudotensor, remains positive for a wide range of normal modes. This supports the validity of the Kerr-CFT conjecture which imposes zero-energy conditions for consistency.
  4. Second Order Perturbations: At second order, the characteristics of solutions become complex, particularly in relation to the Kerr-CFT boundary conditions. This highlights potential nonlinear interactions that may demand further exploration.

Implications and Speculations

The paper's findings provide robust evidence that supports the stability of the NHEK geometry. This stability is crucial for theories that equate quantum gravity in the NHEK geometry with a chiral CFT. However, challenges remain when considering nonlinear perturbations which could breach prescribed boundary conditions. Future work could investigate such nonlinear phenomena and explore whether a consistent framework exists that reconciles all perturbative levels.

The positive energy results bolster the Kerr-CFT conjecture and its implications for holographic duality scenarios, potentially extending such duality frameworks to other extremal black holes. Also, this work opens avenues for further paper into gravitational perturbations in type D spacetimes beyond the Kerr context.

In terms of future directions, it would be prudent to explore the interactions of normal modes that remain within the asymptotic NHEK geometry under nonlinear regimes, considering implications for both classical general relativity and quantum gravity frameworks. Such research could unveil whether Kerr-CFT predictions hold when subjected to complex perturbative landscapes, enhancing the understanding of black hole entropy and holographic principles.

In summary, this paper provides a detailed analysis of the perturbative stability of NHEK geometry, contributing to the broader discourse on the Kerr-CFT conjecture and its theoretical underpinning in gravitational physics.