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Beyond the veil: Inner horizon instability and holography

Vijay Balasubramanian, Thomas S. Levi

Abstract

We show that scalar perturbations of the eternal, rotating BTZ black hole should lead to an instability of the inner (Cauchy) horizon, preserving strong cosmic censorship. Because of backscattering from the geometry, plane wave modes have a divergent stress tensor at the event horizon, but suitable wavepackets avoid this difficulty, and are dominated at late times by quasinormal behavior. The wavepackets have cuts in the complexified coordinate plane that are controlled by requirements of continuity, single-valuedness and positive energy. Due to a focusing effect, regular wavepackets nevertheless have a divergent stress-energy at the inner horizon, signaling an instability. This instability, which is localized behind the event horizon, is detected holographically as a breakdown in the semiclassical computation of dual CFT expectation values in which the analytic behavior of wavepackets in the complexified coordinate plane plays an integral role. In the dual field theory, this is interpreted as an encoding of physics behind the horizon in the entanglement between otherwise independent CFTs.

Beyond the veil: Inner horizon instability and holography

Abstract

We show that scalar perturbations of the eternal, rotating BTZ black hole should lead to an instability of the inner (Cauchy) horizon, preserving strong cosmic censorship. Because of backscattering from the geometry, plane wave modes have a divergent stress tensor at the event horizon, but suitable wavepackets avoid this difficulty, and are dominated at late times by quasinormal behavior. The wavepackets have cuts in the complexified coordinate plane that are controlled by requirements of continuity, single-valuedness and positive energy. Due to a focusing effect, regular wavepackets nevertheless have a divergent stress-energy at the inner horizon, signaling an instability. This instability, which is localized behind the event horizon, is detected holographically as a breakdown in the semiclassical computation of dual CFT expectation values in which the analytic behavior of wavepackets in the complexified coordinate plane plays an integral role. In the dual field theory, this is interpreted as an encoding of physics behind the horizon in the entanglement between otherwise independent CFTs.

Paper Structure

This paper contains 17 sections, 99 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The rotating BTZ black hole and its embedding
  3. The geometry and causal structure
  4. Coordinate systems
  5. The event horizon
  6. The need for wavepackets
  7. The invariants
  8. Computation of the wavepackets and their behavior at the event horizon
  9. Continuing to the other regions and excursions in the complex plane
  10. Continuity of the modes and the wavepacket
  11. The branch cuts and a positive energy condition
  12. The Cauchy horizon
  13. AdS/CFT correlators and detecting the instability
  14. General considerations
  15. Computing the 1-point function
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Penrose diagram detailing the causal structure of the rotating BTZ black hole bthz. The diagram repeats itself indefinitely above and below. The vertical boundaries of the regions $3_{\pm\pm}$ are singularities while the vertical boundaries of the regions $1_{\pm\pm}$ are asymptotically AdS boundaries. We study an instability of the inner horizon at the locations marked $r_-$ that results from the gravitational focusing of wavepackets.
  • Figure 2: The Penrose diagram for regions of interest, detailing some of the horizon structure in the various coordinate systems. The horizons in the other regions are similar.
  • Figure 3: The complex $t$ contour for integration in BTZ coordinates. The wavy lines are the branch cuts that restrict the range of imaginary time. While choosing different locations for the cuts can move these lines, the contour will always either cross or hit the cuts.