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Self-regularized entropy: How much do quantum black holes violate the no-hair theorem?

Shokoufe Faraji, Niayesh Afshordi

Abstract

We compute the canonical (brick wall) entropy of Hawking radiation in a quantum black hole whose exterior is described, to first order in a small quadrupole parameter, by the static $q$-metric, which is an exact vacuum solution of the Einstein equations. Counting near horizon quasinormal modes shows that a modest quadrupolar deformation self-regularizes the ultraviolet divergence: the entropy of Hawking radiation is finite for any non-vanishing quadrupole, without an ad hoc cutoff. Matching this canonical entropy to the Bekenstein-Hawking entropy leads to no-hair violating multipoles, at percent-to-tens-of-percent level, and provides concrete observational targets for the Next Generation Event Horizon Telescope (ngEHT) and the Laser Interferometer Space Antenna (LISA).

Self-regularized entropy: How much do quantum black holes violate the no-hair theorem?

Abstract

We compute the canonical (brick wall) entropy of Hawking radiation in a quantum black hole whose exterior is described, to first order in a small quadrupole parameter, by the static -metric, which is an exact vacuum solution of the Einstein equations. Counting near horizon quasinormal modes shows that a modest quadrupolar deformation self-regularizes the ultraviolet divergence: the entropy of Hawking radiation is finite for any non-vanishing quadrupole, without an ad hoc cutoff. Matching this canonical entropy to the Bekenstein-Hawking entropy leads to no-hair violating multipoles, at percent-to-tens-of-percent level, and provides concrete observational targets for the Next Generation Event Horizon Telescope (ngEHT) and the Laser Interferometer Space Antenna (LISA).

Paper Structure

This paper contains 26 sections, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Spacetime
  3. Echo time and entropy bound
  4. Small-$q \ll 1$ near horizon limit
  5. Tortoise Coordinates
  6. Wave Operator Setup
  7. Separation of variables
  8. Classical turning points and cavity length
  9. Outer turning point.
  10. Tortoise position of the turning point.
  11. Cavity length.
  12. Inner turning point (why one may set $\rho_{\min}=0$).
  13. Mode counting and canonical entropy
  14. WKB radial spectrum
  15. Radial density of states
  16. ...and 11 more sections

Figures (1)

  • Figure 1: (Top, a) Predicted dependence of minimum $q$ parameter on the mass and spin of astrophysical black holes (equation \ref{['eq:accurate-bound']}). (Bottom, b) The range of quantum corrections induced in the multipolar structure of the Kerr metric, for the same span of black hole masses/spins.