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Black hole horizons from within loop quantum gravity

Hanno Sahlmann

TL;DR

This work recasts black-hole horizon degrees of freedom as emergent from the bulk loop quantum gravity data by imposing horizon boundary conditions directly in the quantum theory. Starting from the holonomy-flux algebra, it develops a U(1) boundary model with a positive measure on flat connections that depends only on puncture data, then extends to the full SU(2) case via non-Abelian surface operators, aiming to reproduce horizon physics without a separate horizon quantization. The results suggest that, for spherical horizons, horizon states closely match those found in SU(2) BF theory or ISU(2) Chern-Simons theory, and punctures exhibit anyonic statistics, indicating a rich interrelation between bulk quantum geometry and horizon degrees of freedom. If fully realized, this intrinsic horizon picture could underpin entropy counting and horizon dynamics directly from bulk LQG, with topology playing a decisive role in shaping the effective horizon theory and potentially modifying the area-entropy relation in non-spherical cases.

Abstract

In general relativity, the fields on a black hole horizon are obtained from those in the bulk by pullback and restriction. Similarly, in quantum gravity, the quantized horizon degrees of freedom should result from restricting, or pulling-back, the quantized bulk degrees of freedom. This is not yet fully realized in the - otherwise very successful - quantization of isolated horizons in loop quantum gravity. In this work we outline a setting in which the quantum horizon degrees of freedom are simply components of the quantized bulk degrees of freedom. There is no need to quantize them separately. We present evidence that for a horizon of sphere topology, the resulting horizon theory is remarkably similar to what has been found before.

Black hole horizons from within loop quantum gravity

TL;DR

This work recasts black-hole horizon degrees of freedom as emergent from the bulk loop quantum gravity data by imposing horizon boundary conditions directly in the quantum theory. Starting from the holonomy-flux algebra, it develops a U(1) boundary model with a positive measure on flat connections that depends only on puncture data, then extends to the full SU(2) case via non-Abelian surface operators, aiming to reproduce horizon physics without a separate horizon quantization. The results suggest that, for spherical horizons, horizon states closely match those found in SU(2) BF theory or ISU(2) Chern-Simons theory, and punctures exhibit anyonic statistics, indicating a rich interrelation between bulk quantum geometry and horizon degrees of freedom. If fully realized, this intrinsic horizon picture could underpin entropy counting and horizon dynamics directly from bulk LQG, with topology playing a decisive role in shaping the effective horizon theory and potentially modifying the area-entropy relation in non-spherical cases.

Abstract

In general relativity, the fields on a black hole horizon are obtained from those in the bulk by pullback and restriction. Similarly, in quantum gravity, the quantized horizon degrees of freedom should result from restricting, or pulling-back, the quantized bulk degrees of freedom. This is not yet fully realized in the - otherwise very successful - quantization of isolated horizons in loop quantum gravity. In this work we outline a setting in which the quantum horizon degrees of freedom are simply components of the quantized bulk degrees of freedom. There is no need to quantize them separately. We present evidence that for a horizon of sphere topology, the resulting horizon theory is remarkably similar to what has been found before.

Paper Structure

This paper contains 10 sections, 69 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Heuristic considerations
  3. U(1) boundary conditions
  4. Lebesgue measure on flat connections
  5. Extension to $\mathfrak{A}$
  6. Gauge invariant solutions
  7. Towards quantum isolated horizons with SU(2) boundary conditions
  8. The surface operators $\boldsymbol{W}_S$
  9. A functional on simple loops
  10. Discussion and outlook

Figures (10)

  • Figure 1: A spin network edge punctures $H$ that results in nontrivial holonomy around $\partial S= \alpha$.
  • Figure 2: Relation between a charge network edge and its transformation under a diffeomorphism
  • Figure 3: Relation between a charge network and its tranformation under a diffeomorphism
  • Figure 4: A surface the flux of which does not have a well defined action on surface states
  • Figure 5: Graphs of two gauge invariant charge nets, for the case of $H\simeq S^2$ (left) and $H\simeq T^2$ (right).
  • ...and 5 more figures