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Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons

Jonathan Engle, Karim Noui, Alejandro Perez, Daniele Pranzetti

TL;DR

This work develops a manifestly SU(2) invariant description of Type I isolated horizons within loop quantum gravity, showing that horizon degrees of freedom are governed by an SU(2) Chern-Simons theory coupled to punctures. By deriving a conserved presymplectic structure and translating to Ashtekar-Barbero variables, it provides a clear path from horizon boundary conditions to a horizon Hilbert space built from CS states, enabling explicit state counting. The resulting black hole entropy reproduces the Bekenstein-Hawking relation with a universal logarithmic correction ΔS = -3/2 log(a_H/ℓ_p^2), aligning with Kaul–Majumdar and Carlip analyses, and is robust to choices of boundary variables. The framework offers a gauge-consistent, first-principles route to black hole thermodynamics in LQG and points toward generalizations to distorted horizons and broader horizon classes.

Abstract

A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly SU(2) invariant formulation where the (effective) horizon degrees of freedom are described by an SU(2) Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy.

Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons

TL;DR

This work develops a manifestly SU(2) invariant description of Type I isolated horizons within loop quantum gravity, showing that horizon degrees of freedom are governed by an SU(2) Chern-Simons theory coupled to punctures. By deriving a conserved presymplectic structure and translating to Ashtekar-Barbero variables, it provides a clear path from horizon boundary conditions to a horizon Hilbert space built from CS states, enabling explicit state counting. The resulting black hole entropy reproduces the Bekenstein-Hawking relation with a universal logarithmic correction ΔS = -3/2 log(a_H/ℓ_p^2), aligning with Kaul–Majumdar and Carlip analyses, and is robust to choices of boundary variables. The framework offers a gauge-consistent, first-principles route to black hole thermodynamics in LQG and points toward generalizations to distorted horizons and broader horizon classes.

Abstract

A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly SU(2) invariant formulation where the (effective) horizon degrees of freedom are described by an SU(2) Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy.

Paper Structure

This paper contains 23 sections, 160 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition of isolated horizons
  3. Some extra details for Type I isolated horizons
  4. The main equations
  5. Proof of the main equations
  6. The conserved presymplectic structure
  7. The action principle
  8. The classical results in a nutshell
  9. On the absence of boundary term on the internal boundary
  10. The presymplectic structure in self-dual variables
  11. Presymplectic structure in Ashtekar-Barbero variables
  12. A side remark on the triad as the boundary degrees of freedom
  13. Gauge symmetries
  14. On the first class nature of the IH constraints
  15. The zeroth and first laws of BH mechanics for (spherical) isolated horizons
  16. ...and 8 more sections

Figures (1)

  • Figure 1: The characteristic data for a (vacuum) spherically symmetric isolated horizon corresponds to Reissner-Nordstrom data on $\Delta$, and free radiation data on the transversal null surface with suitable fall-off conditions. For each mass, charge, and radiation data in the transverse null surface there is a unique solution of Einstein-Maxwell equations locally in a portion of the past domain of dependence of the null surfaces. This defines the phase space of Type I isolated horizons in Einstein-Maxwell theory. The picture shows two Cauchy surfaces $M_1$ and $M_2$ "meeting" at space-like infinity $i_0$. A portion of ${\mathfs {I}}^+$ and ${\mathfs {I}}^-$ are shown; however, no reference to future time-like infinity $i^+$ is made as the isolated horizon need not to coincide with the black hole event horizon.