Symmetry and entropy of black hole horizons

TL;DR

In a background‑independent quantum gravity setting, the paper tackles how to reproduce the Bekenstein‑Hawking entropy for horizons and how classical spacetime emerges from quantum horizon states. It employs noiseless subsystems from quantum information theory to identify symmetry‑preserving subspaces of the horizon Hilbert space that survive environmental interactions, linking diffeomorphism‑invariant dynamics to emergent classical symmetries. A key result is that restricting the horizon states to configurations with all punctures carrying the same minimal spin yields a horizon subspace whose entropy–area ratio and quasinormal‑mode spectrum are compatible, yielding a consistent Immirzi parameter value $\gamma$; specifically, $A[\{ j_i \}] = 8\pi \gamma \sum_i \sqrt{j_i(j_i+1)}$ and ${\cal R}=S/A=c/\gamma$, with $\gamma^{min}=\frac{\ln 3}{2\pi\sqrt{j_{\min}(j_{\min}+1)}}$ matching the quasinormal‑mode prediction. This supports a loop quantum gravity picture in which classical spacetime symmetries emerge from the commutant of the horizon–bulk interaction algebra via noiseless subsystems. Overall, the work provides a concrete route to connect quantum horizon geometry with semiclassical thermodynamics and the spectrum of black‑hole perturbations, through symmetry‑driven subspace selection in a diffeomorphism‑invariant theory.

Abstract

We argue, using methods taken from the theory of noiseless subsystems in quantum information theory, that the quantum states associated with a Schwarzchild black hole live in the restricted subspace of the Hilbert space of horizon boundary states in which all punctures are equal. Consequently, one value of the Immirzi parameter matches both the Hawking value for the entropy and the quasi normal mode spectrum of the Schwarzchild black hole. The method of noiseless subsystems thus allows us to understand, in this example and more generally, how symmetries, which take physical states to physical states, can emerge from a diffeomorphism invariant formulation of quantum gravity.

Figures (1)

• Figure 1: The geometry of a quantum horizon is concentrated at a discrete set of punctures. The classical symmetry group SO(3) will thus not act directly on the quantum states representing a Schwarzschild black hole. A discrete group $G_q$, on the other hand, is expected to act on these states. In the classical limit, the action of this discrete group coincides with that of SO(3). For a large number of punctures, the discrete group action shown above would approximate a rotation of the horizon.