The Fermionic Entanglement Entropy of the Vacuum State of a Schwarzschild Black Hole Horizon

TL;DR

This work analyzes the fermionic entanglement entropy of the Schwarzschild black hole horizon for a regularized vacuum state of an asymptotic observer. By separating the Dirac equation on Schwarzschild spacetime and employing the integral propagator representation plus pseudo-differential-operator techniques, the authors reduce the horizon entropy to a per-mode counting problem. They establish a log-enhanced area law in which each occupied angular momentum mode contributes an identical amount to the Rényi entropies, with per-mode constants given by $S_{\kappa,kn}^{\mathrm{BH}} = \tfrac{1}{12}\frac{\kappa+1}{\kappa}$ for $\kappa>2/3$ and $S_1^{\mathrm{BH}} = \tfrac{1}{6}$ for the von Neumann case, ultimately linking horizon entropy to the number of occupied angular modes and yielding an area-proportional scaling when the mode count grows with the horizon area. The analysis relies on a regularized negative-frequency projection, mode-wise reductions, and rigorous control of error terms via Widom-type trace formulas and Schatten-class estimates. The results provide a mathematically precise realization of the enhanced area law for fermionic fields in a black hole background and suggest extensions to more general black hole geometries and dynamical settings with potential implications for holography and quantum gravity. The overall horizon entropy emerges from simple mode counting, illuminating the microscopic structure of black hole entanglement in a controlled quantum-field-theoretic framework.

Abstract

We define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes on the event horizon.

Figures (6)

• Figure 1: Plot of the function $\eta_\kappa$ for $\kappa=0.1$, $1$ and $10$.
• Figure 2: Cross section visualizing the set $\Lambda=\mathcal{K} \times S^2$.
• Figure 3: Illustration with examples of the "vertices" in Theorem \ref{['ThmWidom']}.
• Figure 4: Plot of the function $a$ in \ref{['aDef']}.
• Figure 5: Visualization of the cutoffs and approximations in the first step of the proof of Theorem \ref{['MainThmSec2']} for $C=2$. We start with a function $g$, which is first multiplied by the cutoff-function $\Psi_C$, giving $\tilde{g}$. This function is then approximated by a polynomial $g_\delta$. Multiplying $g_\delta$ by the cutoff function $\Psi_C$ results in a function which is here called $\tilde{g}_\delta$ (but does not directly appear in the proof). The function $\tilde{f}_\delta$ is then given by the difference between $\tilde{g}$ and $\tilde{g}_\delta$.

Theorems & Definitions (57)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6
• Lemma 2.8
• Example 2.9
• Lemma 2.10