Measuring Black Hole Formations by Entanglement Entropy via Coarse-Graining

TL;DR

This work addresses the apparent tension between nonzero horizon entropy in gravity and unitary evolution in the dual CFT by proposing entanglement entropy as a time-dependent coarse-grained measure in holography. It shows that the total von Neumann entropy remains zero for pure-state evolution ($S_{tot}=0$) while the holographic entanglement entropy $S_A$ captures thermal features via the HRT prescription, and introduces a finite coarse-grained entropy $S_{eff}=2(S_A-S_A^{(0)})|_{|A|=|B|}$. The authors construct a solvable 2D CFT toy model—a free Dirac fermion on a circle after a quantum quench—and compute $S_A(t, ext{dim})$ exactly using the replica trick and bosonization, expressing results in terms of theta and eta functions; the limits reproduce known infinite-size results and reveal a finite-size periodicity linked to horizon dynamics. They interpret oscillations of $S_{eff}$ as successive black hole formations and evaporations in AdS, and corroborate this with time-dependent one- and two-point functions that exhibit radiation peaks and recurrence, thereby supporting a unitary, information-preserving picture. This work thus builds a concrete bridge between time-dependent holography and black hole information, highlighting entanglement entropy as a robust diagnostic of BH-like dynamics in finite-size CFTs.

Abstract

We argue that the entanglement entropy offers us a useful coarse-grained entropy in time-dependent AdS/CFT. We show that the total von-Neumann entropy remains vanishing even when a black hole is created in a gravity dual, being consistent with the fact that its corresponding CFT is described by a time-dependent pure state. We analytically calculate the time evolution of entanglement entropy for a free Dirac fermion on a circle following a quantum quench. This is interpreted as a toy holographic dual of black hole creations and annihilations. It is manifestly free from the black hole information problem.

Figures (8)

• Figure 1: The minimal surface $\gamma_A$ for the holographic calculation of entanglement entropy in the AdS black formation (left). Though for an eternal AdS black hole we have $\gamma_A\neq \gamma_B$ (upper right), in our case of black hole formation, we actually find that $\gamma_A=\gamma_B$ (lower right). This is because the horizon vanishes at early time.
• Figure 2: The path-integral calculation of the reduced density matrix $[\rho_A]_{\alpha\beta}$ (left) and the trace Tr$\rho_A^n$ (right).
• Figure 3: The plot of $S_A(\pi/2,\sigma)-S_A(\pi/2,0)$ as a function of $\sigma$ at ${\epsilon}=0.2$.
• Figure 4: The plot of $S_{eff}(t)\equiv 2\left(S_A(t,\pi)-S_A(0,\pi)\right)$ as a function of $t$ at ${\epsilon}=0.2$.
• Figure 5: Quantum black hole creations and annihilations in AdS space by the quantum quench as obtained from Fig.\ref{['figEEt']}.