Black Hole Interior and Time-like Entanglement Entropy

TL;DR

This work introduces Time-like Entanglement Entropy (TEE) as a single-boundary probe of black hole interiors via the Complex-valued Weak Extremal Surface (CWES) prescription. It analyzes Schwarzschild–AdS to establish a Type-I interior baseline with linear-in-$\tau_0$ real part and a physically meaningful imaginary part tied to bulk thermodynamics, then extends to hairy (holographic superconductor) black holes to reveal Type-II interiors with a time-like entanglement phase. A key result is the phase structure controlled by the critical temporal width $\tau_c=-2t_0$, yielding a boundary observable (the time-like entanglement gap) that signals interior causal transitions; as an inner Cauchy horizon appears, $\tau_c\to\infty$, giving pure time-like entanglement. Together, these findings position TEE as a powerful boundary quantum-information measure for detecting hidden interior structure and its relation to cosmic censorship and interior RG-like flows.

Abstract

We establish time-like entanglement entropy (TEE) as a novel tool to characterize the black hole interior from a single-boundary perspective. In the Schwarzschild-AdS black hole, we show that TEE of time-like boundary strips exhibits linear growth as a function of temporal width in the limit of large temporal width, and that its imaginary part carries physical significance rather than being a constant. By analyzing charged, scalar-hairy black holes, we present evidence that TEE detects a hidden "causal phase transition" separating Type-I and Type-II interiors -- distinguished by singularity structure. We identify a critical temporal width $τ_c$ that acts as the order parameter for this transition: for strips narrower than $τ_c$, the system enters a distinct "time-like entanglement phase" dominated purely by time-like contributions, up to a regulator effect; conversely, for strips wider than $τ_c$, space-like entanglement re-emerges. Notably, the existence of a Cauchy horizon drives the $τ_c$ to infinity, leading to pure time-like entanglement. These results suggest that the TEE may supply a novel boundary quantum-information measure to detect structure hidden inside the black hole and suggests a deep connection between TEE and cosmic censorship.

Figures (20)

• Figure 1: A schematic diagram of time-like strip (depicted in red) on the boundary. Here the boundary is $d$-dimensional and spanned by coordinates $\{t,x,\boldsymbol{y}_{d-2}\}$ with $\boldsymbol{y}_{d-2}\in \mathbb{R}^{d-2}$. This strip-configuration serves as the geometric generalization of a time-like interval in higher dimensions.
• Figure 2: Configurations of CWES of time-like strip in SAdS black hole. Blue lines stand for space-like surface and green lines represent time-like surface. (a) There are two types of relevant configurations of CWESs, which are $A_1A_2B_2B_1$ and $A_1B_2A_2B_1$. (b) Comparison of time-like paths: The symmetric surface $A_2B_2$ passes through the bifurcation surface $H$ of the event horizon, whereas $A'_2B_2$ represents a generic path.
• Figure 3: (a) The real part of the CWES area density plotted as a function of the endpoint parameter $v_{A_2}$, where varying this parameter generates a family of candidate extremal surfaces. (b) The minimal extremal configuration of CWES, corresponding to the time-symmetric case $t|_{A_2}=t|_{B_2}=0$.
• Figure 4: The surface $A_1 B'_2$ corresponds to the "turned up" time-reversed image of surface $B_1 B_2$, where geometric constraint requires $t|_{B'_2}=-t|_{B_2}=-t|_{A_2}$.
• Figure 5: The regularized real part of the TEE $\text{Re}\mathcal{A}_{\text{reg}}=\text{Re}\mathcal{A}(\tau_0)$. The solid blue curve represents the full numerical result obtained by parametrically solving Eqs. \ref{['eq:tau_H_final']} and \ref{['eq:area_H_final']}. The dashed black line shows the theoretical linear asymptotic behavior for large $\tau_0$, $\text{Re}\mathcal{A}\approx k\cdot\tau_0$, as predicted in Eq. \ref{['eq:linearReAtau']}. The numerical curve is shown to converge perfectly to this theoretical line.