Aspects of holographic timelike entanglement entropy in black hole backgrounds

TL;DR

The paper advances the holographic understanding of timelike entanglement entropy (tEE) in black-hole spacetimes by formulating tEE as a combination of spacelike and timelike extremal surfaces and analyzing its real and imaginary components.Applying the construction to BTZ and higher-dimensional AdS-Schwarzschild geometries, it reveals a turning-point structure, region-dependent validity, and a volume-plus-area decomposition in the large-subsystem limit, with a complex area term emerging at finite dimensions.A crucial outcome is that, in the large-d limit, the finite part of tEE signals a violation of the timelike area theorem, while near-horizon analysis shows exponential growth rates saturating the MSS bound for both spacelike and timelike branches across various backgrounds.The work highlights the potential of tEE as a diagnostic of temporal correlations and information growth in black-hole spacetimes, while raising questions about dual field-theory interpretations and monotonicity theorems in non-conformal, non-Lorentz-invariant settings.

Abstract

We study the holographic construction of timelike entanglement entropy (tEE) in black hole backgrounds in Lorentzian geometries. The holographic tEE is realized through extremal surfaces consisting of spacelike and timelike branches that encode its real and imaginary components, respectively. In the BTZ black hole, these surfaces extend into the interior of the black hole and reproduce the field-theoretic results. The analysis is further generalized to higher-dimensional AdS-Schwarzschild black holes, where the characteristics of tEE are obtained with increasing size of the boundary subsystem. Besides, we also show that the boundary subsystem length diverges at a dimension-dependent critical turning point. Notably, this critical point moves closer to the black hole horizon as the dimensionality of the bulk increases. For large subsystem lengths, the finite part of the tEE displays a characteristic volume-plus-area structure, with a real volume term and a complex coefficient of the area term approaching constant values at large dimensions. Besides, we also study the monotonicity of a new quantity, timelike entanglement density, which offers insights into a timelike area theorem in specific limits. Subsequently, we investigate the near-horizon dynamics in various black hole backgrounds, where the spacelike and timelike surfaces exhibit exponential growth of the form $e^{\frac{2π}β Δt}$ with inverse black hole temperature $β$.

Figures (6)

• Figure 1: Holographic timelike entanglement entropy for BTZ black hole background with $r_h=1$ and the turning point $r_0=1.5$. The spacelike (green) and the timelike surfaces are demonstrated.
• Figure 2: Depiction of the extremal surfaces associated with the holographic timelike entanglement entropy (tEE) in the BTZ black hole geometry, corresponding to a different position of the turning point $r_0$ relative to the black hole horizon $r_h$. This configuration is characterized by the parameter regime $r_h < r_0 < r_c$, with $r_h = 1$ and $r_0 = 1.2$.
• Figure 3: The spacelike (green) and the timelike (red) surfaces are illustrated in $AdS_6$-Schwarzschild black hole background. The black hole horizon is $r_h=1$ and the turning point of the timelike surface is $r_0=1.1$ which situates in the region $r_0>r_c$. The spacelike surface is extended from the boundary to the interior of the black hole. The timelike and the spacelike surfaces are merged at $r=0$ where the merging condition $g^{\mu\nu}\partial_\mu \Sigma_{\text{Re}}\partial_\nu \Sigma_{\text{Re}}|_{r=0}=g^{\mu\nu}\partial_\mu \Sigma_{\text{Im}}\partial_\nu \Sigma_{\text{Im}}|_{r=0}=0$ is satisfied.
• Figure 4: We demonstrate the spacelike (green) and timelike (red) surfaces in $AdS_4$-Schwarzschild spacetime with the black hole horizon $r_h=1$ and the turning point of the timelike surface $r_0=1.12$. The turning point satisfies the condition $r_0<r_c$. Here we observe, crossing spacelike and timelike surfaces which makes the solution invalid.
• Figure 5: The areas of the spacelike (green-dashed) and the timelike (red-dot-dashed) surfaces are plotted with increasing boundary subsystem size using \ref{['area_int', 't_prime_wbh']}. Here, we have considered $AdS_4$-Schwarzschild spacetime with the horizon $r_h=1$ and the cutoff $\epsilon=10^{-8}$. (a) Here, we plot the total renormalized areas of the surfaces stretched in both the interior and the exterior of the black hole. The small system size corresponds to high values of areas, whereas it decreases with increasing subsystem size. (b) Here, we plot the black hole interior surface areas without any renormalization. We observe a completely opposite characteristic compared to the first case. The spacelike surface area increases, and the timelike surface area saturates in the region of large subsystem size. (c) We plot the renormalized exterior surface areas where an identical characteristic is observed as the total areas depicted in (a). Interestingly, the values of the exterior surface areas dominate the interior surface areas.