Linear Growth of Holographic Time-like Entanglement Entropy and Kasner exponents

Abstract: The holographic time-like entanglement entropy (TEE) extends entanglement to time-like boundary subregions. While its definitive holographic dictionary remains debated, one concrete proposal utilizes piece-wise extremal surfaces. In this work, we adopt this geometric prescription as an exploratory framework to holographically investigate the late-time ($τ_0\to \infty$) growth of TEE in asymptotically AdS black holes with a space-like singularity and no inner horizon. By assuming a Kasner geometry near the space-like singularity and using null energy condition, we analytically show that a critical extremal surface $\mathcal{A}_c$ inside the event horizon completely governs the late-time linear growth of the TEE. This result suggests that the late-time behavior of TEE is tightly constrained by the geometry of black hole interiors. Using numerical results from Einstein-scalar theory, we find a robust behavior: the vacuum Schwarzschild-AdS geometry sets an upper bound on the growth rate of the real part and a lower bound on the imaginary part. We prove these bounds in static planar symmetric case under dominant energy condition and conjecture that it should be true in more general cases.

• The paper demonstrates that the late-time linear growth of TEE is governed by a critical extremal surface subject to the Null Energy Condition.
• It employs the CWES prescription along with asymptotic and Hamilton-Jacobi analyses to derive universal thermodynamic bounds.
• Numerical results in Einstein-scalar models confirm that AdS black hole geometries provide robust bounds on both real and imaginary components of TEE.

Linear Growth of Holographic Time-like Entanglement Entropy and Kasner Exponents

Introduction

This paper investigates the dynamics of time-like entanglement entropy (TEE) in the context of the AdS/CFT correspondence, focusing on its late-time linear growth in asymptotically AdS black hole backgrounds characterized by a space-like singularity and no inner horizon. Unlike the conventional spatial entanglement entropy computed via the Ryu-Takayanagi prescription, TEE extends quantum correlations to time-like boundary subregions. The study employs the Complex-valued Weak Extremal Surface (CWES) prescription, a piece-wise extremal surface construction, as a concrete geometric framework for probing TEE, while acknowledging that the precise holographic dual is still unsettled.

Analytical Framework

The central result is that the late-time growth rate of TEE is entirely governed by the geometry of a critical extremal surface $\mathcal{A}_c$ located inside the event horizon. The authors derive this result using both asymptotic integral analysis and the Hamilton-Jacobi formalism, showing that for a sufficiently large temporal observation window $T_0 \gg 1$, the TEE grows linearly in time, i.e., $S_T \sim v_g(z_c) T_0 + \text{const}$, where $v_g(z_c)$ is dictated by the bulk geometry at the critical surface.

By analyzing the near-singularity asymptotics, the study identifies that the existence of $\mathcal{A}_c$ is guaranteed by the Null Energy Condition (NEC). The NEC is shown to tightly constrain the Kasner exponents that govern the power-law behavior near the singularity. The explicit evaluation in Einstein-scalar gravity confirms that for all physically reasonable parameter ranges (with Kasner exponents determined by the matter content and spacetime dimension), the critical surface $\mathcal{A}_c$ is always present and the late-time TEE growth persists.

Real and Imaginary Components of TEE

The TEE calculated via CWES has both real and imaginary components:

• Real Part: Corresponds to the area of a space-like branch across the horizon that terminates at the singularity. Its late-time growth rate is shown to be determined solely by the critical extremal surface $\mathcal{A}_c$.
• Imaginary Part: Stems from the time-like segment connecting past and future singularities through the bifurcation surface. It is independent of the boundary temporal width and thus probes physics deep inside the black hole. Analytical and numerical evidence demonstrates that the Schwarzschild-AdS geometry provides a universal lower bound for this component.

Numerical Results and Thermodynamic Bounds

The paper presents extensive numerical results for Einstein-scalar models (hairy black holes) in $D=4$ bulk dimensions. Three thermodynamic ensembles are considered:

• Fixed Horizon Radius (Entropy Density): The vacuum Schwarzschild-AdS solution maximizes the real part's growth rate and minimizes the imaginary part.
• Fixed Hawking Temperature: The same bounding behavior is observed.
• Fixed Mass/Energy Density: The bounds depend strongly on the quantization scheme. Under standard quantization, the vacuum bounds hold, whereas under alternative quantization, violations occur. However, when employing a scheme-independent thermodynamic energy (as recently proposed), the bounds are restored universally.

These findings are robust even when local violations of the Dominant Energy Condition (DEC) occur, provided NEC is maintained. This suggests a deeper geometric mechanism protecting the bounds, which is speculated to be a general feature anchored by the NEC and the structure of Kasner exponents in the spacetime.

Implications and Future Directions

The results provide a rigorous geometric underpinning for the linear growth of TEE and establish universal thermodynamic bounds in a broad class of asymptotically AdS spacetimes with space-like singularities. The tight link between the late-time growth rate and the critical extremal surface underscores the role of interior black hole geometry in boundary quantum information dynamics.

Practically, the bounds derived for TEE complement analogous results for spatial entanglement and holographic complexity, opening avenues to explore the maximal rates of quantum information spread in boundary theories. The insensitivity of these bounds to mild quantum gravity corrections suggests robustness against non-classical modifications, and the restoration of bounds by scheme-independent energy definitions hints at deep structural features of holographic thermodynamics.

Possible directions for future research include generalizing to inhomogeneous or arbitrarily shaped time-like boundary strips, analyzing the fate of bounds in cases where NEC is violated (e.g., phantom fields or more exotic singularities), and investigating the interplay between TEE and holographic complexity or temporal quantum information measures.

Conclusion

The study delivers a comprehensive analytic and numeric treatment of time-like entanglement entropy in AdS/CFT, demonstrating linear late-time growth governed by an interior critical extremal surface, with universal bounds imposed by vacuum geometry under standard physical assumptions. The work enhances our theoretical understanding of temporal quantum correlations in strongly coupled systems and lays the groundwork for future inquiries into quantum information limits in holographic settings.