Timelike Holographic Complexity

TL;DR

The paper develops timelike subregion complexity within the CV holographic proposal to extend holographic information measures to timelike boundary intervals. Using AdS$_{d+2}$ and AdS black brane geometries, it computes the bulk volume $V$ enclosed by timelike extremal surfaces anchored on timelike intervals and shows the associated complexity ${\mathcal C}_{\mathrm{T}} = V/(G R)$ remains purely real, with UV divergences matching the spacelike case. In pure AdS, the complexity arises from the region between spacelike and timelike extremal surfaces and exhibits no imaginary component, indicating a genuinely geometric origin. In black brane backgrounds, horizon-penetrating surfaces are allowed yet the CV volume remains real, suggesting timelike complexity encodes causal geometric depth rather than analytic continuation; this contrasts with the complex-valued pseudo-entropy and invites future work on a boundary-field-theory notion of timelike complexity and possible extensions to CA or other time-sensitive holographic probes.

Abstract

Motivated by the pseudo-entropy program, we investigate timelike subregion complexity within the holographic ``Complexity=Volume'' framework, extending the usual spatial constructions to Lorentzian boundary intervals. For hyperbolic timelike regions in AdS geometries, we compute the corresponding bulk volumes and demonstrate that, despite the Lorentzian embedding, the resulting subregion complexity remains purely real. We further generalize our analysis to AdS black brane geometries, where the extremal surfaces can either be constant-time hypersurfaces or penetrate the horizon. In all cases, the computed complexity exhibits the same universal UV divergences as in the spacelike case but shows no imaginary contribution, underscoring its causal and geometric origin. This stands in sharp contrast with the complex-valued pseudo-entropy and suggests that holographic complexity preserves a genuinely geometric and real character even under Lorentzian continuation.

Figures (2)

• Figure 1: The colored region between the spacelike (green) and timelike (red) extremal surfaces represents the bulk volume associated with the timelike subregion complexity.
• Figure 2: Extremal surfaces for a timelike entangling region (blue interval) in Poincaré (left) and global (right) coordinates. In the global case, in addition to the spacelike extremal surface (green), there exists a timelike one (red). The timelike subregion complexity corresponds to the bulk volume enclosed between these two surfaces, illustrated by the shaded (colored) region. This figure essentially reproduces Fig. 3 of Doi:2022iyj for clarity and comparison. Here, $(r, z)$ and $(\tau, \rho)$ denote the Poincaré and global coordinate systems, respectively.