Timelike Holographic Complexity

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.