Timelike Holographic Complexity

TL;DR

This paper investigates timelike subregion complexity in holography, demonstrating that it remains real and purely geometric in Lorentzian spacetimes, contrasting with pseudo-entropy which can be complex.

Contribution

It extends the holographic complexity-equal-volume framework to Lorentzian boundary regions, showing the complexity is real and retains universal divergences even inside black branes.

Findings

Timelike complexity is purely real in pure AdS.

Complexity exhibits universal UV divergences similar to spacelike case.

No imaginary part arises in complexity, unlike pseudo-entropy.

Abstract

Motivated by the pseudo-entropy program, we study timelike subregion complexity within the holographic Complexity-equal-Volume framework, extending previous spatial constructions to Lorentzian boundary intervals. For hyperbolic timelike regions in pure AdS, we compute the enclosed bulk volume and show that, despite the Lorentzian embedding, the resulting complexity is purely real. We generalize the analysis to AdS black brane geometries, where extremal surfaces may either remain entirely outside the horizon or penetrate it, placing their timelike branch inside the black brane interior. In both configurations, the complexity exhibits the same universal UV divergences as the spacelike case, yet it receives no imaginary contribution, highlighting its causal and geometric origin. This reality stands in sharp contrast to the complex-valued pseudo-entropy and indicates that holographic complexity retains a genuinely geometric, real character even under Lorentzian continuation.