More on Complexity in Finite Cut Off Geometry

TL;DR

This work analyzes holographic complexity in finite-cutoff AdS black holes with U(1) charge using the complexity=action proposal. It shows that achieving the expected late-time bound requires a behind-horizon cutoff whose value is fixed by the boundary cutoff, at least for small charged black holes, and it derives the relation between the boundary cutoff r_c and the behind-horizon cutoff r_0 in Einstein-Hilbert-Maxwell theory. The analysis is then extended to Gauss-Bonnet-Maxwell theory, where GB corrections are incorporated in both the quasi-local energy and the WDW action growth; the leading-order r_0–r_c relation remains unchanged in the small-hole limit, with the inner horizon playing the analogous role. The results illuminate how finite-cutoff holography and different proposed complexity bounds interplay for charged systems and highlight the regime of validity being restricted to small GB coupling and late times, suggesting directions for exploring time dependence and larger couplings.

Abstract

It has been recently proposed that late time behavior of holographic complexity in a uncharged black brane solution of Einstein-Hilbert theory with boundary cut off is consistent with Lloyd's bound if we have a cut off behind the horizon. Interestingly, the value of this new cut off is fixed by the boundary cut off. In this paper, we extend this analysis to the charged black holes. Concretely, we find the value of this new cut off for charged small black hole solutions of Einstein-Hilbert-Maxwell theory, in which the proposed bound on the complexification is saturated. We also explore this new cut off in Gauss-Bonnet-Maxwell theory.

Figures (1)

• Figure 1: Penrose diagram of the eternal charged AdS black hole. The theory is defined at a radial finite cut off $r_c$ and the actual WDW patch is shown in green color.