Complexity equals anything for multi-horizon black holes

TL;DR

The paper addresses how holographic complexity can be used to illuminate the interior structure of multi-horizon AdS black holes. By applying the complexity equals anything framework to the Bardeen-AdS class, using both codimension-one (generalized volume) and codimension-zero (CMC-slice based) observables, and comparing with CV, the authors show that complexity can probe all regions where $f(r)<0$, including beyond the outer two horizons. They introduce curvature-weighted weights $a(r)$ and analyze extremal surfaces and CMC slices across one-, two-, and three-horizon cases, revealing interior distinctions and phase-transition-like behavior in the growth of complexity. The results generalize to an arbitrary number of horizons, underscoring the flexibility and diagnostic power of $ abla\mathcal{C}_{\text{gen}}$ for interior geometry, and suggesting avenues to connect with other holographic probes such as codimension-two HM/HRT surfaces. Overall, the work provides a versatile toolkit for diagnosing interior spacetime structure in complex black holes and motivates future explorations across broader matter couplings and horizon configurations.

Abstract

We investigate the ``complexity equals anything" proposal with codimension-one and codimension-zero gravitational observables for multi-horizon black holes, using the Bardeen-AdS class black hole as an example. In particular, we compare the results with the ``complexity equals volume" (CV) proposal and find that the ``complexity equals anything" ($\mathcal{C}_{\text{gen}}$) enables the probing of a more complete black hole interior, that is, all spacetime regions where the blackening factor $f(r)<0$. This is the advantage brought by the flexibility of this holographic complexity conjecture. In addition, we compute the complexity derived from various geometric quantities and show that these constructions can effectively differentiate the distinct interior regions of the black hole.

Figures (21)

• Figure 1: The blackening factor for the Bardeen-AdS class black hole with $l=5, \alpha=1, s=3, q=1.3$. The black dashed curve shows a naked singularity. The brown curve represents that there is a black hole with two horizons and a timelike singularity. The red curve shows that the black hole has still two horizons but the curvature singularity disappears. The blue curve represents that there is a black hole with three horizons and a spacelike singularity. The purple curve represents that there is a black hole with only one horizon and a spacelike singularity.
• Figure 2: (a). The codimension-one extremal surface $\Sigma$ anchored at the boundary time $\tau$. (b). The codimension-zero extremal region $\Sigma_{+} \cup \mathcal{M} \cup \Sigma_{-}$ anchored at the boundary time $\tau$.
• Figure 3: (a) A common effective potential. Here we have two local maxima, one of them appears at $r=r_{f_{R}}$, the other one appears at $r=r_{f_{L}}$. The corresponding values of $P_{v}$ are $P_{\infty,R}$ and $P_{\infty,L}$, respectively. (b) The relation between the conserved momentum $P_v$ and the boundary time $\tau$.
• Figure 4: The blackening factor $f(r)$ for the Bardeen-AdS class black hole with $l = 5,\,\alpha = 1,\,s = 3,\, q = 1.3$, for which there is only one zero point.
• Figure 5: The effective potentials with $l = 5, \alpha = 1, s = 3, q = 1.3, \lambda = \lambda_{1} = 10^{-3}, \lambda_{2} = 10^{-8}$. (a) The effective potentials with only one local maximum. (b) The effective potentials with two local maxima.