Holographic Complexity Bounds

TL;DR

This paper investigates bounds on holographic complexity growth in various AdS black holes, establishing inequalities involving action growth rates, thermodynamic volumes, and horizon topologies, with implications for Lloyd's bound and divergence phenomena.

Contribution

It introduces new bounds on the action growth rate in AdS black holes, including a volume concept for singularities, and explores conditions for Lloyd's bound validity.

Findings

Lower bound inequality for action growth rate in certain black holes.

Relation between action growth and thermodynamic volume differences.

Divergences in volume and action growth in specific black hole solutions.

Abstract

We study the action growth rate in the Wheeler-DeWitt (WDW) patch for a variety of $D\ge 4$ black holes in Einstein gravity that are asymptotic to the anti-de Sitter spacetime, with spherical, toric and hyperbolic horizons, corresponding to the topological parameter $k=1,0,-1$ respectively. We find a lower bound inequality $\frac{1}{T} \frac{\partial \dot I_{\rm WDW}}{\partial S}|_{Q,P_{\rm th}}> C$ for $k=0,1$, where $C$ is some order-one numerical constant. The lowest number in our examples is $C=(D-3)/(D-2)$. We also find that the quantity $(\dot I_{\rm WDW}-2P_{\rm th}\, \Delta V_{\rm th})$ is greater than, equal to, or less than zero, for $k=1,0,-1$ respectively. For black holes with two horizons, $\Delta V_{\rm th}=V_{\rm th}^+-V_{\rm th}^-$, i.e. the difference between the thermodynamical volumes of the outer and inner horizons. For black holes with only one horizon, we introduce a new concept of the volume $V_{\rm th}^0$ of the black hole singularity, and define $\Delta V_{\rm th}=V_{\rm th}^+-V_{\rm th}^0$. The volume $V_{\rm th}^0$ vanishes for the Schwarzschild black hole, but in general it can be positive, negative or even divergent. For black holes with single horizon, we find a relation between $\dot I_{\rm WDW}$ and $V_{\rm th}^0$, which implies that the holographic complexity preserves the Lloyd's bound for positive or vanishing $V_{\rm th}^0$, but the bound is violated when $V_{\rm th}^0$ becomes negative. We also find explicit black hole examples where $V_{\rm th}^0$ and hence $\dot I_{\rm WDW}$ are divergent.