Aspects of holographic complexity and volume of the black holes

TL;DR

The paper tests holographic complexity growth in black holes by comparing complexity-volume (CV) and complexity-action (CA) dualities across BTZ, Schwarzschild, Reissner–Nordström, and Kerr spacetimes. CV analyses reveal that the complexity growth rate is proportional to the product $T_H S_H$, with geometry-dependent factors, while CA produces a universal proportionality between $d\mathcal{C}/dt$ and $T_H S_H$ (up to a constant). It then investigates how the Kerr-complexity rate responds to Penrose processes, superradiance, particle accretion, and Hawking radiation, finding that $\delta\dot{\mathcal{C}}$ generally increases for Penrose/superradiance, can vary in accretion depending on angular momentum, and remains inconclusive for Hawking radiation. The work discusses CV’s limitations (foliation arbitrariness, maximal-slice dependence) and argues CA offers greater universality, while highlighting horizon dynamics and possible hair effects as important factors in non-equilibrium scenarios. Overall, the study strengthens the connection between holographic complexity and interior black-hole physics and outlines avenues for refining these dualities with horizon dynamics and non-equilibrium considerations.

Abstract

In this article, we study the complexity growth rate for Banados Teitlboim Zanelli, Schwarzschild, Reissner Nordstrom, and Kerr black holes using complexity-volume (CV) and complexity-action (CA) dualities and verify that it is proportional to the product of the horizon temperature and entropy of the black holes as conjectured by Susskind. Furthermore, we explore the variation in the complexity growth rate $δ\dot{\mathcal{C}}$ under various physical processes, including the Penrose process, superradiance, particle accretion, and Hawking radiation, and demonstrate that $δ\dot{\mathcal{C}}$ exhibits non-trivial behavior. Under the Penrose process and superradiance, $δ\dot{\mathcal{C}}$ always increases, and under particle accretion, $δ\dot{\mathcal{C}}$ can increase, remain zero, or decrease depending upon the direction of angular momentum of an infalling particle. For the cases of particle accretion, where we find $δ\dot{\mathcal{C}}$ to be negative, we argue that for a reliable estimate, one has to take into account the contribution of the horizon dynamics of the perturbed black hole to the growth of its complexity.