Complexity Growth in Black Holes: A Comparison of the Volume and Action Proposals

Abstract

In this article, we investigate the late-time growth of holographic complexity, defined via the complexity-volume (CV) and complexity-action (CA) prescriptions, for BTZ, Schwarzschild, Reissner-Nordström, and Kerr black holes. Extending previous analyses beyond asymptotically AdS spacetimes, we include asymptotically flat geometries and employ the CV and CA prescriptions as comparative geometric diagnostics of black hole interior dynamics. In all cases considered, the complexity growth rate is governed by horizon thermodynamic data and scales with $T_H S_H$. While the CV prescription exhibits geometry-dependent proportionality constants, the CA prescription yields a universal thermodynamic scaling across all black holes studied, including non-AdS cases. We further analyze variations in the complexity growth rate, $δ\dot{\mathcal{C}}$, under physical processes such as the Penrose process, superradiance, and particle accretion. We find that $δ\dot{\mathcal{C}}$ exhibits non-trivial behavior: it increases under the Penrose process and superradiance, while under particle accretion it can increase, remain unchanged, or decrease depending on the angular momentum of the infalling particle. In quasi-equilibrium regimes, the variation in complexity closely tracks the behavior of the horizon area and interior volume growth, whereas out-of-equilibrium processes render it sensitive to angular momentum transfer and may lead to negative values within an equilibrium approximation. This behavior highlights the limitations of equilibrium-based treatments and motivates a fully dynamical analysis incorporating horizon stresses and transient hair.