Holographic complexity and action growth in massive gravities

TL;DR

This work extends the holographic complexity framework to massive gravity by computing the late-time action growth in the Wheeler–DeWitt patch for neutral and charged AdS black holes, as well as a charged BTZ solution, within massive Einstein–Maxwell gravity. Using the complexity–action duality $\frac{d\mathcal{C}}{dt}=\frac{1}{\pi\hbar}\frac{dS}{dt}$, the authors show that neutral massive AdS black holes still saturate the Lloyd bound with $\frac{dS}{dt}=2M$, while charged massive black holes obey the same Lloyd-like structure $\frac{dS}{dt}=(M-\mu_+Q)-(M-\mu_-Q)$, with $\mu_\pm$ defined at the inner and outer horizons. The graviton-mass terms $m^2\alpha_1$, $m^2\alpha_2$ can lift the action growth relative to the massless case, particularly for charged configurations, indicating that massive charged black holes are faster holographic computers (still respecting the Lloyd bound). The results generalize the universal bound on holographic complexity growth to massive gravity and support the view that black holes are among the fastest information processors in holographic settings, with implications for understanding information storage in gravitational theories.

Abstract

In this paper, we investigate the growth rates of action for the anti-de Sitter black holes in massive-Einstein gravity models and obtain the universal behaviors of the growth rates of action (the rates of holographic complexity) within the "Wheeler-DeWitt"(WDW) patch at the late limit. Furthermore, we find that, for the static neutral cases, when the same mass of black holes is given, the computational speed of the neutral massive black hole is the same as its Einstein gravity counterpart, which is independent with the effect of the graviton mass terms, nevertheless, for the static charged cases, when the same mass and charge parameters of black holes are given, the growth rates of action for the massive charged black holes are always superior to the growth rates of action without graviton mass terms, which directly shows that the massive charged black holes as computers on the computational speeds are faster than their Einstein gravity counterparts.