Time dependence of complexity for Lovelock black holes

TL;DR

This paper investigates how the complexity of holographic states dual to Lovelock black holes evolves over time using the CA proposal, revealing dependencies on higher order couplings and charge, with results differing from Schwarzschild black holes.

Contribution

It provides a detailed analysis of the time dependence of complexity in Lovelock black holes, highlighting differences from Schwarzschild solutions and effects of higher order couplings and charge.

Findings

Early time complexity increases faster with higher order couplings.

Late time complexity growth rate approaches a constant ratio to black hole mass.

Charged Lovelock black holes show similar complexity behavior to Einstein black holes at high charge.

Abstract

We study the general time dependence of complexity for holographic states dual to Lovelock black holes using the "Complexity=Action" (CA) proposal. We observe that at early times, the critical time at which the complexity begins to increase is a decreasing function of the higher order coupling constants, which implies that the complexity evolves faster than that of Schwarzschild black holes. At late times, the rate of change of complexity is essentially determined by the generalised Gibbons-Hawking-York boundary term evaluated at the future singularity. In particular, its ratio to black hole mass is a characteristic constant, independent of the higher order couplings. Thus, in the vanishing coupling limit, the result in general does not reduce to that of Schwarzschild black holes, in spite of that the metric reduces to the latter as well as the gravitational action. In fact, the two differ by a constant during the whole time evolution. Including the next-to-leading order term around late times, we find that as the Einstein case, the late time limit is always approached from above, thus violating any conjectured upper bound given by the late time result. For charged Lovelock black holes, we find that with sufficient charge, the complexity roughly behaves the same as the Einstein case. However, for smaller charges, the two have some significant differences. In particular, unlike the Einstein case, in the uncharged limit the complexity growth rate does not match with the neutral case, differing by a constant in the whole time evolution.