On Complexity for Higher Derivative Gravities

TL;DR

This paper analyzes holographic complexity via the complexity=action proposal for gravitational theories with higher derivatives, specifically F(R) gravity and critical gravity. It shows that for neutral AdS black holes, the late-time CA growth saturates the Lloyd bound when expressed in terms of the physical mass M_HD, with M_F playing a role in F(R) gravity. In critical gravity, the growth rate is modified by the massive spin-2 mode, yielding two butterfly velocities; the massless mode behaves similarly to Einstein gravity, while the massive mode slows the growth, and at the critical point the growth rate vanishes. The work discusses the limitations due to boundary terms, and also contemplates CV-like extensions using Wald-type functionals.

Abstract

Using "complexity=action" proposal we study complexity growth of certain gravitational theories containing higher derivative terms. These include critical gravity in diverse dimensions. One observes that the complexity growth for neutral black holes saturates the proposed bound when the results are written in terms of physical quantities of the model. We will also study effects of shock wave to the complexity growth where we find that the presence of massive spin-2 mode slows down the rate of growth.

Figures (3)

• Figure 1: Two nearby Wheeler-de Witt patches at $t$ and $t+\delta t$ and the part that contributes to the late time behavior of complexity growth. This figure is drawn from Fig 2 of Brown:2015lvg.
• Figure 2: Two nearby Wheeler-de Witt patches for BTZ black hole at $t$ and $t+\delta t$ and the part that contributes to the late time behavior of complexity growth.This figure is drawn from Fig 4 of Brown:2015lvg.
• Figure 3: Kruskal diagram of one-shock geometries. The double black line along $u = 0$ is the shock wave. The green lines show the boundary of WDW patch. The green shaded region is the intersection of WDW and the black hole interior. In this figure a small shift $h(x)$ with $h(x)< v_{0}$ is drown. For a large shift $h(x)$ with $h(x)>v_{0}$ the boundary of WDW intersect the past singularity $uv=1$. This figure is drown from Fig 8 of Brown:2015lvg.