Complexity growth rates for AdS black holes in massive gravity and f(R) gravity

Yan-Yan Li,Shao-Wen Wei,Yu-Xiao Liu,Wen-Di Guo

doi:10.1140/epjc/s10052-017-5466-5

Abstract

The “complexity = action” duality states that the quantum complexity is equal to the action of the stationary AdS black hole within the Wheeler–DeWitt patch at late time approximation. We compute the action growth rates of the neutral and charged black holes in massive gravity and the neutral, charged and Kerr–Newman black holes in f(R) gravity to test this conjecture. Besides, we investigate the effects of the massive graviton terms, higher derivative terms and the topology of the black hole horizon on the complexity growth rate.

Highlights

Introduction where G is theNewton constant and L is a length scale that should be chosen to be either the Anti-de Sitter (AdS) radius or the radius of the black hole horizon
With the help of the thermodynamic quantities (9), we can rewrite it in the following form: dA dt = (M − Ω+ J − μ+ Q) − (M − Ω− J − μ− Q) . (17) Clearly, this is just the action growth rate of a KN AdS black hole in general relativity (GR) speculated in Ref. [32]
The action of a stationary AdS black hole within the WDW patch has been related to the quantum complexity of a holographic state

Summary

Action growth rate of Kerr–Newman black holes in GR

Before dealing with massive gravity and f (R) gravity, we start with the KN black holes in GR. The solutions of the metric and the electric potential are given in Refs. The metric functions are given by r = a2 + r2. Note that this solution is valid only for a2 < l2. The thermodynamical quantities are [37]. The determinant of this metric is sin θ Ξ ρ2. With the help of the thermodynamic quantities (9), we can rewrite it in the following form: dA dt = (M − Ω+ J − μ+ Q) − (M − Ω− J − μ− Q) . (17) Clearly, this is just the action growth rate of a KN AdS black hole in GR speculated in Ref.

Action growth rate of black holes in massive gravity

Conclusion and discussion