Investigating two counting methods of the holographic complexity

We investigated the distinction between two kinds of “complexity equals action” (CA) conjecture counting methods which are separately provided by Brown et al. and Lehner et al. separately. For the late-time CA complexity growth rate, we show that the difference between two counting methods only comes from the boundary term of the segments on the horizon. However, both counting methods give the identical late-time result. Our proof is general, independent of the underlying theories of higher curvature gravity as well as the explicit stationary spacetime background. To be specific, we calculate the late-time action growth rate in Schwarzschild anti-de Sitter black hole for F(Ricci) gravity, and show that these two methods actually give the same result. Moreover, by using the Iyer-Wald formalism, we find that the full action rate within the Wheeler-DeWitt patch can be expressed as some boundary integrations, and the final contribution only comes from the boundary on singularity. Although the definitions of the mass of black hole has been modified in F(Ricci) gravity, its late-time result has the same form with that of Schwarzschild anti-de Sitter black hole in Einstein gravity.

Figure 1

Wheeler-DeWitt patch at the late time of spacetime which shares the same Penrose diagram with SAdS black hole, where the dashed line denotes the cutoff surface at the singularity, satisfying the asymptotic symmetries.