# Investigating two counting methods of the holographic complexity

## TL;DR

This paper compares two methods of calculating holographic complexity growth in black holes, showing they yield identical late-time results despite differences in boundary term contributions, and extends the analysis to F(Ricci) gravity.

## Contribution

It demonstrates the equivalence of two holographic complexity counting methods at late times across different gravity theories, including higher curvature gravity.

## Key findings

- Both methods produce the same late-time complexity growth rate.
- The difference between methods arises only from boundary terms on the horizon.
- Full action rate can be expressed as boundary integrals, mainly from the singularity boundary.

## Abstract

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 SAdS 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 WDW 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 SAdS black hole in Einstein gravity.