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.
Highlights
Quantum computational complexity theory is used to figure out what the implications of quantum physics to computational complexity theory are [1]
This theory never played a crucial role in gravitational physics until Susskind noticed one thing that there is a relationship between complexity and the deep inner structure of the stretched horizon as well as the extreme long-time behavior of black holes
He argued that the difference between “easy” operators and “hard” operators is related to the time evolution of a certain measure of complexity which associated with the stretched horizon of Alice’s black hole [2]
Summary
Quantum computational complexity theory is used to figure out what the implications of quantum physics to computational complexity theory are [1]. Whereafter, Susskind proposed the complexity equals volume (CV)-conjecture and tried to build a bridge of complexity and the volume of the Einstein-Rosen bridge (ERB) [3,4] Speaking, this duality can be described by. Soon afterward, Susskind proposed the “complexity equals action” conjecture which is known as CA-duality This new “bridge” connects the complexity with the action in bulk. The BRSSZ method is put forward to calculating the complexity growth rate at the late time for CA duality. We show that the difference between the two methods only comes from the boundary term of the segments near the horizon for the late-time action growth rate within the WDW patch.
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