Complexity growth of massive black hole with a probe string

In this work, we study the computational complexity of massive gravity theory via the “Complexity = Action” conjecture. Our system contains a particle moving on the boundary of the black hole spacetime. It is dual to inserting a fundamental string in the bulk background. Then this string would contribute a Nambu–Goto term, such that the total action is composed of the Einstein–Hilbert term, Nambu–Goto term and the boundary term. We shall investigate the time development of this system, and mainly discuss the features of the Nambu–Goto term affected by the graviton mass and the horizon curvature in different dimensions. Our study could contribute interesting properties of complexity.