Abstract
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.
Highlights
We shall study the influence of horizon curvatures, the graviton mass and the dimension of spacetime on the velocity dependent complexity growth, which is dual to Nambu–Goto action growth in massive black hole with a probe string
We studied the complexity growth in the dynamical system with a Wilson line operator, which is holographically dual to the massive black holes with a probe string
We focused on the Nambu–Goto action growth and employed the CA conjecture to explore the effect of string on the complexity growth in a boundary gauge theory with momentum relaxation
Summary
I where R is the scalar curvature, l is the AdS radius and f is a fixed symmetric tensor. Modeling systems via translationally invariant quantum field theories always comes across problems unless the effects of momentum dissipation is incorporated. Massive gravity is a completive candidate in which the momentum dissipation is involved, and it is an effective bulk theory that does not conserve momentum without borrowing additional fields. In the action (1), the last terms represent massive potentials associated with the graviton mass which breaks the diffeomorphism invariance in the bulk, which produces momentum relaxation in the dual boundary theory. Where hi j d xi d x j is a line element of Einstein space with constant curvature n(n − 1)k and k = 1, 0, −1 corresponds to a spherical, Ricci flat, and hyperbolic horizon for black hole. Where the integral constant m0 is the black hole mass parameter
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