Complexity of four-dimensional hairy anti-de-Sitter black holes with a rotating string and shear viscosity in generalized scalar–tensor theories

Abstract

In four dimensions, we consider a generalized scalar–tensor theory where the coupling functions only depend on the kinetic term of the scalar field. For this model, we obtain a set of hairy anti-de-Sitter black hole solutions, allowing us to calculate the computational complexity, according to the Complexity equals Action conjecture. To perform this, the system contains a particle moving on the boundary, corresponding to the insertion of a fundamental string in the bulk. The effect string is given by the Nambu–Goto term, analyzing the time development of this system. Together with the above, we calculate the shear viscosity, where the viscosity/entropy density ratio can violate the Kovtun–Son–Starinets bound for a suitable choice of coupling functions.

Holographic Complexity of Braneworld in Horndeski Gravity

“…Thus, U(𝜔) is expected to be Gaussian like, which in turn leaves the AdS space non-conforming, that is, it implies an energy configuration like E(L) ∼ L. Furthermore, such an energy configuration is presented in this way, to agree with the linear confinement of m 2 n growing with n, where L ∼ m 2 n . We solve numerically Equation (16) and introduced these solutions graphically on the Figure (3). As expected, near the braneworld origin (where brane is usually located), graviton potential behave as a gaussian-like, which ensures the AdS space non-conformality.…”

“…On the other hand, such growth is a generic property of any linear confining gauge theory. Thus, we can formulate and calculate the computational complexity, according to the Complexity equals Action conjecture, [1,3] in order to assess the warp factor influence on the action complexity growth. Now, following the prescription of [13,37] and considering the total action(Equation 20) with only null boundaries present (therefore, ignoring any joint contributions to the total action integral), we have that the NG action of probe string is given by:…”

“…On the other hand, such growth is a generic property of any linear confining gauge theory. Thus, we can formulate and calculate the computational complexity, according to the Complexity equals Action conjecture, [ 1,3 ] in order to assess the warp factor influence on the action complexity growth. Now, following the prescription of [13, 37] and considering the total action(Equation ) with only null boundaries present (therefore, ignoring any joint contributions to the total action integral), we have that the NG action of probe string is given by: $$$$\begin{align} {S}_{\textit{NG}}=-{T}_{s}\int \chi d\sigma \sqrt{-\textit{det}{g}_{\textit{ind}}}, \end{align}$$$$ $$$$\begin{align} \chi =1+2\gamma {g}^{\textit{MN}}{\nabla}_{M}\phi {\nabla}_{N}\phi \end{align}$$$$To analyze the evolution of the string moving within the warped geometry of a braneworld, we need to construct an induced metric using the parameters τ and σ in the world‐sheet of the fundamental string.…”

“…[1] Also, beyond Horndeski theories are now being developed and incorporated. [3,4] Aforementioned black hole entropy is the thermodynamic quantity, which is extracted through the AdS/BCFT correspondence as presented by the authors. [2,4] This correspondence [5][6][7][8] is the most important realization of the holographic principle, [9,10] which relates a gravity theory in a higher dimensional anti-de Sitter (AdS) bulk to a conformal field theory (CFT), but without gravity living on the bulk boundary.…”

AdS/BCFT correspondence and Horndeski gravity in the presence of gauge fields: from holographic paramagnetism/ferromagnetism phase transition

“…where R, G µν and Λ are the scalar curvature, the Einstein tensor, and the cosmological constant respectively, φ = φ(r) is a scalar field, α and γ are coupling constants, while that κ = 1/(16πG N ), where G N is the Newton Gravitational constant. The Lagrangian (1) has been exhaustively explored from the perspective of hairy black hole configurations [27][28][29][30][31], boson and neutron stars [33][34][35], Hairy Taub-NUT/Bolt-AdS solutions [36], as well as holographic applications such that quantum complexity and shear viscosity [37][38][39][40][41].…”