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

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.