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We investigate black brane solutions in asymptotically Lifshitz spacetime in 3+1dimensional Horndeski gravity, which parameters are related to the cosmological constant as α = γΛ and depend on arbitrary values of the dynamical critical exponent z since Λ = −(1 + 2z)/L 2 . For the case z = 1 we recover black brane solutions in asymptotically AdS 4 spacetime. We also investigate the shear viscosity in the 2+1dimensional dual boundary field theory via holographic correspondence. We show that for arbitrary values of AdS radius L, only two specific critical exponents are allowed: z = −0.3 or z = 3.3. At the former value, we find that the bound for viscosity to entropy density ratio η/s ≥ 1/(4π) is violated.

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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.