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In this study we apply two different methods in the context of $AdS/CFT$ correspondence and calculate the diffusion coefficient and $DC$ conductivity of a four-dimensional spatially anisotropic static black hole. First, the \emph{modified} transport coefficients is obtained by stretched horizon method and Fick's law in the context of the membrane paradigm. In order to do such calculation, we use the Maxwell equations with electromagnetic gauge field propagating in two dimensions. Two dimensional propagating gauge field leads to the complex transport coefficients which is proved by present paper. In second step, we explain electro-thermal method and employ an effective vector field and extract retarded Green's function on the classical boundary. Then, $DC$ conductivity and diffusion coefficient are obtained by using Kubo formula. Our calculation can be applied on two well-known examples of anisotropic black holes as the Einstein-Maxwell-dilaton-axion model and AdS-Einstein-Maxwell-dilaton-axion in massive gravity., Comment: 17 pages

In this paper, we take the slowly rotating black three-brane background and perturb it by introducing a vector gauge field. We find the components of the gauge field through Maxwell equations and Bianchi identities. Using currents and some ansatz we find Fick's first law at long wavelength regime. An interesting result for this non-trivial supergravity background is that the diffusion constant on the stretched horizon which emerges from Fick's first law is a complex constant. The pure imaginary part of the diffusion constant appears because the black three-brane has angular momentum. By taking the static limit of the corresponding black brane the well known diffusion constant will be recovered. On the other hand, from the point of view of the Fick's second law, we have the dispersion relation $\omega=-iDq^{2}$ and we found a damping of hydrodynamical flow in the holographically dual theory. Existence of imaginary term in the diffusion constant introduces an oscillating propagation of the gauge field in the dual field theory.

In this paper, we determine thermodynamical quantities for one kind of higher dimensional black holes. We take two charged and neutral blackfold which are higher dimension. We investigate the thermodynamical properties of such black hole at near horizon. By expanding those metrics around horizon, considering periodicity condition of imaginary time and applying some thermodynamical constraints, we extract stress energy tensor of blackfolds. We compare the thermodynamical properties of charged and neutral blackfold. We see here that the stress energy tensor of charged blackfold has brane-tension component in addition to thermal properties. Then by knowing that the spacetime of neutral blackfold is Ricci-flat and there is a nice relation between Ricci-flat and AdS spacetimes we determine the AdS form of Ricci-flat. Also we calculate the corresponding temperature which is satisfied by our previous calculation. Finally, we check the phase transition of blackfolds and show that there is not any critical point for them. So, the thermodynamical stability of both blackfolds will be proven, this result agree with literature., Comment: 11 pages

In this paper we consider charged and neutral blackfold and extract the Brown-York stress energy tensor. Also, we show that the neutral blackfold spacetime is Ricci- flat and the other spacetime is not. This Ricci-flat condition gives us opportunity to calculate the AAdS spacetime. In order to have dual theory one can consider the AAdS in Fefferman- Graham coordinates. This frame gives correct form of stress tensor in the boundary. The corresponding tensor with using this frame will be traceless and conserved. Such stress tensor is same as perfect fluid and it proves the dual renormalized theory exists for the neutral blackfold ., Comment: 12 pages