An inhomogeneous Jacobi equation for minimal surfaces and a perturbative change of Holographic Entanglement Entropy

The change in Holographic entanglement entropy (HEE) for small fluctuations about pure AdS are obtained by a perturbative expansion of the area functional in terms of the change in the bulk metric and the embedded extremal surface. However, it is known that change in the embedding appears at second order or higher. It was shown that these changes in the embedding can be calculated in the 2+1 dimensional case by solving a 'generalized geodesic deviation equation'. We generalize this result to arbitrary dimensions by deriving an inhomogeneous form of the Jacobi equation for minimal surfaces. The solutions of this equation map a minimal surface in a given space-time to a minimal surface in a space-time which is a perturbation over the initial space-time. Using this we perturbatively calculate the changes in HEE up to second order for the boosted black brane like perturbations over AdS4 .