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

TL;DR

This paper derives a generalized inhomogeneous Jacobi equation for minimal surfaces in arbitrary dimensions to compute second-order perturbations in Holographic Entanglement Entropy caused by small metric fluctuations.

Contribution

It extends the 2+1 dimensional geodesic deviation approach to arbitrary dimensions by formulating an inhomogeneous Jacobi equation for minimal surfaces.

Findings

Derived an inhomogeneous Jacobi equation for minimal surfaces in any dimension.

Applied the equation to compute second-order HEE changes for boosted black brane perturbations.

Provided a perturbative method for analyzing HEE in perturbed AdS spacetimes.

Abstract

The change in Holographic entanglement entropy (HEE) for small fluctuations about pure anti De Sitter (AdS) is 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 in 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 upto second order for boosted black brane like perturbations over $AdS 4$ .