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

Understand

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.