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

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 $AdS4$ .

31 Pages, Numerical results modified, typos corrected, two figures and comments added on the positivity of relative entropy

References in corpus (6)

• Relative entropy and the Bekenstein bound
• On shape dependence of holographic mutual information in AdS4
• Entanglement Entropy in Quantum Gravity and the Plateau Problem
• Metric Perturbations of Extremal Surfaces
• Generalized geodesic deviation equations and an entanglement first law for rotating BTZ black holes
• Perturbations of spiky strings in flat spacetimes

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• Informational properties of holographic Lifshitz field theory
• Shape dependence of renormalized holographic entanglement entropy
• More of the Bulk from Extremal Area Variations
• Geometrical tools for embedding fields, submanifolds, and foliations
• Metric Perturbations of Extremal Surfaces
• Entanglement entropy at higher orders for the states of a = 3 θ = 1 Lifshitz theory
• Holographic entanglement entropy for $Li{f}_4^{(2)}\times S^1\times S^5$ spacetime with string excitations
• Entanglement Entropy for D3-, M2- and M5-brane backgrounds