Metric Perturbations of Extremal Surfaces

Abstract

Motivated by the HRRT-formula for holographic entanglement entropy, we consider the following question: what are the position and the surface area of extremal surfaces in a perturbed geometry, given their anchor on the asymptotic boundary? We derive explicit expressions for the change in position and surface area, thereby providing a closed form expression for the canonical energy. We find that a perturbation governed by some small parameter $\lambda$ yields an expansion of the surface area in terms of a highly non-local expression involving multiple integrals of geometric quantities over the original extremal surface.

Example added, see subsection 3.6 and section 6

References in corpus (6)

• Deriving covariant holographic entanglement
• Entanglement equilibrium for higher order gravity
• An Inhomogeneous Jacobi equation for minimal surfaces and perturbative change of Holographic Entanglement Entropy
• Generalized geodesic deviation equations and an entanglement first law for rotating BTZ black holes
• A relationship between scalar Green functions on hyperbolic and Euclidean Rindler spaces
• More on Holographic Volumes, Entanglement, and Complexity

Cited by in corpus (7)

• Towards Bulk Metric Reconstruction from Extremal Area Variations
• The Holographic Shape of Entanglement and Einstein's Equations
• Surface Theory: the Classical, the Quantum, and the Holographic
• Holographic excited states in AdS Black Holes
• An Inhomogeneous Jacobi equation for minimal surfaces and perturbative change of Holographic Entanglement Entropy
• More of the Bulk from Extremal Area Variations
• Geometrical tools for embedding fields, submanifolds, and foliations