Metric Perturbations of Extremal Surfaces

TL;DR

This paper derives explicit formulas for how extremal surfaces in holography change under small geometric perturbations, linking these changes to the canonical energy and providing insights into entanglement entropy calculations.

Contribution

It introduces a closed-form expression for the surface area variation of extremal surfaces under perturbations, connecting geometric changes to canonical energy in holographic settings.

Findings

Explicit formulas for extremal surface perturbations

Expansion of surface area involving non-local integrals

Connection between perturbations and canonical energy

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.