Inhomogeneous Jacobi equation for minimal surfaces and perturbative change in 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 at second order or higher. It was shown that these changes in the embedding can be calculated in the 2 þ 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 up to second order for boosted black brane like perturbations over AdS 4 .

Holographic subregion complexity of boosted black brane and Fisher information

“…Thus at second order one cannot work with the same t = constant embedding for stationary asymptotically AdS spactimes. But as shown in [48] one can still work with the t = constant slice in the boosted black brane spacetime but only for the perpendicular case. This is due to the fact that the minimal surface still remains in the same time slice.…”

Surface theory: the classical, the quantum, and the holographic

“…In this section, we give a unified treatment of classical surface theory [11][12][13][14][15][16][17][18][19][20][21][22], presented to maximize ease in generalizations to quantum extremal surfaces (of these references, we would highlight [16] for a very pleasant introduction to the topic of embedded surfaces). To begin, let us review some basic definitions and properties of surfaces with a particular emphasis on formalism that will be useful to later deriving properties of perturbations of extremal surfaces.…”

“…. , d on (at least a portion of) M ; then the map ψ : Σ → M which embeds Σ in M is described by the d embedding functions X µ (y) 21 . Any tensor field V a 1 ···a k b 1 ···b l which is a functional of Σ can therefore be expressed in this coordinate system as a functional of the X µ (y):…”

Entanglement entropy at higher orders for the states of a = 3 θ = 1 Lifshitz theory

“…The boundary of such extremal surfaces coincides with the boundary of the subsystem in the CFT. Recently it has been observed that the excitations in the CFT follow entanglement laws similar to the black hole thermodynamic laws [5,6,7]; see also [9], [12], [10], [13]. It is understood now that the entanglement entropy (S E ) and the energy of small excitations (E) in AdS spacetime obey a definite relation…”