[Paper Review] Inhomogeneous Jacobi equation and Holographic subregion complexity

Avirup Ghosh, Rohit Mishra|arXiv (Cornell University)|Jul 26, 2019

Black Holes and Theoretical Physics2 references4 citations

TL;DR

This paper develops a perturbative variational method to compute holographic subregion complexity in asymptotically AdS spacetimes by solving an inhomogeneous Jacobi equation that governs deformations of the minimal surface dual to a boundary subregion. The approach yields the linear-order change in volume complexity for strip and disk-like regions under boosted black brane perturbations over pure AdS₄, with the key result that the linear change vanishes for spherical subregions in 3+1 dimensions.

ABSTRACT

We derive a general expression for obtaining Holographic subregion complexity for asymptotically $AdS$ spacetimes, pertubatively around pure $AdS$ using a variational technique. An essential step in finding subregion complexity is to identify the bulk minimal surface of the entangling subregion. Our method therefore heavily relies on solutions of an inhomogeneous version of Jacobi equation, used to study deformations of the entangling surface for perturbations of the bulk metric. Using this method we have obtained the change in complexity for a strip and a circular disk like subsystem for \emph{boosted} black brane like perturbations over pure $AdS_4$. As a corollary, we find that for spherical subsytems in $3+1$ dimensional bulk, the linear change of subregion complexity for \emph{boosted} black brane like perturbations over pure $AdS_4$ , vanishes.

Motivation & Objective

• To develop a general perturbative method for computing holographic subregion complexity in asymptotically AdS spacetimes beyond pure AdS.
• To identify the bulk minimal surface for a boundary subregion under metric perturbations, which is essential for complexity calculations.
• To derive the linear-order change in subregion complexity for specific geometries—strip and circular disk—under boosted black brane-like perturbations.
• To show that for spherical subregions in 3+1 dimensional bulk spacetimes, the linear change in subregion complexity vanishes under such perturbations.
• To establish a framework based on the inhomogeneous Jacobi equation for studying surface deformations in perturbed gravitational backgrounds.

Proposed method

• The method employs a variational technique to compute changes in the volume of a maximal co-dimension one bulk surface bounded by the minimal co-dimension two surface dual to the boundary subregion.
• It relies on solving an inhomogeneous Jacobi equation that describes the first-order deformation of the entangling surface under metric perturbations.
• The derivation accounts for variations in both the embedding of the surface and the bulk metric, using a perturbative expansion around pure AdS₄.
• The approach incorporates the extrinsic curvature and mean curvature vector variations, with tangent and normal components treated via Gauss and Codazzi equations.
• The formalism uses the symplectic structure of gravity and its relation to boundary quantum state complexity, aligning with the Complexity=Volume conjecture.
• Key identities involving the Riemann tensor, metric variation, and connection changes are derived to handle second-order perturbations and surface terms.

Experimental results

Research questions

• RQ1 How can holographic subregion complexity be computed perturbatively in asymptotically AdS spacetimes with non-trivial bulk geometry?
• RQ2 What is the role of the inhomogeneous Jacobi equation in determining the deformation of the minimal surface under metric perturbations?
• RQ3 How does the linear change in subregion complexity behave for different boundary subregion shapes—specifically strip and circular disk—under boosted black brane-like perturbations?
• RQ4 Why does the linear change in complexity vanish for spherical subregions in 3+1 dimensional bulk spacetimes under such perturbations?
• RQ5 What is the contribution of tangent variations of the extrinsic curvature and mean curvature vector to the overall complexity variation?

Key findings

• The linear-order change in holographic subregion complexity for a strip-like subregion in a boosted black brane background over pure AdS₄ is non-zero and computable via the inhomogeneous Jacobi equation.
• For a circular disk-like subregion under the same perturbations, the linear change in complexity is also non-zero and derived using the same variational framework.
• The method successfully computes the complexity change by solving the inhomogeneous Jacobi equation for surface deformations, providing a systematic perturbative approach.
• For spherical subregions in 3+1 dimensional bulk spacetimes, the linear change in subregion complexity vanishes under boosted black brane-like perturbations, indicating a symmetry-protected cancellation.
• The formalism accounts for both metric and embedding variations, with tangent contributions shown to not affect the final inhomogeneous Jacobi equation, simplifying the computation.