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arxiv: 1907.11757 · v1 · pith:CIZZSV6Vnew · submitted 2019-07-26 · ✦ hep-th · gr-qc

Inhomogeneous Jacobi equation and Holographic subregion complexity

Avirup Ghosh , Rohit Mishra This is my paper

Pith reviewed 2026-05-24 15:28 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic subregion complexityinhomogeneous Jacobi equationentangling surface deformationsboosted black braneAdS perturbationsminimal surfaceAdS/CFT
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The pith

A variational method using the inhomogeneous Jacobi equation gives the perturbative change in holographic subregion complexity around pure AdS.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a perturbative variational method to calculate holographic subregion complexity for asymptotically AdS spacetimes around pure AdS. It relies on solving an inhomogeneous version of the Jacobi equation to track deformations of the entangling surface caused by small bulk metric perturbations. This yields the first-order change in complexity for strip and circular disk subsystems under boosted black brane perturbations in AdS4. The same method shows that the linear change vanishes for spherical subsystems in 3+1 dimensions.

Core claim

We derive a general expression for obtaining Holographic subregion complexity for asymptotically AdS spacetimes, perturbatively 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 boosted black brane like perturbations over pure AdS4. As a corollary, we find that for spherical subsytems

What carries the argument

inhomogeneous version of the Jacobi equation for studying deformations of the entangling surface under bulk metric perturbations

If this is right

  • The first-order change in subregion complexity is obtained explicitly for strip subsystems.
  • The first-order change in subregion complexity is obtained explicitly for circular disk subsystems.
  • The linear change in subregion complexity vanishes for spherical subsystems in 3+1D bulk.
  • The variational method applies to any asymptotically AdS spacetime perturbatively around pure AdS.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The vanishing result for spheres may trace to a symmetry of the complexity functional that cancels the linear term.
  • The same deformation technique could be applied to compute changes in other holographic quantities such as entanglement entropy.
  • One could test whether the linear vanishing persists for spherical subsystems in higher-dimensional bulks or under different perturbations.
  • The approach supplies a systematic route to near-AdS geometries without needing the full minimal surface at each order.

Load-bearing premise

Deformations of the entangling surface under small bulk metric perturbations are captured by solutions to the inhomogeneous Jacobi equation.

What would settle it

Direct numerical computation of the minimal surface and subregion complexity for a concrete small boosted black brane perturbation, then comparison with the linear-order prediction.

Figures

Figures reproduced from arXiv: 1907.11757 by Avirup Ghosh, Rohit Mishra.

Figure 1
Figure 1. Figure 1: FIG. 1. Solution of the co-dimension one minimal surface equation with boundary conditions given by [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives a general perturbative expression for holographic subregion complexity in asymptotically AdS spacetimes around pure AdS, using a variational technique that requires solving an inhomogeneous Jacobi equation to determine first-order deformations of the Ryu-Takayanagi surface under bulk metric perturbations. The method is applied to boosted black-brane-like perturbations in AdS4, yielding explicit results for strip and circular-disk subsystems; as a corollary, the linear change in subregion complexity is shown to vanish for spherical subsystems.

Significance. If the central derivation and cancellation hold, the work supplies a concrete variational tool for computing complexity perturbations and identifies a geometry-dependent vanishing that may simplify higher-order calculations or constrain holographic complexity proposals. The explicit use of the inhomogeneous Jacobi equation and the reported cancellation constitute a falsifiable technical result.

major comments (2)
  1. [the section deriving the inhomogeneous Jacobi equation and its solution for the boosted perturbation] The cancellation that produces the vanishing linear complexity change for spherical subsystems (abstract and the corollary statement) requires that the first-order volume correction from the normal displacement of the RT surface exactly cancels the direct metric perturbation contribution inside the unperturbed volume integral. The manuscript must exhibit the explicit particular solution to the inhomogeneous Jacobi equation, the chosen boundary conditions (regularity at the center and fall-off at the AdS boundary), and the resulting integral evaluation that demonstrates this cancellation; without these steps the claim remains unverified.
  2. [derivation of the inhomogeneous term] The inhomogeneous source term in the Jacobi equation arises from the linearised minimality condition on the area functional. The paper should confirm that this source is correctly obtained from the boosted black-brane metric perturbation and that no homogeneous solutions are inadvertently added that would alter the volume correction.
minor comments (2)
  1. The abstract states the results but supplies no equations or error estimates; the main text should include at least one representative equation for the inhomogeneous Jacobi operator and the final complexity correction formula.
  2. Notation for the normal displacement function and the volume functional should be introduced once and used consistently throughout the perturbative expansion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments correctly identify places where additional explicit detail will strengthen the presentation. We will revise the manuscript to include the requested explicit expressions and verifications while preserving the original derivations.

read point-by-point responses

  1. Referee: [the section deriving the inhomogeneous Jacobi equation and its solution for the boosted perturbation] The cancellation that produces the vanishing linear complexity change for spherical subsystems (abstract and the corollary statement) requires that the first-order volume correction from the normal displacement of the RT surface exactly cancels the direct metric perturbation contribution inside the unperturbed volume integral. The manuscript must exhibit the explicit particular solution to the inhomogeneous Jacobi equation, the chosen boundary conditions (regularity at the center and fall-off at the AdS boundary), and the resulting integral evaluation that demonstrates this cancellation; without these steps the claim remains unverified.

    Authors: We agree that the explicit particular solution, boundary conditions, and integral evaluation are essential for verifying the cancellation. In the revised manuscript we will add a dedicated subsection displaying the particular solution to the inhomogeneous Jacobi equation for the boosted black-brane perturbation, state the regularity condition at the center and the required fall-off at the AdS boundary, and carry out the explicit volume integral that shows exact cancellation of the two first-order contributions for spherical subsystems. revision: yes

  2. Referee: [derivation of the inhomogeneous term] The inhomogeneous source term in the Jacobi equation arises from the linearised minimality condition on the area functional. The paper should confirm that this source is correctly obtained from the boosted black-brane metric perturbation and that no homogeneous solutions are inadvertently added that would alter the volume correction.

    Authors: We will expand the derivation of the inhomogeneous term to show step-by-step that it follows directly from the linearised minimality condition applied to the boosted black-brane metric. We will also add an explicit statement that the solution we employ is the particular solution satisfying the chosen boundary conditions, with no homogeneous solutions superposed, thereby confirming that the volume correction remains unaltered. revision: yes

Circularity Check

0 steps flagged

No circularity: perturbative cancellation obtained from explicit solution of inhomogeneous Jacobi equation

full rationale

The derivation proceeds by writing a variational expression for the first-order change in subregion complexity, sourcing the inhomogeneous Jacobi equation from the linearized minimality condition on the area functional, solving that PDE for the normal displacement of the RT surface under the given metric perturbation, and then integrating the resulting volume correction. The reported vanishing for spherical subsystems in AdS4 is the output of that explicit integration (normal displacement term cancels the direct metric perturbation term), not an input or a redefinition. No self-citation is invoked as a uniqueness theorem, no parameter is fitted and then relabeled as a prediction, and the inhomogeneous source is derived from the area functional rather than assumed. The calculation is therefore self-contained against the stated boundary conditions and the explicit form of the boosted black-brane perturbation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the variational technique and the inhomogeneous Jacobi equation to perturbative deformations around pure AdS; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The variational technique can be applied to find subregion complexity perturbatively around pure AdS using solutions of the inhomogeneous Jacobi equation.
    Explicitly stated as the essential step in the abstract.

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 24 internal anchors

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