Geometric Entropies and their Hamiltonian Flows

Donald Marolf; Pratik Rath; Xi Dong

Geometric Entropies and their Hamiltonian Flows

T0 review · 0 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The geometric entropy's Hamiltonian flow is not a pure boundary-condition-preserving kink once higher-derivative terms are included.

desk verdict Higher-derivative corrections break the BCP-kink form of geometric entropy flow; the paper shows this cleanly with two explicit 2D examples and a field-redefinition argument. read the letter →

arxiv 2501.12438 v2 pith:FRDNYEQC submitted 2025-01-21 hep-th gr-qc

Xi Dong , Donald Marolf , Pratik Rath This is my paper · ORCID

classification hep-thgr-qc

keywords geometricentropyhigher-derivativegravityHamiltonianflowPeierlsbracketBCPkinktransformationJTholographicentanglementmodular

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what Hamiltonian flow the geometric entropy generates on the phase space of a gravitational theory with higher-derivative corrections. In Einstein gravity and JT gravity with minimal couplings, the answer is a boundary-condition-preserving (BCP) kink-transformation: the flow adds a delta-function boost to the extrinsic curvature at the HRT surface and leaves all other Cauchy data alone. The paper's central claim is that this pure geometric form fails once perturbative higher-derivative interactions are included: the flow still adds the extrinsic-curvature kink, but it also produces delta-function jumps in the dilaton and matter fields, localized at the HRT surface, because the kink alone no longer satisfies the constraint equations. The claim is shown explicitly in two JT-gravity-plus-scalar theories using Peierls brackets, backed by a general field-redefinition argument and by Poisson/Dirac bracket checks. If correct, holographic modular flow should generically induce such non-geometric singularities whenever the HRT surface is not the bifurcation surface of a Killing horizon.

What carries the argument

The load-bearing object is the Peierls bracket of the geometric entropy $\sigma$ with arbitrary observables: one deforms the action by $-s\sigma$, solves the retarded and advanced linearized solutions, and takes their difference to get the Hamiltonian vector field of $\sigma$. The geometric entropy is the response of the Euclidean action to a conical defect at the extremal HRT surface (the extremal codimension-2 surface whose area gives holographic entanglement entropy), and the BCP kink-transformation is the operation that adds $2\pi s\,\delta_\Sigma(\gamma)$ to the extrinsic curvature component $K_{\perp\perp}$ while leaving other Cauchy data fixed. In two dimensions the retarded solution is a Lorentzian conical defect with conformal factor $\omega=\pi s\,\theta(-u)\theta(v)$, and the flow is read off by comparing Cauchy data on slices just before and just after the defect. A second, more general argument uses perturbative field redefinitions: since $\sigma$ transforms as a scalar while field redefinitions can mix $K_{\perp\perp}$ with non-metric fields, the flow cannot be universally the pure geometric kink.

What would settle it

Compute the Peierls bracket of the geometric entropy in Theory 1 at first order in $\lambda$ on a non-Killing HRT surface and check whether the term $\delta\dot{\varphi} = -2\pi\lambda s (\nabla\psi)^2 \delta_\Sigma(\gamma)$ appears; if the pure BCP kink data, with only $K_{\perp\perp}$ shifted, satisfied the constraints, the paper's central claim would be wrong. The broader falsifier is to exhibit any perturbative higher-derivative theory whose geometric entropy flow is exactly the BCP kink on a non-Killing spacetime, which would contradict the field-redefinition covariance argument.

Watch

Extended reading notes

Core claim

This paper establishes that, in a general gravitational theory with perturbative higher-derivative terms, the Hamiltonian flow generated by the geometric entropy $\sigma$ does not coincide with the boundary-condition-preserving (BCP) kink transformation, except when the HRT surface is the bifurcation surface of a Killing horizon. In the two worked examples, inserting $\sigma$ into the action makes the geometry a Lorentzian cone with the usual kink in $K_{\perp\perp}$, but in the original fields the flow also changes $\dot{\varphi}$ and, in the second theory, $\dot{\psi}$ and $\dot{\rho}$ by terms proportional to $\lambda$ times matter gradients contracted with a delta function supported on the HRT surface, for example $\delta\dot{\varphi} = -2\pi\lambda s (\nabla\psi)^2 \delta_\Sigma(\gamma)$ in Theory 1. The pure BCP kink data fails the constraint equations, and the extra singularities are exactly what restores them. For Killing horizons the extra terms vanish because the Killing symmetry removes the delta function by a coordinate shift, and the flow is the BCP kink generated by Wald entropy; Appendix E proves this for general theories, not just the examples. The covariant Peierls bracket is the tool that makes the computation tractable, and Appendix B reproduces the results with Dirac brackets.

Load-bearing premise

The load-bearing premise is that the higher-derivative coupling $\lambda$ is perturbatively small, so all fields are expanded in powers of $\lambda$ and no new degrees of freedom enter; if $\lambda$ were not small, the Ostrogradsky instability and new modes would require a different phase space.

Editorial extensions

If this is right

The BCP kink transformation alone does not preserve the constraint equations in higher-derivative gravity; the geometric entropy flow must add dilaton and matter singularities localized at the HRT surface.
In the Killing-horizon case the extra singularities vanish, so the flow reduces to the BCP kink and agrees with the Wald entropy flow; this is the regime where the type II von Neumann algebra argument for generalized entropy holds.
If the holographic correspondence is right, boundary modular flow in a state without Killing symmetry should produce non-geometric bulk singularities of the same kind, in addition to the Weyl shocks already known in Einstein gravity.
In the example theories, the susceptibility construction recovers the geometric entropy only after adding total-derivative terms, so the Lorentzian analogue of the Euclidean derivation requires a non-obvious choice of presymplectic potential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: because the paper's field-redefinition argument is general, any theory obtained from Einstein gravity by a field redefinition that mixes the kink direction $K_{\perp\perp}$ with matter should show non-geometric singularities, so the two examples should be generic rather than accidents of JT gravity.
Inference: in settings without a Killing horizon, the type II algebra boost generator may need to include matter and dilaton dressing, not just the geometric boost; checking this would require extending the calculation to higher dimensions and beyond the perturbative coupling regime.
Inference: the existence of shockwave-free one-parameter families of generators suggests that identifying the geometric entropy flow in a general theory may require outside information, such as the boundary modular Hamiltonian, rather than a purely local bulk rule.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the headline: the paper answers a real question in holographic entanglement, whether geometric entropy flow is always a BCP kink transformation, and the answer is no once higher-derivative couplings are included. The evidence is two explicit 2D examples plus a field-redefinition argument, and the Killing-horizon case is cleanly separated.

What's new and good: the two JT-plus-scalar computations are the first explicit geometric entropy flows in higher-derivative theories. The Peierls-bracket method is well suited, and Appendix B's Dirac-bracket cross-check makes the examples trustworthy. The field-redefinition argument in the introduction is a nice conceptual explanation for why the BCP form should fail generically. Appendix E giving the Killing-horizon result for general theories is a useful extension of earlier work. The susceptibility analysis in Appendix C is a neat tool, and the results are internally consistent, with no fitting and no circularity.

Soft spots, in proportion: the perturbative-in-lambda assumption is the load-bearing one, stated plainly at the start of Section 4.1. It means the conclusions apply to theories with perturbative higher-derivative corrections and no new degrees of freedom. That is a real restriction, but it is exactly the regime where the question was open. The examples are two-dimensional and special; the general field-redefinition argument is persuasive but not a theorem. The paper itself flags the possible O(s^2) delta-function products when integrating the flow, shows they do not arise in these examples, but leaves the general question open. The discussion of modular flow leans on a forthcoming paper [25] and is suggestive, not a proof. None of these undermines the main claim.

Bottom line: this is a paper I would send to a serious referee. The computations are explicit and cross-checked, the caveats are honestly scoped, and it advances the subfield. I'd cite it for the explicit higher-derivative flow examples, and it's a good reading-group candidate for a holography group.

Referee Report

0 major / 3 minor

Summary. The paper studies the Hamiltonian flow generated by the geometric entropy in two-dimensional dilaton-gravity theories with perturbative higher-derivative matter couplings. Using the Peierls bracket formalism, the authors compute the linearized flow for two JT-gravity-plus-scalar models, showing that the flow is not simply the boundary-condition-preserving (BCP) kink transformation: in addition to the extrinsic-curvature delta function (1.3), the dilaton and matter fields acquire distributional singularities localized at the HRT surface, e.g. Eqs. (4.33)-(4.35) and (4.62)-(4.66). The results are cross-checked with an independent Dirac-bracket calculation in Appendix B, and Appendix E shows that for Killing-horizon bifurcation surfaces the flow reduces to the BCP form and is generated by the Wald entropy. The computations are performed explicitly under the stated perturbative-in-λ assumption (Sec. 4.1), and the paper discusses connections to modular flow, type II algebras, and the Lorentzian derivation of geometric entropy.

Significance. If the results hold, the paper resolves an open question about the universality of the BCP kink transformation: it demonstrates by explicit construction that in higher-derivative theories the geometric entropy flow generically acquires additional, non-geometric singularities, while reproducing the known BCP form in the Killing-horizon case. This is important for the program of deriving type II von Neumann algebras from gravitational constraints and for the holographic interpretation of modular flow. The paper's strengths include the use of two independent formalisms (Peierls and Dirac brackets) that agree, the explicit order-by-order nature of the computations, the clearly stated perturbative regime, and the Killing-horizon cross-check in Appendix E. The perturbative-in-λ assumption is a genuine limitation for inferring non-perturbative behavior, but it is transparently stated and does not affect the validity of the claims within that regime.

minor comments (3)

[Eq. (4.21)] The field-redefinition notation is confusing: the displayed equality 'φ̃ = φ − (λ/2)∇_αψ∇^αψ = φ − λ ∇_uψ∇_uψ' cannot be correct as written, since the last term is not equal to the contraction; in conformal gauge the intended expression is φ − λ e^{-2ω}∂_uψ∂_vψ (or the second equality should be removed). Because Eq. (4.28) and the subsequent computation are consistent, this is a typographical issue, but it should be corrected to avoid a mismatch for readers.
[Sec. 5 (Modular flow)] The discussion of modular flow relies on a forthcoming companion paper [25] for a key claim about the relation between BCP kink transformations and modular flow more generally. Since the present paper is otherwise self-contained, the authors should either clarify the extent to which this discussion depends on unpublished work or mark those statements as conjectural.
[Throughout] A careful proofreading pass is recommended: the text as provided contains several typographical and OCR artifacts (e.g., 'Bar bara' in the affiliation, 'Jack iw-Teitelboim' in the abstract, 'umambiguously' in Sec. 2.2), and a few equations have broken or inconsistent spacing. These do not affect the results but would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geometric-entropy flow is computed from a stipulated sigma via Peierls brackets, with no fitted input or self-referential reduction.

full rationale

The paper's derivation chain is self-contained. The geometric entropy sigma is taken as a stipulated input, either from the known JT result or from Ref. [23] in the higher-derivative examples, and the flow is then computed by solving the Peierls-bracket equations of motion with sigma added as a source. The central results, such as Eqs. (4.32)-(4.35) and Eqs. (4.62)-(4.66), are derived by explicit solution of the sourced equations of motion, not by fitting a parameter and calling it a prediction. The Appendix C susceptibility calculation reconstructs sigma from the flow, but the paper explicitly presents this as an alternate computation of the generator given a transformation, and it reproduces the same sigma rather than being used to define it; this is a consistency check rather than a circular reduction. The field-redefinition argument in Sec. 1 cites Ref. [31] for the transformation property of geometric entropy, but the central claim does not rest on that citation: the explicit two-dimensional examples and the Killing-horizon check in Appendix E stand independently. Self-citations to Refs. [23, 25, 31] are either prior parameter-free inputs with stated assumptions that do not include the target result, or contextual remarks about future work. No equation in the paper is defined in terms of the target result, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. The perturbative-in-lambda restriction is a clearly stated scoping assumption, not a circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It relies on the previously derived expression for σ (Ref. [23]) and on standard assumptions about perturbativity, extremality, and the Peierls bracket formalism.

assumptions (4)

domain assumption The geometric entropy σ for each theory is taken from the Euclidean analysis of Ref. [23]: σ = 2πφ − πλ(∇ψ)^2 in Theory 1 and σ = 2πφ − πλρ(∇ψ)^2 in Theory 2.
Used as the observable whose flow is computed; the paper does not re-derive σ from the action.
domain assumption The higher-derivative coupling λ is treated perturbatively, so no new degrees of freedom are introduced and the Ostrogradsky instability is avoided.
Stated in Sec. 4.1; all results are formal power series in λ.
domain assumption The HRT surface is an extremum of σ, so variations of the surface location vanish (∂σ/∂x = 0).
Used repeatedly to drop delta-function terms in constraint equations, e.g., after Eq. (4.18).
standard math The Peierls bracket is equivalent to the Poisson bracket on the phase space of classical solutions, and the flow is defined via dO/ds = {O, σ}.
Framework reviewed in Sec. 2.2 and used throughout.

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Cite this review

Pith. "Pith review of Geometric Entropies and their Hamiltonian Flows." pith.science (2026). https://pith.science/paper/FRDNYEQC

@misc{pith2026250112438,
  author       = {Pith},
  title        = {Pith review of: Geometric Entropies and their Hamiltonian Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRDNYEQC}},
  note         = {Machine review of arXiv:2501.12438}
}

read the original abstract

In holographic theories, the Hubeny-Rangamani-Takayanagi (HRT) area operator plays a key role in our understanding of the emergence of semiclassical Einstein-Hilbert gravity. When higher derivative corrections are included, the role of the area is instead played by a more general functional known as the geometric entropy. It is thus of interest to understand the flow generated by the geometric entropy on the classical phase space. In particular, the fact that the associated flow in Einstein-Hilbert or Jackiw-Teitelboim (JT) gravity induces a relative boost between the left and right entanglement wedges is deeply related to the fact that gravitational dressing promotes the von Neumann algebra of local fields in each wedge to type II. This relative boost is known as a boundary-condition-preserving (BCP) kink-transformation. In a general theory of gravity (with arbitrary higher-derivative terms), it is straightforward to show that the flow continues to take the above geometric form when acting on a spacetime where the HRT surface is the bifurcation surface of a Killing horizon. However, the form of the flow on other spacetimes is less clear. In this paper, we use the manifestly-covariant Peierls bracket to explore such flows in two-dimensional theories of JT gravity coupled to matter fields with higher derivative interactions. The results no longer take a purely geometric form and, instead, demonstrate new features that should be expected of such flows in general higher derivative theories. We also show how to obtain the above flows using Poisson brackets.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Canonical quantization for effective theories with perturbations altering degrees of freedom: a covariant phase space approach
hep-th 2026-06 unverdicted novelty 5.5 of 10

Covariant phase space formalism enables perturbative canonical quantization of higher-derivative perturbed theories, verified on a solvable 2D charged particle model where perturbative results match exact expansion.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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X. Dong, D. Marolf and P. Rath, The JLMS Formula, Modular Flow and the Area Operator , to appear

work page

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