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Gravitation in flat spacetime from entanglement

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under holographic assumptions, the first law of entanglement is equivalent to linearized gravitational equations around Minkowski spacetime in three and four dimensions.

desk verdict A genuine flat-space extension of gravity-from-entanglement, but the central equivalence is conditional: the proof imports boundary components of Einstein's equations as 'holographic' input, and one 3d integration constant is never shown to vanish. read the letter →

arxiv 1908.02044 v3 pith:P567RBXF submitted 2019-08-06 hep-th gr-qc

Victor Godet , Charles Marteau This is my paper · ORCID
classification hep-thgr-qc
keywords flatholographyentanglemententropyRyu-TakayanagiprescriptionfirstlawofCarrolliangeometryMinkowskispacetimelinearizedgravityBMSsymmetry
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 works within flat-space holography, where gravity in asymptotically flat spacetime is dual to a quantum system living on null infinity, and entanglement entropies of certain boundary regions are computed by an analog of the Ryu-Takayanagi formula. For linearized perturbations of three- and four-dimensional Minkowski spacetime, it claims that the first law of entanglement, the equality between entropy change and modular-energy change, holds if and only if the linearized gravitational field equations hold in the bulk. This matters because it extends to flat space the AdS/CFT result that gravity emerges from entanglement thermodynamics, without needing a cosmological constant or an AdS boundary. The cost is a set of assumptions about the dual theory, and in particular boundary stress-tensor conservation and trace conditions that are the only remaining holographic input.

What carries the argument

The central object is the generalized Rindler transformation: a symmetry of null infinity that maps the domain of dependence of a boundary region A to a spacetime with an imaginary-time circle, so that the generator of the circle is the modular flow. Its bulk extension is a Killing vector of Minkowski spacetime that vanishes on the Rindler bifurcation surface, and the RT surface is the union of an infalling light sheaf with the portion of the bifurcation surface it bounds. The identity dχ = −2ξ^a δE_ab ε^b carries the argument: integrating χ between the boundary region and the RT surface turns the first law into vanishing integrals of components of the equations of motion, and bulk isometries generate enough such integrals to force those components to zero. The remaining constants are killed by Carrollian boundary stress-tensor conservation and trace conditions, which the paper derives in three dimensions from the flat limit of AdS and assumes in four dimensions.

What would settle it

Take a linearized perturbation of 3d Minkowski with δErr = δErφ = 0 but δEur = C0(u,φ) and δEφφ = −r²C0(u,φ), and check whether the boundary trace conditions (5.29)–(5.30) permit C0 ≠ 0; if they do, the paper's conclusion δEur = 0 fails while its stated assumptions still hold, which is exactly the gap the paper flags in Sec. 4.2.

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Extended reading notes

Core claim

The central claim is that, for linearized perturbations of Minkowski spacetime, the gravitational equations of motion are equivalent to the first law δS^grav_A = δE^grav_A for all boundary regions A in a special class. In three dimensions, the proof uses RT surfaces made of two light rays and a curve on the Rindler bifurcation surface; variations of interval size, bulk isometries, and deformations of the infalling light sheaf force δErr = δErφ = 0, and the remaining components vanish once the conservation equation is combined with Carrollian boundary stress-tensor conditions. In four dimensions, the entangling regions are generalized watermelon slices and the same strategy yields δErr = δErθ = 0 everywhere, after imposing the analogous boundary stress-tensor conditions at leading asymptotic order. The result is phrased for general theories of gravity through Wald's Noether charge, so it covers higher-derivative corrections to Einstein gravity.

Load-bearing premise

The load-bearing premise is that the boundary of the putative dual theory obeys stress-tensor conservation and trace conditions; in three dimensions these conditions are themselves large-radius components of the linearized Einstein equations, so the derivation of those components is not independent of what it aims to prove.

Editorial extensions

If this is right

  • If correct, linearized gravity around Minkowski needs no cosmological constant or AdS boundary to emerge from entanglement; the boundary dual is Carrollian and lives on null infinity.
  • The derivation applies to any diffeomorphism-invariant theory of gravity because both entropy and energy are defined through Wald's Noether charge, so it covers higher-derivative corrections.
  • The RT prescription is not unique: the freedom in choosing the infalling light sheaf changes the entropy by a regulator-dependent amount, which the paper interprets as a choice of UV cutoff in the dual theory.
  • Positivity of von Neumann entropy imposes constraints on allowed perturbations, selecting a code subspace on which the modular Hamiltonian is bounded below.
  • In four dimensions the equivalence holds for entangling regions shaped like deformed watermelon slices rather than spheres, showing the first-law logic does not rely on conformally flat boundary geometry.

Reading between the lines

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

  • Editorial inference: the admitted three-dimensional gap means the cleanest honest reading is that first law plus boundary Einstein equations implies bulk Einstein equations; if the boundary conditions are regarded as part of the holographic dictionary rather than as consequences of bulk dynamics, the result still stands as an equivalence.
  • Editorial inference: the light-sheaf dependence of the entropy suggests that flat-space entanglement entropy is inherently cutoff dependent, so quantitative matches with conjectured dual theories should specify the sheaf; this may be a feature rather than a bug.
  • Editorial inference: one could test the positivity constraints directly by computing δE_A for explicit Bondi perturbations and asking whether the relevant modular-flow integral can become negative for some region A; a negative value would not refute the paper but would tell which perturbations admit a consistent modular Hamiltonian.
  • Editorial inference: the four-dimensional proof uses perturbations with vanishing gravitational-wave aspect Cij, so closing the remaining gap would involve checking whether the first law also forces Cij = 0 or whether the equivalence extends to those modes as well.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes an analog of the Ryu-Takayanagi prescription for asymptotically flat spacetimes in three and four dimensions, building on prior work by Jiang-Song-Wen and others. For a special class of boundary regions, the authors define gravitational entropy and energy in terms of Wald's Noether charge, and study the implication of the first law of entanglement δS_A = δE_A for linearized perturbations of Minkowski spacetime. The main claimed result is that this first law is equivalent to the linearized gravitational equations of motion. The proof proceeds by using the first law over a family of RT surfaces generated by bulk isometries and light-sheaf deformations to set certain components of δE_{ab} to zero, then invoking conservation equations and boundary stress-tensor constraints to kill the remaining integration constants. The paper also develops a Carrollian stress-tensor formalism for 3d flat space via the flat limit of AdS_3, and proposes a 4d generalization, with explicit computations for on-shell perturbations in Bondi gauge and a positivity constraint on allowed perturbations.

Significance. If the central claim were fully established, this would be a valuable extension of the 'gravity from entanglement' program to flat space, with several novel ingredients: a refined RT prescription featuring a light-sheaf choice, a derivation of Carrollian stress-tensor conservation laws from the flat limit, and explicit first-law checks for both 3d and 4d. The paper is careful and explicit in its analytic steps, and it is commendably transparent about the places where assumptions are needed (Sec. 4.2, Sec. 6.3). The Carrollian stress-tensor decomposition and the flat-limit analysis in Sec. 5 are useful contributions in their own right. However, the central theorem as stated in the abstract and Sec. 2 is stronger than what is actually proven: the derivation passes part of the target equations through the boundary stress-tensor conservation and trace conditions, and the 4d proof relies on an unproven assumption about boundary conditions. The paper is therefore best read as establishing that the first law, together with asymptotic boundary Einstein constraints, implies the full bulk linearized equations, not that the first law alone does.

major comments (3)
  1. [§4.2 and §5.2] The 3d proof is circular in a load-bearing way. After the light-sheaf and radial-translation arguments give δE_rr = δE_rφ = 0 (eqs. (4.29), (4.32)), the conservation equation ∇_a δE^{ab}=0 leaves the remaining components as δE_ur = C_0(u,φ), δE_φφ = −r²C_0, δE_uφ = C_2(u,φ)/r, and δE_uu = C_1(u,φ)/r. The paper then uses the Carrollian conservation equations (5.12)–(5.13) and trace conditions (5.29)–(5.30) to set C_1=C_2=0 and expects C_0=0. But Eq. (5.14) identifies C_1=0 and C_2=0 with the (u,u) and (u,φ) components of the linearized Einstein equations at leading order in r. Thus the proof assumes the large-r components of the very equations it aims to derive. Moreover, C_0=0 is not demonstrated; Sec. 4.2 explicitly states 'we have not been able to show it conclusively.' The abstract's claim of equivalence between the first law and the full linearized equations is therefore not established; the established statement is that the first law plus asymptotic boundary Einstein constraints implies the bulk equations.
  2. [§6.3] The 4d proof is conditional on an unproven and unconstructed boundary input. After reducing δE_rr and δE_rθ to zero, the remaining components are solved in terms of integration constants C_0(u,φ), C_1(u,θ,φ), C_2(u,θ,φ), C_3(u,θ,φ). The paper states 'we will assume that these boundary conditions ensure the vanishing of the components δE_ua at leading asymptotic order' and expects the trace condition to imply C_0=0, but no 4d Carrollian stress tensor, conservation equations, or trace identities are derived. Unlike the 3d case, where Sec. 5 provides explicit flat-limit equations, the 4d boundary conditions are purely assumed. This is a load-bearing gap: without these conditions, the first law only constrains a subset of the linearized Einstein equations. The conclusion in Sec. 7 that 'we have shown that the first law of entanglement is equivalent to the linearized gravitational equations of motion' is too strong for the 4d case.
  3. [§3.3–3.4] The corner-regulated prescription (3.48) is an additional input that is not derived from the five assumptions in Sec. 2. The paper motivates it by the discrepancy between δS_A=0 and δE_A≠0 for the on-shell perturbation (3.44), and argues that a smooth curve ~A_ε arbitrarily close to the cornered surface produces the correct first law. However, this is a new rule for computing the entropy, not a consequence of the stated assumptions, and the first-law equivalence δS_A=δE_A depends on it. The claim that the corner has a non-trivial contribution is supported only by an analogy with polar coordinates (footnote 6) and a formal limit argument, not by a systematic derivation from the presymplectic structure. Since the central theorem relies on this prescription, it should either be derived from the assumptions or explicitly incorporated as a sixth working assumption; the current status is a gap between the stated axioms and the proof.
minor comments (4)
  1. [Footnote 1] Footnote 1 cites reference [27] as 'To appear' with no author list or preprint number; this should be completed or removed before publication.
  2. [§3.3, Eq. (3.41)] The Bondi-gauge perturbation in Eq. (3.41) contains the term '−2r²U dudr', which appears to be a typo for '−2r²U dudφ'; the same pattern appears in the flat-limit metric (5.1).
  3. [§4.2, Eq. (4.31)] Eq. (4.31) displays an expression without an equality sign or a right-hand side; it should read '= 0' for the constraint following from the radial translation.
  4. [§6.2] The derivation of the 4d light-sheaf parametrization (6.38) is quite dense; the claim that modular-flow tangency forces each light sheaf to intersect the bifurcation surface at a single point (Sec. 6.2, after Eq. (6.37)) would benefit from an explicit algebraic derivation, as the current text leaves room for doubt.

Circularity Check

3 steps flagged · score 6.0 of 10

The claimed equivalence is only partial: the r-components of the linearized Einstein equations are derived from the first law, but the u-components are imported through holographic conservation/trace conditions that are themselves large-r components of the target equations.

  1. self definitional [Sec. 4.2 (eqs. (4.40)-(4.42)) and Sec. 5.2 (eq. (5.14))]
    "To be more precise, we have that the (u,u) and (u,φ)-components of the linearized Einstein equations scale with r as δEuu = C1(u,φ)/r and δEuφ = C2(u,φ)/r, such that C1 = 0 ⇔ (5.12) and C2 = 0 ⇔ (5.13). These conditions are the holographic input we need for the proof of Sec. 4."

    The proof fixes the integration constants C1 and C2 by invoking the boundary stress-tensor conservation equations (5.12) and (5.13). But eq. (5.14) says that C1 and C2 are exactly the coefficients of the leading r-behavior of the target linearized Einstein components δEuu and δEuφ, and that C1 = 0 is equivalent to (5.12) while C2 = 0 is equivalent to (5.13). Thus the conclusions δEuu = 0 and δEuφ = 0 are not derived from the first law; they are imposed by assuming the very components of the gravitational equations that the theorem claims to establish. The genuinely first-law-derived content is limited to δErr = δErφ = 0.

  2. self definitional [Sec. 4.2 (after eq. (4.37)) and Sec. 5.2 (eqs. (5.29)-(5.30))]
    "We expect that the trace conditions (5.29) and (5.30) imply that C0 = 0 although we have not been able to show it conclusively. Assuming that this is the case, we obtain δEur = δEφφ = 0. ... Equations (5.29) and (5.30) are the third holographic input that we have to impose for the proof in Sec. 4."

    The remaining radial component δEur, together with δEφφ = -r²δEur, is eliminated by an unproven expectation about trace conditions, and the paper explicitly lists (5.29)-(5.30) as an additional holographic input rather than as a consequence of the first law. The paper itself flags that the implication C0 = 0 was not shown conclusively, so this step is an assumed boundary/Einstein constraint rather than a derived equation.

1 more flagged steps
  1. self definitional [Sec. 6.3 (paragraph after eq. (6.76))]
    "From now on, we will assume that these boundary conditions ensure the vanishing of the components δEua at leading asymptotic order. The trace condition, similar to (5.29) and (5.30) in 3d, should imply that C0 = 0, leading to δEur = δEθθ = δEφφ = 0, everywhere in the bulk."

    In the 4d generalization the paper does not construct the Carrollian stress tensor or prove its conservation/trace identities; it simply assumes boundary conditions that kill δEua at leading asymptotic order, and it expects stress-tensor conservation to kill the constants C1, C2, C3. Since those constants are the leading coefficients of δEuθ, δEuφ, and δEuu, the 4d proof explicitly assumes target components of the linearized Einstein equations rather than deriving them from the first law. The paper itself identifies the needed analysis as future work.

full rationale

The paper contains a genuinely non-circular core: from the first law of entanglement and bulk isometries it derives δErr = δErφ = 0 everywhere in 3d, and analogous r-components in 4d. These are real constraints on the linearized equations of motion that do not reduce to the input assumptions. However, the full claim of equivalence to all linearized gravitational equations is not established from the first law alone. The remaining components are fixed by conservation and trace conditions of the holographic stress tensor, and the paper explicitly identifies those conditions as 'the holographic input we need for the proof'. In the flat Bondi analysis, these conditions are not independent data: eq. (5.14) states that the constants C1 and C2 are exactly the leading large-r pieces of δEuu and δEuφ, with C1 = 0 equivalent to the (u,u) Einstein component and C2 = 0 equivalent to the (u,φ) Einstein component. Therefore, setting C1 = C2 = 0 is assuming the target equations' u-components, not deriving them. The C0 = 0 step is even weaker, being an unproven expectation from trace conditions that are themselves declared an additional holographic input. In 4d the same structure is explicit: the paper assumes the vanishing of δEua at leading asymptotic order and expects the boundary stress-tensor conservation to kill the remaining constants, while leaving the construction of the 4d Carrollian stress tensor to future work. Thus the central claim should be read as: first law plus boundary/large-r Einstein constraints implies the remaining bulk linearized Einstein equations. The paper is transparent about these inputs, and there is no evidence of self-citation being load-bearing; the circularity is the importation of part of the target equations through the holographic stress-tensor assumptions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No constants are fitted to data; the only parameters are geometric sizes of entangling regions. The central claim rests on five working assumptions about flat-space holography plus boundary stress-tensor conditions imported from the flat limit of AdS. The 4d proof explicitly assumes the boundary conditions, and the 3d proof leaves C0 undetermined. No new particles or forces are postulated.

assumptions (7)
  • domain assumption There exists a quantum system on I+ with Hilbert space H on each slice of constant retarded time u, and a state for each bulk configuration.
    Assumption 1 in Sec. 2; necessary to define entanglement entropy for boundary regions.
  • domain assumption For a special class of subregions A of ∂Σ, a density matrix ρA exists and the Hilbert space factorizes on subregions.
    Assumption 2 in Sec. 2; needed to define reduced density matrices.
  • domain assumption A generalized Rindler transformation exists such that ρA = U^{-1} e^{-K_A} U, with K_A bounded from below on a code subspace.
    Assumption 3 in Sec. 2; provides the modular Hamiltonian and vacuum annihilation.
  • domain assumption δ<KA> equals the Iyer-Wald energy δE^grav_A associated with the bulk Killing vector ξA.
    Assumption 4 in Sec. 2; connects boundary energy variation to bulk gravitational energy.
  • ad hoc to paper The von Neumann entropy SA equals the integral of Wald's functional over the RT surface A~ homologous to A and fixed by the bulk modular flow.
    Assumption 5 in Sec. 2; the analog of the RT prescription. For Einstein gravity it reduces to area over 4G on γ.
  • ad hoc to paper Boundary stress-tensor conservation equations (5.12)-(5.13) and trace conditions (5.29)-(5.30) hold for off-shell perturbations in 3d, and analogous conditions are assumed in 4d.
    Used in Sec. 4.2 and 6.3 to set integration constants C0, C1, C2 to zero. The 3d conditions are derived from the flat limit of AdS but are not consequences of the first law; in 4d they are simply assumed (Sec. 6.3).
  • ad hoc to paper The entanglement entropy for perturbed geometries is computed with the regulated corner prescription (3.48), integrating Wald's functional over a smooth curve A~ε approaching the cornered RT surface.
    Introduced in Sec. 3.3 to reconcile the first law with the nonzero energy variation for perturbations (3.44); the corner contributes a nontrivial piece.

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Pith. "Pith review of Gravitation in flat spacetime from entanglement." pith.science (2026). https://pith.science/paper/P567RBXF

@misc{pith2026190802044,
  author       = {Pith},
  title        = {Pith review of: Gravitation in flat spacetime from entanglement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P567RBXF}},
  note         = {Machine review of arXiv:1908.02044}
}
read the original abstract

We explore holographic entanglement entropy for Minkowski spacetime in three and four dimensions. Under some general assumptions on the putative holographic dual, the entanglement entropy associated to a special class of subregions can be computed using an analog of the Ryu-Takayanagi formula. We refine the existing prescription in three dimensions and propose a generalization to four dimensions. Under reasonable assumptions on the holographic stress tensor, we show that the first law of entanglement is equivalent to the gravitational equations of motion in the bulk, linearized around Minkowski spacetime.

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Forward citations

Cited by 1 Pith paper

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    hep-th 2019-08 conditional novelty 6.0 of 10

    In flat-space holography, the first law of entanglement entropy of a BMS-invariant field theory is shown to imply the linearized Einstein equations in the three-dimensional bulk.

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Reviewed August 14, 2026 · model on record in the stance chip above.