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REVIEW 4 major objections 3 minor 29 references

The gravitational entropy of a horizon is the Noether charge of a null generator with a universal normalization, yielding the area law without invoking temperature.

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2026-08-04 17:22 UTC pith:SBY4X7GU

load-bearing objection Careful re-derivation of the area law via a temperature-free Noether charge, but the central normalization conjecture is underdetermined for non-Killing horizons, so the advertised generality is not yet earned. the 4 major comments →

arxiv 2509.10921 v3 pith:SBY4X7GU submitted 2025-09-13 hep-th gr-qc

Gravitational Entropy

Sangmin Choi , Malcolm J. Perry This is my paper
classification hep-th gr-qc MSC 83C5783C40 PACS 04.70.Dy04.20.Fy
keywords gravitational entropyNoether chargecovariant phase spacehorizonBekenstein-Hawking entropyde Sitter horizonKerr-Newmanlightsheet
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper attempts to put gravitational entropy on a footing that does not require the Hawking temperature. It argues that for any horizon, the entropy is the Noether charge of a null vector field that generates the horizon's lightsheet, normalized so that its non-affinity parameter is 2π/l_p. The formalism adds a correction to the covariant phase space method that accounts for the fact that this generating vector field itself varies with the spacetime parameters. Tested on Schwarzschild, Kerr, Kerr-Newman, de Sitter, and Kottler spacetimes, the construction reproduces the Bekenstein-Hawking entropy S=A/(4Gℏ) on any section of the horizon. A sympathetic reader would care because it extends entropy from black holes to cosmological horizons and arbitrary lightsheets, suggesting that gravitational entropy is a geometric property rather than a thermal effect.

Core claim

The paper's central claim is that the gravitational entropy of a horizon can be computed as the Noether charge associated with a null generator ξ satisfying ξ^a∇_a ξ^b = (2π/l_p) ξ^b, integrated over any section of the horizon. This universal normalization replaces the Killing normalization tied to surface gravity and Hawking temperature. The key technical step is the inclusion of the variation δξ of the vector field in the covariant phase space formalism; without it, the charge is not integrable when the surface gravity varies across configuration space. With this correction, the entropy takes the form S = (l_p/ℏ) Q_ξ, and in every case examined it equals A/(4Gℏ). The method applies not onl

What carries the argument

The central object is the null generator ξ of the horizon, normalized by the universal condition ξ^a∇_a ξ^b = (2π/l_p) ξ^b (eq. 4.14), together with the covariant phase space formalism extended to allow the vector field to vary with the configuration (δξ ≠ 0). The new term in the Noether charge formula, involving ˆF[δξ], ensures integrability and yields the entropy as an integral of the two-form ˆF[ξ] over any section of the horizon.

Load-bearing premise

The universal normalization of the null generator ξ, fixing its non-affinity to 2π/l_p, is a conjecture chosen to reproduce the known area law; if this normalization is not the correct prescription for general lightsheets or for horizons without a well-defined surface gravity, the claimed generality does not follow.

What would settle it

Take a Vaidya spacetime with a time-dependent mass, whose horizon is a null surface but not a Killing horizon. Apply the universal normalization (4.14) to its null generator and compute the Noether charge. If the resulting integral does not equal A/(4Gℏ) at each instant, or if the consistency condition (2.15) fails so that δQ_ξ is not integrable, the claimed generality beyond stationary horizons fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any horizon with a null generator satisfying the universal normalization, the entropy is the integral of the Noether charge over any section, giving S = A/(4Gℏ).
  • The entropy is defined without reference to temperature, so it applies to cosmological horizons and, potentially, to more general lightsheets.
  • The first law δM = T δS + Ω δJ + Φ δQ is reproduced as a consistency condition, using Smarr-type identities for each spacetime.
  • The method yields the inner horizon entropy of Kerr-Newman as well, giving S_- = A_-/(4Gℏ).
  • The construction suggests that gravitational entropy is a geometric property of lightsheets, not tied to black hole microstates or to a thermal bath.

Where Pith is reading between the lines

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

  • If the universal normalization holds for arbitrary lightsheets, the same Noether charge construction could assign an entropy to any spatial region, turning the holographic lightsheet proposal into a concrete charge formula.
  • The correction term δξ suggests that for dynamical horizons or non-Killing horizons, the entropy may deviate from the area law or fail to be integrable; the paper only tests stationary cases, so this extension is untested.
  • The inner horizon result hints that the same formula could give meaning to entropy behind the horizon, though the physical interpretation of any negative temperatures or entropies there remains to be explored.
  • A testable extension is to apply the normalization to the Rindler horizon or to a general null surface and compare the resulting charge with the expected entanglement entropy of a region in flat space.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper proposes that the classical gravitational entropy of a horizon is the Noether charge conjugate to a null vector field ξ that generates the lightsheet of the region, with ξ normalized by the universal condition ξ^a∇_a ξ^b = (2π/l_p) ξ^b (Eq. 4.14). The covariant phase space formalism is extended to allow the generating vector field to vary in configuration space, yielding an extra term F[δξ] in the charge variation. The formula is tested on Schwarzschild, Kerr, Kerr-Newman (outer and inner horizons), de Sitter, and Kottler spacetimes, and in each case reproduces S = A/(4Gℏ). The paper argues that this gives a temperature-independent definition of gravitational entropy, related to Bousso's lightsheet proposal and applicable beyond black-hole horizons.

Significance. If the proposed normalization were well-defined, the paper would unify Wald-type Noether charge methods with Bousso's lightsheet program and would extend entropy computations to cosmological and other non-black-hole horizons. The explicit phase-space computations in the stationary Kerr-Newman family are careful and correctly reproduce the first law and area law. However, the central equation (4.14) is a conjecture rather than a derivation, and, as I detail below, it does not uniquely determine ξ for general non-Killing horizons. All worked examples are stationary Killing horizons, where the Killing symmetry resolves the ambiguity. The consistency condition (2.15) is verified only through Smarr-type identities, not through a general proof. Thus the paper's broad claims are not yet supported; its useful content is a set of consistent checks for stationary horizons.

major comments (4)
  1. [Sec. 4, Eq. (4.14)] The normalization condition (4.14) is underdetermined for non-Killing null surfaces. If l^a is an affinely parameterized null generator with affine parameter λ, then any ξ = f0(x^A) e^{(2π/l_p) λ} l^a satisfies (4.14) for an arbitrary positive function f0 on an initial cross-section. The Noether charge Q_ξ is generically not invariant under this rescaling. Only for a Killing horizon does the symmetry force f0 to be constant, after which (4.14) fixes the normalization. Since all examples in the paper are stationary Killing horizons, the claimed applicability to general lightsheets or non-black-hole horizons is not established. The authors should either restrict the claim to Killing horizons or provide an additional condition that selects f0.
  2. [Sec. 2, Eqs. (2.14), (3.7)] The variation δξ of the vector field is used throughout, but no coordinate-independent definition of δξ on field space is given. Comparing vector fields on different spacetimes requires an identification of the tangent bundles of nearby configurations, i.e., a connection on field space. The computations implicitly choose such an identification for the Kerr-Newman family via adapted coordinates, but a general prescription is absent. Without it, the terms F[δξ] and the consistency condition (2.15) are ambiguous for arbitrary variations. This is a load-bearing issue for the claimed generality.
  3. [Sec. 5, Eq. (5.12); Sec. 6.2, Eq. (6.25)] The consistency condition (2.15) is verified only by using the first law of black hole thermodynamics and the Smarr formula. This is a self-consistency check, not a derivation. The paper explicitly notes (after Eq. (2.16)) that if such a C does not exist, then Q_ξ does not exist. No general criterion is given for when the last two terms of (2.14) are a total variation. The worked examples all satisfy Smarr-type relations, so the method's validity beyond stationary, axisymmetric (or spherically symmetric) cases is untested.
  4. [Sec. 6.3] The inner-horizon computation is introduced with the caveat that S′ 'will have to be the boundary of a timelike slice' and that the authors 'shall not concern ourselves with this for now.' This is an admitted missing support. Since the inner horizon is not a boundary of the asymptotically flat exterior region, the interpretation of the integral as a Noether charge is not established. The calculation is a formal application and should be clearly labeled as provisional, or the required slice should be constructed.
minor comments (3)
  1. [Sec. 7] Typo: 'wold construct' should be 'would construct'.
  2. [Sec. 4, Eq. (4.8)] The relation S = l_p/ℏ Q_ξ is introduced abruptly; a comment on the engineering dimensions of Q_ξ and the factors of l_p and ℏ would improve readability.
  3. [Sec. 9] The concluding 'Speculations' paragraph correctly says the picture 'appears' to apply to more general regions. Given the underdetermination of Eq. (4.14) outside Killing horizons, this statement should be made more cautious or explicitly tied to a future resolution of the ambiguity.

Circularity Check

2 steps flagged

The universal normalization (4.14) is calibrated on the Schwarzschild area law; subsequent derivations of S=A/4Gℏ partly reuse the first law and Smarr formula, so the central construction is partially input.

specific steps
  1. fitted input called prediction [Section 4, Eqs. (4.12)-(4.14)]
    "The choice λ=2π(κl p )−1 = 8πGM l−1 p then gives δQ ξ =δ(4πGM 2 l −1 p ) and S= l p ℏ Q ξ = 4πGM 2 ℏ = A 4ℏG . ... We are therefore led to conjecture that in general ξ a should be chosen such that it is null geodesic generating the lightsheet ... Its parametrisation is fixed by ξ a ∇ a ξ b = 2π l p ξ b ."

    The free scale λ of the null generator is not independently derived; it is selected so that δQξ = δ(4πGM²/l_p), which is exactly the Bekenstein-Hawking area law for Schwarzschild. Equation (4.14) is then conjectured as a universal rule, but it is just this Schwarzschild-fitted choice rewritten as a differential equation. Thus the Schwarzschild case is not a prediction of the formalism; it is the input used to calibrate the formalism. All later applications inherit this calibration.

  2. renaming known result [Section 5, Eqs. (5.12)-(5.20)]
    "This is a consequence of the first law of black hole thermodynamics δM=T H δS+ Ω +δJ, and the Smarr formula [25], which for the Kerr black hole takes the form M= 2T H S+ 2Ω +J. ... After putting T H =ℏκ/2π, we obtain the entropy S= l p ℏ Q ξ = M 2T H − Ω + J T H = A 4Gℏ ."

    The consistency condition (2.15) that makes the Noether charge well-defined is verified using the first law and the Smarr formula, both of which already contain the area law. The final equality S=A/4Gℏ is then obtained by substituting the Smarr relation. Thus the Kerr computation does not independently derive the area law from the Noether formalism; it restates the known thermodynamic identity that was used to establish consistency.

full rationale

The central circularity is in Section 4: the normalization of ξ is chosen so that the Noether charge integrates to the known Schwarzschild entropy, and the same choice is then elevated to the conjectured universal condition (4.14). Therefore the Schwarzschild 'test' is a fit, not an independent prediction. For Kerr and Kerr-Newman, the consistency condition and the final identification with A/4Gℏ are established using the first law and Smarr formulas, which already encode the area law; these checks are therefore partially restatements of the known thermodynamics. The de Sitter and Kottler computations are genuine evaluations of the ansatz with no additional fitted parameter, and they provide some independent support. There is no load-bearing self-citation; the cited Wald/Iyer and Bousso results are external. A separate validity gap, not itself circularity, is that (4.14) fixes the exponential growth of ξ along each null generator but not its arbitrary initial value on a cross-section, so the claimed applicability beyond Killing horizons is underdetermined; all worked examples are stationary Killing horizons where the symmetry removes this ambiguity. Because the main construction is calibrated to the target result but is also tested on several non-Schwarzschild cases, a score of 6 (partial circularity) is appropriate.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The paper introduces no new fields, particles, or dimensions. Its central input is the conjectural normalization (4.14), which acts as an effective free parameter tuned to the known area law.

free parameters (2)
  • normalization constant 2π/l_p in eq. (4.14) = 2π/l_p
    The constant is chosen so that the Noether charge integrates to the known Bekenstein-Hawking entropy A/4Gℏ in the Schwarzschild case; a different constant would change the entropy coefficient.
  • U(1) gauge parameter ε for Kerr-Newman = ε = (ℏ/l_p T_H) Φ_+
    The gauge parameter is selected proportional to the horizon electric potential to satisfy the consistency condition; this choice is analogous to the gravitational normalization and is fitted to the known first law.
axioms (4)
  • standard math Covariant phase space formalism and the presymplectic structure (Section 2)
    The paper builds on standard Noether charge methods from Wald and Iyer-Wald.
  • domain assumption Bousso's proposal that gravitational entropy of a region is determined by the lightsheet at its boundary (ref [12])
    This motivates the choice of the null generator as the relevant vector field, but it is not established.
  • ad hoc to paper Universal normalization of the null generator, eq. (4.14)
    This is the paper's conjecture, introduced to reproduce known entropy results; it is not derived from any deeper principle.
  • domain assumption First law of black hole thermodynamics and Smarr formula (used in eq. 5.12 and 6.25)
    These relations are used to prove the consistency condition; they are standard but already encode the area law that the paper aims to derive.

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0 comments
read the original abstract

We formulate the classical gravitational entropy of a horizon as a Noether charge that does not require the notion of a temperature, and which is applicable to horizons that are not necessarily associated with black holes. This introduces a correction to the covariant phase space formalism that accounts for the configuration-dependence of the generating vector field conjugate to the charge. The vector field is related to the proposal of Bousso that the gravitational entropy of a region is determined by the lightsheet at its boundary. We test the formula on various black hole and cosmological horizons.

discussion (0)

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Reference graph

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