Consistency Problems of Conformal Killing Gravity

Bayram Tekin; Emel Altas

REVIEW 3 major objections 4 minor 21 references

Consistency Problems of Conformal Killing Gravity

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

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Pith's one-line read This paper argues that Conformal Killing Gravity, a modified theory that inherits all General Relativity solutions, cannot assign a mass to Schwarzschild or Kerr black holes because the conserved current it constructs vanishes identically…

desk verdict A correct algebraic observation about one conserved-current construction in CKG, but the broad conclusion that the theory lacks any conserved charges is not proven. read the letter →

arxiv 2502.06262 v2 pith:VVZBKMSV submitted 2025-02-10 gr-qc astro-ph.COhep-th

Emel Altas , Bayram Tekin This is my paper · ORCID

classification gr-qcastro-ph.COhep-th PACS 04.20.-q04.50.Kd04.70.Bw

keywords conformalKillinggravityrank-3fieldequationsconservedchargesEinsteinmanifoldsSchwarzschildmassblackholethermodynamicsmodifiedNoetherprocedure

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

Conformal Killing Gravity is a proposed modification of General Relativity whose field equations are a rank-3 tensor built from derivatives of the Ricci tensor, with a source given by derivatives of the energy-momentum tensor. This paper argues that, because those equations do not come from a diffeomorphism-invariant action, the theory has no way to define conserved charges for its black hole solutions. The paper constructs a conserved current and shows that on Einstein manifolds---including Schwarzschild and Kerr---the current vanishes identically. Hence the parameter $m$ in the Schwarzschild metric cannot be interpreted as a mass, and no analogue of the quadrupole formula exists for gravitational waves. If this is right, a theory proposed as a dark-energy alternative fails a basic consistency requirement for any gravity theory.

What carries the argument

The central object is the rank-3 H-tensor $H_{\mu\nu\sigma}=\nabla_\mu R_{\nu\sigma}+\nabla_\nu R_{\mu\sigma}+\nabla_\sigma R_{\mu\nu}-\frac{1}{3}(g_{\nu\sigma}\nabla_\mu R+g_{\mu\sigma}\nabla_\nu R+g_{\mu\nu}\nabla_\sigma R)$, which defines the vacuum field equations $H_{\mu\nu\sigma}=0$. The load-bearing computation is the second divergence $\nabla^\nu\nabla^\mu H_{\mu\nu\sigma}$, which the paper rewrites as a gradient plus a divergence of a symmetric tensor; that tensor defines $\Phi_{\mu\sigma}$ and hence the current $J^{\mu}=\sqrt{-g}\,\xi_\sigma\Phi^{\mu\sigma}$. On Einstein manifolds all curvature terms in $\Phi_{\mu\sigma}$ cancel, so the current vanishes identically for Schwarzschild and Kerr.

What would settle it

Exhibit a diffeomorphism-invariant action, auxiliary fields allowed, whose metric variation yields $H_{\mu\nu\sigma}=0$, and compute the Noether charge of the Schwarzschild metric; a nonzero mass would falsify the paper's conclusion.

Watch

Extended reading notes

Core claim

The paper's central finding is that the H-tensor field equations of Conformal Killing Gravity do not support conserved charges that could give meaning to the integration constants in known solutions. Taking two covariant divergences of the H-tensor yields an identity from which a divergence-free second-rank tensor $\Phi_{\mu\sigma}$ and a conserved current $J^{\mu}=\sqrt{-g}\,\xi_{\sigma}\Phi^{\mu\sigma}$ are constructed for any Killing vector $\xi$. For every Einstein manifold, which includes the Schwarzschild and Kerr spacetimes in vacuum, this current is identically zero. The paper concludes that a black hole carries the same energy and angular momentum as the empty background, so the theory is ill-defined as a theory of gravity, in the same way as the Cotton gravity case.

Load-bearing premise

The argument rests on the premise that no diffeomorphism-invariant action exists for the rank-3 field equations, because if such an action did exist the Noether procedure would supply charges and the inconsistency claim would collapse.

Editorial extensions

If this is right

The Schwarzschild mass parameter $m$ cannot be identified with a conserved mass in Conformal Killing Gravity; a black hole is indistinguishable from the vacuum in conserved charge.
Kerr black holes have zero conserved angular momentum in this theory, so the first law of black hole thermodynamics cannot be formulated.
No quadrupole formula can be derived for gravitational waves from compact sources, because the weak-field source is a derivative of the energy-momentum tensor rather than the tensor itself.
Any rank-3 tensor theory without a diffeomorphism-invariant action faces the same test: its conserved charges must not vanish on the solutions whose integration constants are meant to be physical.

Reading between the lines

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

Beyond the paper: the zero-charge obstruction is generic for Einstein manifolds, so it would also affect cosmological solutions with a cosmological constant, making the difficulty broader than black hole physics alone.
Beyond the paper: the decisive unresolved question is whether a diffeomorphism-invariant action with auxiliary fields exists; a targeted search for such an action would settle whether the theory is truly inconsistent.
Beyond the paper: the result suggests a screening criterion for modified-gravity proposals---a theory that cannot assign charges to its own vacuum solutions cannot sustain a thermodynamic or gravitational-wave program.

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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

The thing to know: this is a useful, mostly correct critique of Conformal Killing Gravity, but it doesn't prove what the abstract claims. The new algebraic work—computing the off-shell divergences of the H-tensor and constructing a conserved current from them—is solid. The leap from "this current vanishes on Einstein manifolds" to "CKG has no conserved charges" is not.

The computation in Eqs. (39)–(49) is the real content. The authors show the first and second divergences of H_mu_nu_sigma are nonzero off-shell, then use the second divergence to define Phi_mu_sigma, which is divergence-free on-shell. On Einstein manifolds, Phi_mu_sigma is identically zero. That is a legitimate new result, and it does undermine this particular Deser-Tekin-style charge construction for Schwarzschild and Kerr in CKG. The paper also honestly notes that the source term being a derivative of T_mu_nu complicates any quadrupole formula, and the discussion of on-shell versus off-shell Bianchi identities is clear and valuable.

But the strong conclusion—"clearly the theory is ill-defined"—doesn't follow. The authors construct a bulk current density J^mu = sqrt(-g) xi_sigma Phi^mu_sigma. A conserved charge is not the integral of the bulk current over a hypersurface; it is a boundary integral of a superpotential. The vanishing of the bulk integrand does not imply the boundary integral vanishes. Komar's charge is the standard counterexample: in vacuum GR the bulk integrand vanishes, yet the boundary integral gives Q = m. To rule out charges in CKG, one must show that no superpotential can be built from the linearized H-tensor, or that any such boundary term vanishes under the relevant fall-off conditions. The paper doesn't do that. The stress-test note makes this point, and I think it is right.

Similarly, the assertion after Eq. (10) that a rank-3 H-tensor "cannot come from an action" assumes the metric is the only dynamical field. Actions with auxiliary fields are not excluded; if one exists, Noether charges become available. That is a second gap in the exhaustiveness argument.

These soft spots are significant but not fatal to the paper's core algebra. This is a legitimate consistency warning, not a no-go theorem. The modified-gravity community—especially people working on CKG and Cotton gravity—should read it for the explicit divergence identities and the cautionary tale about naive charge constructions. It deserves a serious referee and, after revision that tempers the conclusions, publication. I would send it to peer review, with a referee who knows the superpotential literature and will push on the boundary-term loophole.

Referee Report

3 major / 4 minor

Summary. The manuscript argues that Conformal Killing Gravity (CKG), which is defined by rank-3 field equations, is inconsistent as a physical theory because its integration constants, such as the mass parameter m in Schwarzschild, cannot be interpreted as conserved charges. The paper reviews diffeomorphism invariance of tensor field equations, derives the first and second covariant divergences of the H-tensor, and constructs a conserved current from the second divergence. It then observes that this current vanishes on Ricci-flat Einstein manifolds, including Schwarzschild and Kerr, and concludes that black holes in CKG have the same conserved charges as the vacuum. The paper additionally asserts that the rank-3 H-tensor cannot arise from a diffeomorphism-invariant action with the metric as the only dynamical field, and it argues that the absence of such an action blocks the Noether construction of charges and the standard quadrupole formula.

Significance. If the central claim could be established, the paper would provide a strong and useful viability constraint on Conformal Killing Gravity, a theory that has attracted recent interest. The explicit algebraic derivations are a real strength: the divergence identities in Eqs. (39)-(49) and the construction of the current in Eqs. (50)-(55) are self-contained and can be checked directly, and the paper does not rely on fitted parameters or assumed conclusions. The main weakness is that the decisive inference, from the vanishing of one constructed current to the nonexistence of any conserved charge, is not supported by an exhaustiveness argument. The paper is therefore best read as a conditional critique whose central conclusion requires additional justification.

major comments (3)

[Section II.B.1, Eqs. (50)-(55)] The paper shows that its constructed current J^mu vanishes for Ricci-flat backgrounds, but this does not establish that no conserved charges exist in CKG. The charge formulas in Eqs. (31)-(33) are boundary integrals of a superpotential F^{mu nu}, and a bulk current can vanish on-shell while the boundary charge is nonzero; the Komar charge for Schwarzschild, where the bulk integrand vanishes in vacuum but the boundary integral gives Q=m, is the standard example. To conclude that CKG is ill-defined, the authors must prove that no superpotential or alternative charge construction can assign nonzero charges to Schwarzschild and Kerr, or they must state explicitly why any valid charge must come from a current of the form in Eqs. (50)-(55). As written, the transition from "this current vanishes" to "the theory is ill-defined" is a logical gap.
[Section II, after Eq. (10)] The assertion that a rank-3 H-tensor cannot come from an action "with the metric being the dynamical field" excludes actions with auxiliary fields, Lagrange multipliers, or connection variables. The paper provides no proof of this exclusion, and the claim is load-bearing: if such an action exists, the Noether procedure could supply conserved charges and undermine the central inconsistency argument. The statement should either be proven for a clearly defined class of actions or softened to the weaker claim that no metric-only diffeomorphism-invariant action is known.
[Section II.B.1, Eq. (50)] The statement that the current is identically zero for all Einstein manifolds is too broad. For an Einstein metric with R_{mu nu} = lambda g_{mu nu}, the tensor G_{mu nu} defined in Eq. (42) equals (Lambda - lambda) g_{mu nu}, and substituting into Eq. (50) does not give a vanishing Phi_{mu sigma} for generic lambda and Lambda; the scalar term, for example, contains -(1/3) R^2. The conclusion for Schwarzschild and Kerr is unaffected because those spacetimes are Ricci-flat with Lambda = 0, but the claim as written overstates the result and should be restricted to metrics with G_{mu nu} = 0, i.e., cosmological Einstein manifolds with the chosen Lambda.

minor comments (4)

[Section II.B.1, paragraph after Eq. (51)] The sentence "This is not acceptable since no black hole spacetime cannot be allowed to have the same energy and angular momentum as a black hole spacetime" contains a double negative and should be rewritten, for example as "No black hole spacetime can be allowed to have the same energy and angular momentum as the vacuum background."
[Section II.B.1, Eq. (51)] The phrase "not worrying about the overall dimensional factor" is imprecise; if the current is defined only up to a constant, the normalization should be stated explicitly, since the numerical value of a conserved charge depends on that normalization.
[Section II.B, Eqs. (28)-(33)] The background-charge formalism is presented only for theories with a symmetric second-rank field equation. Because CKG has a third-rank equation, the paper should clarify which steps of this formalism fail and why no analogous superpotential can be constructed from the linearized H-tensor; this clarification is directly relevant to the main charge-conservation argument.
[References] Reference [17] would benefit from the full article identifier or DOI, and the comparison with Cotton gravity in Ref. [19] is cited rather than summarized; a short explanation of the parallel would help readers assess the strength of the analogy.

Circularity Check

0 steps flagged · score 1.0 of 10

The conserved-charge computation is self-contained; the paper's stronger conclusion rests on an unproven exhaustiveness assumption, but that is a logical gap, not circularity.

full rationale

The derivation chain is self-contained: the H-tensor is defined in Eq. (9), the on-shell Bianchi-type identities and the second-divergence identity in Eq. (49) are obtained by direct tensorial algebra using the contracted Bianchi identity and the Ricci identity, and the conserved current Phi in Eqs. (50)-(51) is constructed from that identity. No parameter is fitted, no quantity is secretly defined in terms of the target conclusion, and the claim that the current gives identically zero charges on Einstein manifolds is a computation rather than an input. The paper's citations to the authors' earlier work ([1,2,4,17,19]) supply the standard background-charge framework and an analogy with Cotton gravity, but they are not load-bearing uniqueness theorems: the Conformal Killing Gravity-specific vanishing result is established here. The more serious concerns are correctness risks rather than circularity: the assertion after Eq. (10) that a rank-3 H-tensor cannot come from an action assumes the metric is the only dynamical field, and the step from 'the constructed current vanishes' to 'no conserved charges exist at all' assumes exhaustiveness of this particular charge construction without ruling out alternative boundary superpotentials or Noether charges from an auxiliary-field action. These are gaps a referee should weigh, but they are not reductions of the derivation to its own inputs. Accordingly, no specific circular step can be quoted, and the score is kept at 1.0 only for the minor self-referential framing.

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

No free parameters or invented entities. The argument leans on three background assumptions and one unproved claim (no action), which limits the strength of the negative conclusion.

assumptions (4)

ad hoc to paper The H-tensor, being rank-3 with the metric as the only dynamical field, cannot come from a diffeomorphism-invariant action.
Stated in Section II after Eq. (10) as 'It is clear that with 3 indices, the H-tensor cannot come from an action'; no proof is given and the restriction to metric-only variation is not justified.
domain assumption A viable gravity theory must allow integration constants in black hole solutions to be interpreted as conserved charges defined through a Noether-type or background-subtraction procedure.
Stated in the Introduction as guidelines; this normative premise is used to judge CKG ill-defined.
domain assumption The Deser-Tekin background-subtraction charge construction is the appropriate tool for theories without an action.
Invoked in Section II.B using reference [17]; alternative charge definitions are not considered.
standard math Spacetime is pseudo-Riemannian and sufficiently differentiable; standard Bianchi and Ricci identities hold.
Used throughout Sections II.A-II.B.

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Cite this review

Pith. "Pith review of Consistency Problems of Conformal Killing Gravity." pith.science (2026). https://pith.science/paper/VVZBKMSV

@misc{pith2026250206262,
  author       = {Pith},
  title        = {Pith review of: Consistency Problems of Conformal Killing Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVZBKMSV}},
  note         = {Machine review of arXiv:2502.06262}
}

abstract

We show that gravity field equations based on a tensor with rank greater than 2 have consistency problems in the sense that integration constants in the solutions, such as the parameter $m$ in the Schwarzschild metric, do not allow for an interpretation in terms of conserved quantities in the theory. The recently introduced Conformal Killing Gravity, an interesting extension of General Relativity that inherits all the solutions of the latter, and defined with a rank-3 tensor field equation that does not arise from a diffeomorphism-invariant action, is plagued with this problem. In this theory, it is not clear at all how one can define the energy and angular momentum for black hole solutions, or define the analogues of the formulas, such as the quadrupole formula, in the weak field limit for gravitational waves emitted by compact sources.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages

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Direct approach Due to these obstacles, let us try to get a conserved current b y directly manipulating the ﬁeld equations in their explicit form. Let us ﬁrst consider o nly the matter-free case. We found above that the ﬁrst divergence of the H-tensor is ∇µ H µνσ = □ Rνσ + 2Rν λ Rλσ + 2Rµλ Rµνσλ − 1 3 gνσ □ R + 1 3∇σ ∇ν R. (39) Taking one more divergence ...

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for this and the relevant references therein, as well as t he second derivative of the equation that brings constraints on perturbation theory, so-called the Taub-ch arges, about a given background solution. 3 invariance of the Ricci curvature, Ricci( ϕ ∗ g) = ϕ ∗ Ricci(g), the contracted Bianchi identity follows: ∇µ Gµν = 0 with Gµν := Rµν − 1 2 gµν R [6...

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Let us ﬁrst consider o nly the matter-free case

Direct approach Due to these obstacles, let us try to get a conserved current b y directly manipulating the ﬁeld equations in their explicit form. Let us ﬁrst consider o nly the matter-free case. We found above that the ﬁrst divergence of the H-tensor is ∇µ H µνσ = □ Rνσ + 2Rν λ Rλσ + 2Rµλ Rµνσλ − 1 3 gνσ □ R + 1 3∇σ ∇ν R. (39) Taking one more divergence ...

work page

[2] [2]

Deser and B

S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev. D 67, 084009 (2003)

work page 2003

[3] [3]

Deser and B

S. Deser and B. Tekin, Gravitational energy in quadratic curvature gravities, Phys. Rev. Lett. 89, 101101 (2002)

work page 2002

[4] [4]

3 invariance of the Ricci curvature, Ricci( ϕ ∗ g) = ϕ ∗ Ricci(g), the contracted Bianchi identity follows: ∇µ Gµν = 0 with Gµν := Rµν − 1 2 gµν R [6]

for this and the relevant references therein, as well as t he second derivative of the equation that brings constraints on perturbation theory, so-called the Taub-ch arges, about a given background solution. 3 invariance of the Ricci curvature, Ricci( ϕ ∗ g) = ϕ ∗ Ricci(g), the contracted Bianchi identity follows: ∇µ Gµν = 0 with Gµν := Rµν − 1 2 gµν R [6...

work page

[5] [5]

Harada, Gravity at cosmological distances: Explaini ng the accelerating expansion without dark energy, Phys

J. Harada, Gravity at cosmological distances: Explaini ng the accelerating expansion without dark energy, Phys. Rev. D 108, 044031 (2023)

work page 2023

[6] [6]

Altas and B

E. Altas and B. Tekin, Second Order Perturbation Theory i n General Relativity: Taub Charges as Integral Constraints, Phys. Rev. D 99, no.10, 104078 (2019)

work page 2019

[7] [7]

J. L. Kazdan, Another proof of Bianchi’s identity in Riem annian geometry, Proceedings of the American Mathematical Society 81.2 (1981): 341-342. 11

work page 1981

[8] [8]

D. M. DeTurck, Existence of metrics with prescribed Ricc i curvature: local theory. Inventiones mathematicae 65.2 (1981): 179-207

work page 1981