Static Spherically Symmetric Solutions in Nonconservative Unimodular Gravity

Anuar Idrissov; Hernando Quevedo

REVIEW

Static Spherically Symmetric Solutions in Nonconservative Unimodular Gravity

Anuar Idrissov, Hernando Quevedo

General Relativity and Quantum Cosmology · Submitted 12 Sep 2026

Pith's verdict: A clean identification of the trace-sector parameter C in unimodular gravity, with a useful classification of which energy conditions survive the reconstruction freedom. The observational part is honest about its limitations and the key parameter window is super-extremal, so the paper's practical impact is modest.

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arXiv ID2609.13628 (version 2)

Pith IDJJDBT6R2, Pith's permanent record of this paper and its review

CitationAnuar Idrissov, Hernando Quevedo. "Static Spherically Symmetric Solutions in Nonconservative Unimodular Gravity." arXiv:2609.13628 (2026). https://pith.science/paper/JJDBT6R2

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Editor's note

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This paper does one thing clearly and one thing well. The clear thing: it identifies the coefficient C of the 1/r^3 term as the carrier of the trace sector in unimodular gravity, and shows that within the Rastall restriction the nonconservation measure is kappa*lambda*C. The well-done thing: it proves that only the two null projections of the energy conditions are split-invariant, which settles why the wormhole no-go results in this theory are robust and why claims based on the strong or dominant conditions are not. The derivations are explicit, the text is honest about the ansatz being imported from the parametrized post-Schwarzschild program, and the algebraic control of the curvature invariants is solid. I appreciated the careful separation between the two roles of Λ and the frank discussion of the coordinate-invariance objection. Credit where due: the paper is a well-organized theoretical exercise that respects its own limits.

The soft spots are in proportion. The observational proposal (combination 9δ1 − 4δ2 as a null test for C) is a correct statement of principle, but the authors themselves give the numbers showing that present precision is two orders of magnitude short and that spin degeneracy is currently unresolvable without a stationary generalization. So that section is honest but not actionable. The dissolution of the light-ring obstruction (Proposition 2) lives entirely in a super-extremal parameter window (Q/M between 1.15 and 1.17). The authors say as much, but a reader skimming the paper might miss that the‘counterexample’is not astrophysical. The Rastall identification is sharp only within the ray Λ = κλR, which the paper acknowledges. For a general nonconstant Λ, C is not the sole carrier of nonconservation. So the headline claim is qualified correctly but tightly.

Who is this for? People working on unimodular gravity and parametrized black hole tests. It is a useful reference for the energy-condition invariance point, which transfers to any metric. The mathematical results are sound and the literature is cited responsibly. The paper deserves a serious referee and should be published after minor revision. I would ask the authors to add a sentence at the start of Sec. IX stating clearly that the observational test cannot be applied to rotating black holes without the stationary generalization, and to flag the super-extremal nature of the window (46) more prominently in the abstract or conclusions. Otherwise it is ready.

I would accept peer review on this paper and would cite it in my own work if I were writing about reconstruction freedom in unimodular gravity.

What would settle it

A direct test of the paper's central claim would be to measure the strong-field observables of a static black hole candidate with sufficient precision to extract the combination 9δ1 - 4δ2 = 7C/($18M^{2}$). If C is found to be nonzero and independent of the electric charge measurement (from weak-field tests), that would confirm the presence of the trace-sector parameter. However, distinguishing the nonconservative interpretation from a general-relativistic conserved source requires independently constraining the Rastall parameter λ (e.g., through cosmological observations), because within the reconstruction family the same C can be assigned to conserved matter (λ = 0). A measurement of κλC ≠ 0 would falsify the general-relativistic reading and support the nonconservative branch.

Referee report

Minor objections (5)

[Sec. II C] The heading reads "he Rastall restriction" — should be "The Rastall restriction."
[Sec. IV B, after Eq. (17)] The phrase "r 5 Λ = 2µC" has a missing superscript; it should read "r^5 Λ = 2µC."
[Table I] The row for arbitrary Λ(r) uses "∂ rΛ/κ" without specifying that this is the radial component of the divergence; a brief note like "∇_μ T^{μr} = ∂_r Λ/κ" would improve clarity.
[Sec. V B, Theorem 1 proof] The proof of part (iii) states that at r = r_⋆ the admissible split is unique. This follows from the conditions ρ = p_t = 0, but the statement could be expanded slightly to show that the system of equations forces Λ_5 = C/2.
[Sec. IX B, Eq. (76)] The phrase "9δ_1 − 4δ_2 = 7C/18 M^2" uses δ_1 and δ_2 defined earlier as the shifts of r_ph and r_isco. It might help to restate those definitions in the same paragraph for readability.

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The paper

The paper claims

In nonconservative unimodular gravity, the inverse-cubic parameter of a static spherically symmetric black hole carries the entire trace sector and controls nonconservation through the product of its coefficient with the Rastall coupling…

The reading

This paper investigates the reconstruction freedom in nonconservative unimodular gravity for a static, spherically symmetric metric that includes an inverse-cubic term (parameter C) as the leading deformation beyond Reissner–Nordström. The theory allows the source to be split between matter and vacuum in infinitely many ways, yet the paper shows that the geometry alone fixes the trace sector entirely through C: the Ricci scalar is R = 2C/$r^{5}$. Within the Rastall restriction of the reconstruction family, the nonconservation (energy exchange between matter and vacuum) is proportional to κλC, so the product κλC is the invariant measure. Among the six standard energy-condition combinations, only the two null projections (ρ + p_r and ρ + p_t) are independent of the split; all other combinations can be given either sign by choosing the split. This yields a general criterion: no-go results based on the null energy condition are robust in unimodular gravity, while those based on the strong or dominant conditions are not. Applied to wormhole throats, the criterion reproduces the universal violation of the null condition. The paper also derives strong-field observables (horizon, photon sphere, shadow, innermost stable circular orbit, deflection angle, Lyapunov exponent) and identifies a charge-independent combination that isolates C, though current observational precision is insufficient to measure it.

What carries the argument

The reconstruction family of sources parameterized by Λ(x) in Eq. (2), which expresses the unimodular field equations as κT^μ_ν = G^μ_ν + Λ(x)δ^μ_ν. For the metric (11) with f = 1 - A/r + B/$r^{2}$ - C/$r^{3}$, the Ricci scalar is R = 2C/$r^{5}$, showing that C carries the entire trace sector. The Rastall restriction Λ = κλR fixes Λ to be proportional to C/$r^{5}$, and then the divergence of the source is proportional to κλC. The key identity is Eq. (13), which gives R, G^t_t, G^θ_θ in terms of B and C alone, with A cancelling. This identification allows the paper to link the 1/$r^{3}$ term to the nonconservative sector.

Extended reading notes

Core claim

The paper's central claim is that the parameter C (coefficient of the 1/$r^{3}$ term in the metric) carries the complete trace sector of the geometry, since R = 2C/$r^{5}$, and that within the Rastall restriction Λ = κλR of the reconstruction family, the product κλC measures the nonconservation. At the level of the field equations, the Rastall prescription corresponds to a restricted choice within the unimodular reconstruction family, and the split selected on energy-condition grounds (traceless matter) coincides with the degenerate Rastall value κλ = 1/4. Of the six standard combinations entering the energy conditions, only the two null projections ρ + p_r and ρ + p_t survive the reconstruction freedom; all other combinations depend on the choice of Λ. This gives a general criterion: arguments resting on the null energy condition are reconstruction invariant, while those resting only on the strong or dominant conditions are not. Applied to wormhole throats, the criterion reproduces the null-energy no-go result for every split. For the black-hole family, the inverse-cubic parameter carries the complete trace sector, and the nonconservative contribution is controlled by the product κλC, so that conservation is recovered exactly when that product vanishes, which can happen either through λ = 0 (general relativity with conserved matter) or through C = 0 (Reissner–Nordström geometry).

Load-bearing premise

The metric ansatz (11) with g_tt g_rr = -1 and the specific polynomial form f = 1 - A/r + B/$r^{2}$ - C/$r^{3}$ is imported from the parametrized post-Schwarzschild program and is not derived from the field equations; the paper acknowledges this in Sec. I.

Editorial extensions

If this is right

If the paper is correct, any no-go theorem about traversable wormholes in unimodular gravity that relies on the null energy condition (NEC) is robust, because NEC is split-invariant; the recent NEC-violation results are unreachable.
The parameter C becomes a direct observational target for testing nonconservation in strong-field gravity. The combination 9δ1 - 4δ2 (where δ1 is the shift of the photon sphere and δ2 the shift of the ISCO) isolates C independently of the charge parameter B, providing a null test for the trace sector.
Within the Rastall restriction, a measurement of C combined with an independent determination of λ would test the nonconservative exchange amplitude. If κλC is found nonzero, it signals energy transfer between matter and the vacuum sector.
The paper's criterion resolves conflicting claims about energy conditions in unimodular black holes: conclusions based on NEC are trustworthy; those based on SEC or DEC depend on the split choice and are not inherent to the geometry.
For black holes with three exterior photon spheres, the exotic shell predicted in general relativity is not forced in unimodular gravity; the optimal split (traceless Maxwell source) satisfies all energy conditions on the exterior within window (46), albeit at super-extremal charges.

Reading between the lines

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

The identification of C as the carrier of the trace sector suggests that any static spherically symmetric metric in unimodular gravity with a nonvanishing Ricci scalar will similarly have a single geometric parameter controlling the nonconservative sector, generalizing beyond the “post-Schwarzschild” truncation.
The stationary generalization of this metric (e.g., a rotating black hole with a 1/r^3 term) is necessary before the null test can be applied to astrophysical sources like M87* or Sgr A*. The spin degeneracy highlighted in Sec. IX C implies that even a modest spin of a/M = 0.1 produces shifts that swamp the C signature within the admissible parameter range, so joint fitting of a and C will be esse
The paper’s energy-condition criterion may extend to other theories with reconstruction freedom (e.g., f(R) gravity or scalar-tensor theories), providing a systematic way to identify which energy condition statements are invariant under the freedom.

Original abstract

We study static and spherically symmetric black holes in nonconservative unimodular gravity, where the traceless field equations determine the source of a given geometry only up to a pure-trace term. We consider the Reissner-Nordstr\"om metric with an additional $1/r^{3}$ term and show that its coefficient fixes the Ricci scalar and, when the freedom in the source is restricted to the Rastall form, controls the nonconservation of energy and momentum. We find that only the null projections of the source are independent of how the geometry is divided between matter and vacuum. Therefore, no-go results based on the null energy condition hold in unimodular gravity, while those based on the strong or dominant conditions are generally split dependent. We identify the traceless source as the optimal split with respect to the energy conditions and determine the region in which all standard energy conditions can be satisfied. We also identify a combination of the shifts of the photon sphere and of the innermost stable circular orbit that measures the $1/r^{3}$ coefficient independently of the charge, although its detection requires a precision far beyond current observations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,319 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.42 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The traceless field equations of unimodular gravity determine the source of a given geometry only up to a pure-trace contribution. We study the consequences of this reconstruction freedom for a static, spherically symmetric black-hole geometry containing the leading inverse-cubic deformation of Schwarzschild not fixed by weak-field data. We show that among the standard combinations entering the energy conditions, only the two null projections are independent of the reconstruction, whereas the energy density and the strong and dominant combinations depend on the chosen split between matter and vacuum. This yields a general criterion: conclusions based on the null energy condition are reconstruction invariant, while those based only on the strong or dominant conditions are not. Applied to wormhole throats, this criterion reproduces the null-energy no-go result for every split. For the black-hole family, the inverse-cubic parameter carries the complete trace sector. Within the Rastall restriction of the reconstruction family, nonconservation is controlled by the product of this parameter with the Rastall coupling, and the traceless split is selected by the energy conditions. We also derive strong-field observables and identify a charge-independent combination of photon-sphere and innermost-stable-orbit shifts that isolates the trace-sector parameter, while current observational precision remains insufficient to measure it.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The traceless field equations of unimodular gravity determine the source of a given geometry only up to a pure-trace contribution.	background 1.00	0.09	0.06	asserted 1.00
s1	We study the consequences of this reconstruction freedom for a static, spherically symmetric black-hole geometry containing the leading inverse-cubic deformation of Schwarzschild not fixed by weak-field data.	contribution 0.98	0.79	0.05	asserted 0.78
s2	We show that among the standard combinations entering the energy conditions, only the two null projections are independent of the reconstruction, whereas the energy density and the strong and dominant combinations depend on the chosen split between matter and vacuum.	result 0.99	0.93	0.26	asserted 1.00
s3	This yields a general criterion: conclusions based on the null energy condition are reconstruction invariant, while those based only on the strong or dominant conditions are not.	interpretation 0.69	0.88	0.68	asserted 1.00
s4	Applied to wormhole throats, this criterion reproduces the null-energy no-go result for every split.	result 0.97	0.87	0.13	asserted 1.00
s5	For the black-hole family, the inverse-cubic parameter carries the complete trace sector.	result 0.96	0.75	0.08	asserted 1.00
s6	Within the Rastall restriction of the reconstruction family, nonconservation is controlled by the product of this parameter with the Rastall coupling, and the traceless split is selected by the energy conditions.	result 0.85	0.84	0.08	asserted 1.00
s7	We also derive strong-field observables and identify a charge-independent combination of photon-sphere and innermost-stable-orbit shifts that isolates the trace-sector parameter, while current observational precision remains insufficient to measure it.	result 0.74	0.94	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Physically allowed region in the (

**Figure 2.** FIG. 2. Outer horizon (a) and photon sphere (b) against

**Figure 3.** FIG. 3. Critical impact parameter, which is also the shadow radius (a), and its relative deviation from the Schwarzschild value

**Figure 4.** FIG. 4. Deflection angle. (a) the exact integral (

**Figure 5.** FIG. 5. (a) null effective potential

Figures reproduced from arXiv:2609.13628 by the authors.

References

Discussion

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