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REVIEW 4 major objections 6 minor 59 references

Signatures of cubic gravity in the strong regime

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read In Einsteinian cubic gravity, a positive coupling shrinks black hole horizons, and Sgr A* keeps its horizon only if the coupling is below roughly 0.1.

desk verdict Solid, code-backed phenomenology of Einsteinian cubic gravity; the main results hold within the single-function branch, but the ansatz needs scrutiny before trusting the quantitative bound. read the letter →

arxiv 2502.01747 v2 pith:GY2DZYZZ submitted 2025-02-03 gr-qc astro-ph.GAphysics.optics

Flavio C. Sánchez , Armando A. Roque , Javier Chagoya This is my paper · ORCID
classification gr-qcastro-ph.GAphysics.optics MSC 83C5783D0583C10
keywords Einsteiniancubicgravityhigher-curvatureblackholehorizonsphotonspherelightbendingshadowsangulardifferenceSgrA*
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

The paper claims that Einsteinian cubic gravity, a higher-curvature theory chosen to keep general relativity's particle spectrum, leaves visible marks in the strongest gravitational fields. Working with a static, spherically symmetric metric, it shows that the sign of the coupling λ controls the horizon: positive λ shrinks the horizon, negative λ enlarges it, and for λ ≳ 0.1 a Sgr A*-mass object would have no black hole horizon. The same coupling moves the photon sphere by tens of percent, so the shadow radius and a recently proposed angular-difference observable become tests of the theory. The paper concludes that the coupling is constrained to λ ≲ 0.1 by the existence of Sgr A*'s horizon and photon sphere, and that the cubic terms' strongest signals are close to the source.

What carries the argument

The load-bearing object is the integrated vacuum field equation (7), obtained from the single-function ansatz ds² = -f dt² + $f^{{-1}}$ dr² + r² dΩ², which reduces the six-derivative cubic field equations to one ordinary differential equation containing the ADM mass integration constant C0. At a horizon, f(rh) = 0 and f'(rh) ≥ 0 turn this equation into the cubic relation (24) for f'(rh), which the paper uses to classify horizon existence and sign dependence. For observables, two tools carry the argument: the effective potential VEff = f L²/r², whose maximum locates the photon sphere, and the triangular-array angular difference α from null geodesics, computed numerically using the near-horizon, asymptotic, and weak-coupling series to seed integrations.

What would settle it

Solve the full vacuum field equations without imposing the reciprocal metric relation and compare the resulting horizon curve with Fig. 1; any discrepancy would overturn the derived thresholds. Alternatively, a shadow-radius measurement of Sgr A* accurate to a few percent would expose or rule out the claimed ~50% photon-sphere shift.

Watch

Extended reading notes

Core claim

Einsteinian cubic gravity admits exact de Sitter-like solutions with an effective cosmological constant set by λ and Λ, and, away from maximal symmetry, static spherically symmetric solutions whose metric function is governed by one integrated equation with mass integration constant C0. In these solutions the horizon radius is smaller than the Schwarzschild radius for λ > 0 and larger for λ < 0, with λ = 0 recovering Schwarzschild. For C0 fixed to the mass of Sgr A* and ΛEff fixed to the observed cosmological constant, λ ≈ 0.1 is the threshold above which the non-cosmological horizon disappears, leaving a naked singularity with divergent Ricci scalar at r = 0. The photon sphere follows the horizon: positive (negative) λ moves it inward (outward) by up to about 50% for |λ| < 0.1, and the angular difference α between triangular null-geodesic arrays in cubic gravity and in the Kottler or Schwarzschild background grows as the array approaches the source, reaching thousands of arcseconds at impact parameter 10 rs for λ = -5 and about $10^{4}$ arcsec near threshold for positive λ.

Load-bearing premise

Everything downstream assumes the static spherically symmetric solution has the reciprocal relation g_tt = 1/g_rr (the single-function ansatz); if the full cubic equations force a second independent metric function, the horizon radii, photon-sphere positions, and angular differences could all change.

Editorial extensions

If this is right

  • If the claim is right, a positive λ reduces the event-horizon radius below the Schwarzschild value for a fixed mass, so mass estimates based on horizon-crossing observables would be systematically shifted unless the cubic term is included.
  • For a Sgr A*-mass object, λ ≳ 0.1 removes the black hole horizon entirely; since Sgr A* is observed to have one, the coupling is observationally bounded from above at about 0.1.
  • The photon sphere moves by about 50% relative to Schwarzschild for |λ| just under 0.1, implying changes in black hole shadow size of a similar order.
  • The angular difference α between cubic gravity and the Kottler or Schwarzschild solution grows sharply when the geodesic triangle approaches the source, so future high-precision deflection measurements near a compact object can isolate the cubic contribution.
  • In the Solar System the expected cubic contribution to α is about 10^-9 arcsec, which is below currently planned micro-arcsecond mission sensitivities.

Reading between the lines

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

  • A direct extension not made in the paper would be to compute the predicted shadow image for these non-asymptotically flat metrics and compare it with existing millimetre-wavelength shadow observations of Sgr A*, converting the λ ≲ 0.1 bound into an actual posterior constraint.
  • Because the single-function ansatz is assumed, a natural stress test is to solve the full vacuum equations with two independent metric functions; if they differ, the horizon and photon-sphere relations change, especially on the naked-singularity branches.
  • The same triangular-array observable could be applied to rotating cubic-gravity black holes, where the photon-sphere displacement should be azimuthal and could be separated from the spherically symmetric part.
  • If higher-order curvature terms are added, the horizon-shift and photon-sphere relations likely become polynomial in the additional couplings, so the sign-and-magnitude pattern found here may be the first member of a family.
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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

4 major / 6 minor

Summary. This paper studies static, spherically symmetric vacuum solutions in Einsteinian cubic gravity with a cosmological constant, using the single-function metric ansatz ds² = -f(r)dt² + dr²/f(r) + r²dΩ². The authors derive approximate weak-coupling, asymptotic, and near-horizon solutions, integrate the field equations numerically, and report that a positive coupling λ shrinks the horizon while a negative λ enlarges it, with a naked-singularity branch appearing for λ ≳ 0.09–0.1 for a SgrA*-mass object. They then compute the angular difference α in triangular null-geodesic arrays and the photon-sphere position, concluding that strong-field observables can constrain the coupling to λ ≲ 0.1 and that the strongest cubic effects occur near the source. The manuscript includes publicly available numerical codes.

Significance. If the results are correct, the paper would provide concrete, falsifiable strong-field predictions for Einsteinian cubic gravity: a horizon-size dependence on the sign of λ, a photon-sphere shift, and angular differences measurable in principle with laser-ranging or interferometric missions. The analytic work extends earlier studies by expressing approximate solutions in terms of a single integration constant C0 and by covering both signs of λ and non-asymptotically flat backgrounds. Credit is due for the explicit asymptotic and near-horizon series, the comparison of these series with numerical integrations, and the public release of the numerical codes. However, the central quantitative claims are currently conditional on an unproved metric ansatz and on numerical outputs without stated uncertainty, and the headline λ ≲ 0.1 bound is derived for a single mass value.

major comments (4)
  1. [Section II, Eq. (4) and throughout] The single-function ansatz g_tt = 1/g_rr is load-bearing for all results, but the paper does not show that this ansatz is exhaustive for the solutions studied. The statement that the densities C and C' are trivial in static spherically symmetric vacuum, together with Refs. [11,12], establishes the existence of a one-function branch, not uniqueness: the general static spherically symmetric metric has two independent functions, and there is no coordinate freedom to impose g_tt g_rr = -1 while retaining r as the areal radius. Because Eq. (5), the integrated master equation (7), the near-horizon series (20), the horizon equation (24), and all numerical metrics behind Table II and Figs. 5–7 rely on this ansatz, the claimed horizon, photon-sphere, and angular-difference results are conditional on the branch. The paper should either derive the two-function field equations and prove (or verify numerically for a two-function metric with the same boundary conditions) that a solution branch with g_tt g_rr = -1 exists and is the one relevant for black holes, or explicitly restrict every conclusion to the one-function branch and note that the general solution is not covered.
  2. [Section II, Eq. (7)] The master integrated equation (7), from which nearly all subsequent analytical and numerical results are obtained, is stated with the phrase 'can be integrated' but no derivation is provided. There is no appendix or reference that shows how Eq. (5) is reduced to Eq. (7), and this equation is the starting point for the weak-coupling, asymptotic, near-horizon, and numerical solutions. This gap is load-bearing for the paper's central claims, because any error or missing integration constant in Eq. (7) would propagate through the horizon properties, angular differences, and the λ ≲ 0.1 constraint. The authors should include a derivation of Eq. (7) (or a detailed reference that contains it) and a direct consistency check, such as substituting the final expression back into Eq. (5).
  3. [Table II and Figs. 5–6] The reported angular differences α are quoted to several significant figures without error bars, convergence tests, or tolerances. In particular, the claim that the results for Kottler and Schwarzschild backgrounds are the same 'under double-precision floating-point numerical resolution' is not accompanied by any estimate of the numerical resolution, and the constant row b1 = 10^4 rs in Table II, where α = 2.04 × 10^-8 arcsec for all λ, could be a numerical floor rather than a physical result. Since α is the central observational signature, the paper should report convergence with respect to the Runge-Kutta tolerances, the grid spacing used for the central-difference derivatives in Eq. (32), the placement of the outer boundary where the asymptotic solution is matched, and the resulting uncertainty on each α value and on the location of the λ ≈ 0.1 threshold.
  4. [Section II F and Section III C] The constraint λ ≲ 0.1 and the associated naked-singularity threshold are derived for a single value of the mass, C0 = 4 × 10^6 M⊙ (later 4.3 × 10^6 M⊙), with no exploration of the SgrA* mass uncertainty or the general dependence on C0. The horizon equation (24) and the near-horizon relation (22) depend on C0, so the threshold is expected to shift with the mass; the text itself quotes λ ≈ 0.09 in one place and λ ≲ 0.1 in the conclusion. The authors should compute the threshold over the observationally allowed mass range for SgrA*, present the dependence of the critical λ on C0, and specify the numerical criterion used to decide that the maximum of the effective potential disappears in Fig. 7.
minor comments (6)
  1. [Section II F, Eq. (27) and figure captions] The dimensionless rescaling (27) means that values such as λ = 0.05 are only meaningful in the chosen units; the plot axes and table captions should state these units explicitly and distinguish them from dimensionful values used in astrophysical contexts.
  2. [Section II F, Fig. 3] The left panel of Fig. 3 uses C0 = 4 × 10^21 M⊙ and is described in the text as a non-physical illustrative solution, but the caption does not clearly say this; adding a sentence to the caption would avoid confusion.
  3. [Section III B, Table II] Table II and the surrounding text use C0 = 4.3 × 10^6 M⊙, while Section II F and Figs. 2, 5, and 6 use C0 = 4 × 10^6 M⊙; these values should be reconciled, since the angular differences depend on C0.
  4. [Section III A, Eq. (32)] The notation ar g_{\phi\phi} and ar g_{rr} in the tangent formula is not defined; if these are the optical metric components, they should be introduced before Eq. (32).
  5. [Section III B] The statement that the Kottler and Schwarzschild backgrounds give the same α for b1 < 10^4 rs because the cosmological constant contributes only when b1 > 10^10 rs is asserted without derivation; a quantitative estimate of this threshold would make the statement easier to verify.
  6. [Throughout] There are several typographical or notation inconsistencies, including 'SgrA∗' versus 'Sgr A*', 'Red. [17]' instead of 'Ref. [17]', and the use of 'and' in footnote 1; these should be cleaned up during revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the horizon, angular-difference, and photon-sphere claims are computed consequences of the stated action and single-function ansatz, not fitted outputs or self-citation loops.

full rationale

I walked the derivation chain from the action (1) through the field equations (3), the single-function reduction (4)-(7), the approximate solutions (12)-(15) and (20)-(23), the horizon equations (24)-(25), the numerical integrations in Section II F, and the angular-difference/photon-sphere analysis in Section III. Each quantitative claim (smaller/larger horizon for positive/negative lambda, Table II alpha values, photon-sphere shift, lambda <= 0.1 constraint) is obtained by solving the field equations with stated boundary conditions; no parameter is fitted to the output that it is then said to predict. The only self-citation that plays a substantive role is Ref. [22] for the angular-difference method; that method is independently checkable, as the paper itself notes that in the Schwarzschild-versus-Minkowski limit it reproduces the textbook deflection angle, so it is real evidence rather than a circularity. The main caveat is the single-function ansatz (4), which is asserted and supported by Refs. [11,12] but not proven exhaustive for all branches explored; if the full ECG vacuum equations required a second independent metric function, the horizon and photon-sphere numbers could change. That is a limitation or validity risk, not a circular reduction of the paper's outputs to its inputs. I therefore find no significant circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central results rest on the ECG action, the single-function SSS ansatz, the effective-theory validity regime, the choice Lambda_Eff equals the observed cosmological constant, and the branch-dependent numerical seeding procedure. No new particles, forces, or dimensions are introduced. The main free parameters are the coupling lambda, the input Lambda_Eff, and the near-horizon coefficient a2; none are fitted to the signatures being predicted, but they are chosen by hand and scanned.

free parameters (3)
  • Coupling constant lambda = Scanned by hand: -5.0, -1.5, 0.05, 0.1 (dimensionless units of 16 C0^4 / M_Pl^3)
    The theory parameter whose sign and magnitude drive all central claims: horizon shift, angular difference alpha, and photon sphere position. It is not fitted to the signatures, but it is varied freely and the threshold lambda approximately 0.1 is inferred from requiring a SgrA*-like horizon.
  • Effective cosmological constant Lambda_Eff = 1.1056 x 10^-52 m^-2 (Planck 2018 observed value)
    Fixed to the observational cosmological constant before solving Eq (16) for the bare Lambda. This input sets the asymptotic behavior of all numerical solutions and therefore directly shapes the lensing and photon sphere results.
  • Near-horizon coefficient a2 = f''(r_h)/2 = Not given in closed form; selected to match the asymptotic branch for lambda < 0
    The near-horizon series Eq (20) has a2 as its only free parameter. For negative coupling, a2 is tuned so that the numerical solution connects to the asymptotic solution Eq (15); different choices change the near-horizon metric and hence horizon and lensing predictions.
assumptions (6)
  • domain assumption The Einsteinian cubic gravity action (1) with P as the only relevant cubic density for static spherically symmetric vacuum solutions.
    The paper states that densities C and C' contribute only trivially for SSS vacuum, leaving P as the unique non-trivial combination, citing Refs [11,12]. This is load-bearing because Eq (5) and Eq (7) follow from this restriction.
  • domain assumption The static spherically symmetric metric has a single metric function f(r) with g_tt = 1/g_rr, as in Eq (4).
    This ansatz reduces the field equations to one independent ODE. If a second independent metric function were needed, every derived horizon, photon sphere, and lensing result could change. The paper cites prior work for the inversely-related property but does not prove it for all branches.
  • domain assumption The theory is treated as an effective low-energy theory with validity regime |lambda P / M_Pl^3| much less than |R| and r_h much greater than (lambda / M_Pl^5)^{1/4}.
    The authors explicitly exclude known Ostrogradsky and ghost instabilities by assuming extra modes are infinitely heavy and by restricting to the effective regime. This is stated in the introduction and Section II, and it limits the physical applicability of all solutions.
  • ad hoc to paper Lambda_Eff is fixed to the observed cosmological constant and Eq (16) is inverted to determine the bare Lambda.
    This modeling choice, described in Section II C, differs from earlier works that fix Lambda and compute Lambda_Eff. It ensures the asymptotic de Sitter behavior matches observation but makes the 'self-tuning' of the cosmological constant an imposed condition rather than an independent prediction.
  • standard math Null geodesics obey the weak equivalence principle, giving the effective potential V_Eff = f(r) L^2 / r^2.
    Used in Section III C to locate photon spheres. This is the standard geodesic derivation for a static spherically symmetric metric and is not specific to cubic gravity.
  • domain assumption Numerical solutions are seeded from the asymptotic solution for lambda > 0 and from the near-horizon solution for lambda < 0.
    The paper states that for positive coupling the near-horizon seed is invalid because f'(r_h) and f''(r_h) are of the same order, while for negative coupling the equation is stiff and requires a near-horizon seed. The final metrics, and therefore alpha and photon sphere results, depend on the success of this branch-dependent construction.

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Pith. "Pith review of Signatures of cubic gravity in the strong regime." pith.science (2026). https://pith.science/paper/GY2DZYZZ

@misc{pith2026250201747,
  author       = {Pith},
  title        = {Pith review of: Signatures of cubic gravity in the strong regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GY2DZYZZ}},
  note         = {Machine review of arXiv:2502.01747}
}
read the original abstract

We investigate the effects of Einsteinian cubic gravity in the strong gravitational regime. In the first part, we explore analytical solutions for a static, spherically symmetric metric, establishing the existence of maximally symmetric de Sitter solutions, as well as asymptotically de Sitter solutions, with an effective cosmological constant. We also study, analytically and numerically, how the horizon properties are affected by cubic gravity. Our results reveal that a positive coupling constant reduces the horizon size, while a negative one increases it. In the second part, we analyze potential observational signatures of cubic terms, focusing on their effects on the bending of light. Specifically, we investigate the angular difference, related to the deflection angle but valid near the source, along with the behavior of the photon sphere. Our findings show that the strongest effects of the cubic terms occur in the strong gravity regime, and there exists a direct relationship between the value of the coupling constant and the photon sphere position, opening up the possibility to constrain cubic gravity with black hole shadows.

Figures

Figures reproduced from arXiv: 2502.01747 by the authors.

Figure 1
Figure 1. shows a discrete sample of the solution space of Eq. (24), constrained to rh ≥ 0 and f ′ (rh) ≥ 0 within the range −5 ≤ λ ≤ 5. It is important to note that not all roots (represented by circular markers) exhibit the expected asymptotic behavior. For instance, when λ = 0.05, the only root that satisfies this condition is de￾noted with a triangular marker. (The complete numerical solution is shown in [PITH_FULL_IMAGE… view at source ↗
Figure 2
Figure 2. presents the numerical solutions for two pos￾sible values of the positive coupling constant: λ = 0.05 (left panel) and λ = 0.1 (right panel). We used an adap￾tive explicit fifth-order Runge-Kutta routine for the nu￾merical integration, with errors estimated using a fourth￾order routine [37–39]. To ensure that the solutions are ghost-free (see Section II C for a discussion) and re￾produce the expected asymptotic beha… view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: , but in this case for λ ∈ [−10, 0] and b1 = 10rs. The last change is due to the exterior horizon moving to the right for this branch, and we need to ensure that the triangular array is not inside it. In this case, we always have an exterior event horizon (greater than…
Figure 7
Figure 7. Figure 7: illustrates the relationship between the ef￾fective potentials (per unit squared angular momentum) and the coupling constant λ, with C0 fixed to the mass of SgrA∗ . As expected, when λ ̸= 0, the potential profile undergoes changes compared to the Schwarzschild case (λ …

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