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REVIEW 4 major objections 5 minor 106 references

Reissner-Nordstr\"om and Kerr-like solutions in Finsler-Randers Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Finsler-Randers gravity changes how massive particles orbit black holes, while leaving photons on their general-relativity paths.

desk verdict The photon result is a definitional artifact, the Kerr solution is a partial numeric sketch, and the timelike RN effective potential is the only solid piece. read the letter →

arxiv 2505.08009 v1 pith:ETUDG5Y7 submitted 2025-05-12 gr-qc astro-ph.HEhep-th

Georgios Miliaresis , Konstantinos Topaloglou , Ioannis Ampazis , Nefeli Androulaki , Emmanuel Kapsabelis , Emmanuel N. Saridakis , Panayiotis C. Stavrinos , Alkiviadis Triantafyllopoulos This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th
keywords Finsler-RandersgravityReissner-NordströmblackholeKerrgeodesicseffectivepotentiallightliketimelikeorbitstangentbundlegeometry
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 asks what happens to two classic black-hole spacetimes, charged Reissner-Nordström and rotating Kerr, when gravity is described by Finsler-Randers geometry, in which the metric also depends on the direction of motion through a small one-form field. It constructs explicit Finsler-Randers versions of both solutions and derives the corresponding geodesic equations. The central result is a clean separation: curves classified as lightlike by the underlying Riemannian interval are identical to their general-relativity counterparts, while timelike curves acquire a modified energy integral and an effective potential that shifts bound orbits and precession. If this holds, the theory's observational signature is concentrated in massive-particle dynamics around black holes, with photon deflection, shadows, and lensing left untouched. The paper thus offers a concrete way to test whether local spacetime anisotropy can exist without disturbing current light-based gravitational tests.

What carries the argument

The engine of the construction is the Randers one-form $A_\gamma(x)$ added to the Riemannian square root, $F = \sqrt{g_{\mu\nu}y^\mu y^\nu} + A_\gamma y^\gamma$, with $|A| \ll 1$; solving the tangent-bundle field equations fixes $A$ for each background. The second mover is the interval label $\sigma$, defined by $\sigma^2 = g_{\mu\nu}\dot x^\mu\dot x^\nu$ and used to classify geodesics. Every $A$-dependent term in the Euler-Lagrange equations carries a factor $\sigma$, so $\sigma = 0$ (lightlike) removes the Randers contribution identically while $\sigma = 1$ (timelike) leaves it active.

What would settle it

Evaluate null geodesics with the full Finsler geodesic equation for a curve satisfying $F = 0$ in the Reissner-Nordström-Randers spacetime; if such curves differ from the general-relativity null geodesics, the paper's invariance claim fails. Observational check: measure a photon ring or lensing deflection and a stellar orbit around the same black hole—if the photon observables deviate from GR while the orbit also deviates, the assumption behind the claim is violated.

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Extended reading notes

Core claim

The paper claims that the Reissner-Nordström and Kerr metrics remain exact Riemannian parts of a Finsler-Randers spacetime, with the anisotropy encoded in a Randers one-form $A_\gamma$ determined by the modified field equations. For the charged spherical case the one-form is $A_0 = Q/r + \tilde A_0\sqrt{f(r)}$, and the geodesic equations acquire a modified energy integral $f(r)\dot t = -\sigma F(r) + \mathcal{E}$ and radial equation $(\dot r)^2 + f(r)(l^2/r^2 + \sigma^2) = \mathcal{E}^2 + \sigma^2 F(r)^2 - 2\sigma\mathcal{E}F(r)$. For a rotating spacetime, the components $A_5$ and $A_6$ vanish analytically while $A_4$ and $A_7$ are found numerically in a bounded region. Because every Randers correction in the geodesic equations is multiplied by $\sigma$, and lightlike curves are defined by $\sigma^2 = 0$, photons follow exactly the GR geodesics while massive particles feel a shifted effective potential and altered orbits.

Load-bearing premise

The conclusion that light is untouched hinges on calling a curve lightlike when the Riemannian interval $\sigma^2 = g_{\mu\nu}\dot x^\mu\dot x^\nu$ vanishes, rather than when the full Finsler metric function $F$ vanishes; the paper chooses this explicitly to keep light speed observer-independent.

Editorial extensions

If this is right

  • In the Finsler-Randers Reissner-Nordström spacetime, the modified energy integral and effective potential shift the orbits of massive test particles, while the photon-sphere radius and light deflection remain at their general-relativity values.
  • Bound timelike orbits acquire additional precession beyond the standard Reissner-Nordström prediction, controlled by the charge $Q$ and the Randers magnitude $\tilde A_0$, and they reduce to Schwarzschild orbits when both vanish.
  • In the Kerr-like case, timelike trajectories deviate from classical Kerr geodesics in the region where the numerically determined $A_4$ and $A_7$ components are trusted, whereas the lightlike sector remains identical to Kerr.
  • Quasinormal modes of background scalar fields coincide with their Riemannian Kerr modes when the fields are taken to be independent of the tangent-space coordinates.
  • Observational quantities built only from photons, such as shadows and lensing, should match general relativity exactly, so the distinctive Finsler-Randers signature must be sought in the motion of massive tracers near black holes.

Reading between the lines

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

  • Because the lightlike sector matches general relativity exactly under this construction, any observed anomaly in photon observables would weigh against the model rather than support it; its observational window is necessarily in massive-particle dynamics.
  • A natural extension would define null curves by the full Finsler metric function $F = 0$ instead of the Riemannian interval $\sigma = 0$, which would bring the Randers anisotropy into photon motion and change shadows and lensing.
  • If the complete nonperturbative Kerr-Randers solution retains the divergence of $A_4$ and $A_7$ at spatial infinity, the model would require an infrared regulator or a different vacuum before it can be applied at cosmological scales; the present claims are local to a finite region around the black hole.
  • A direct test would compare stellar-orbit precession with a photon-ring measurement around the same black hole: the former can be fit by a nonzero $\tilde A_0$ while the latter must stay unchanged, a pairing that cleanly separates this framework from modified gravities that alter both sectors.
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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 / 5 minor

Summary. The manuscript extends a Finsler-Randers gravitational framework, previously applied to Schwarzschild and Schwarzschild-de Sitter spacetimes, to Reissner-Nordström and Kerr black holes. For the Reissner-Nordström case it infers the Randers one-form from the earlier spherically symmetric solution, derives energy and radial equations for geodesics, and concludes that lightlike geodesics are unaffected while timelike geodesics acquire a modified energy integral and effective potential. For the Kerr case it solves a subset of the field equations analytically for two components of the Randers one-form, obtains the remaining two components numerically in a finite radial region using an Euler-type scheme with a Fourier fit, and integrates timelike geodesics numerically. A final section argues that quasinormal modes of background fields are unchanged. The stated headline results are that photons follow GR geodesics while massive particles deviate from GR.

Significance. If the derivations were sound, the paper would offer a concrete class of Finsler-Randers black-hole spacetimes with potentially observable deviations in massive-particle orbits, and it would be useful to have the numerical Kerr sector spelled out. However, the main claims are largely consequences of definitions rather than derived results: the lightlike-geodesic conclusion follows from setting the Riemannian interval to zero, and the Reissner-Nordström one-form is an inferred ansatz rather than a verified solution. The Kerr one-form is explicitly admitted to be unverified against all field equations. The paper is clearly organized and reproduces the standard GR limits in the appropriate regimes, and the numerical illustrations are instructive, but the central physical conclusions are not established by the calculations presented.

major comments (4)
  1. [Section III, Eq. (42)] The Reissner-Nordström Randers one-form is introduced as 'inferred from the general solution of the spherically symmetric problem [12]' rather than derived by substituting into the field equations (59) or (21)-(23) for the RN metric. Since all subsequent RN geodesic results depend on this A_0, the inference must be checked explicitly; as written, the central RN solution is an ansatz, not a solution, and no residual equation is presented.
  2. [Section III, Eqs. (43)-(54)] The result that lightlike geodesics are unaffected is a tautology. The geodesic equations (44)-(45) contain the Randers one-form only through terms multiplied by sigma, and Eq. (54) is obtained by setting sigma=0. Because sigma is defined as the Riemannian interval, the Randers term is removed at the level of the causal classification, not by any dynamical property. In a Randers geometry the Finsler metric function is F = sqrt(g_mu_nu y^mu y^nu) + A_mu y^mu, and the natural lightlike condition is F=0, which depends on A. The statement that sigma=0 'preserve[s] the invariance of the speed of lightlike signals' is itself the point at issue. Moreover, Eq. (44) is derived assuming sigma is a nonzero constant along the curve; substituting sigma=0 after deriving the equation is not a controlled null limit. The paper's central conclusion that photons are unaffected is therefore not established; it is an artifact of the adopted definition.
  3. [Section IV.A, Eqs. (63) and Fig. 4] The Kerr one-form components A_4 and A_7 are obtained with an Euler-type integration over a finite grid, r/R_s in (2,10), and then fitted to a Fourier series. The authors explicitly state that 'these solutions have not been verified to satisfy all choices of indices (mu,nu) in (59)'. Without a residual check, convergence study, or boundary conditions, these functions cannot serve as reliable input for the geodesic integrations in Fig. 5. In addition, the normalization is arbitrary and the divergence at infinity introduces a free cutoff, so the Kerr phenomenology is not quantitatively controlled.
  4. [Section IV.C] The conclusion that quasinormal modes are unchanged is also a consequence of an imposed ansatz rather than a derived result. The field is assumed constant along the vertical directions of the tangent bundle, which immediately reduces the Lagrangian to its Riemannian form. This does not follow from the Finsler-Randers dynamics; it is an additional restriction, so the section does not provide a test of the framework's predictions for perturbations.
minor comments (5)
  1. [Section III, Eq. (42)] The absolute value |1 - 2M/r + Q^2/r^2|^{1/2} makes the one-form non-smooth where the RN metric component changes sign; the domain of validity of the solution should be stated explicitly.
  2. [Section IV.A] The numerical solution should specify the exact radial cutoff, the boundary data, and the grid resolution used for the Euler integration; currently only the range r/R_s in (2,10) is given.
  3. [Figures 2 and 5] The plots would be more informative with quantitative comparisons, such as the radial period or precession angle, rather than the qualitative statement that orbits are 'quickly precessing Keplerian ellipses'.
  4. [Section III, after Eq. (43)] The text mentions 'the spacial metric'; this should read 'the spatial metric'.
  5. [Section III, Eq. (45)] The coupling to the electromagnetic tensor is stated as applying to 'a unit charge particle', but the charge-to-mass ratio does not appear explicitly in the equation; the convention should be stated where the equation is introduced.

Circularity Check

2 steps flagged · score 6.0 of 10

Lightlike-geodesic invariance is an artifact of defining null curves by the Riemannian interval σ²=0; the timelike modifications are parametric, not circular.

  1. self definitional [Section III, Eqs. (43)-(54) (definition of lightlike via Riemannian interval; derivation of Eq. (54))]
    "We characterise, however, these geodesics as timelike, lightlike or spacelike depending on the sign of the Riemannian line element: ... Lightlike: σ²=0 along the curve ... This choice is made in order to preserve the invariance of the speed of lightlike signals... In the former case we observe that all of our equations return to their GR counterparts as all of the additional, perturbing terms are weighted by σ and are therefore lost."

    Lightlike trajectories are defined by σ² = g_μν ẋ^μ ẋ^ν = 0, and the Randers one-form enters the geodesic equations (44)-(45) only through factors of σ (e.g., σ g^{μν}Φ_{νρ}ẋ^ρ). Setting σ=0 therefore removes the Finsler-Randers correction by stipulation, so Eq. (54) restates the chosen definition rather than deriving a property of the Finsler-Randers dynamics. The natural Randers null condition F = sqrt(g y y) + A_μ y^μ = 0 depends on A, and the σ→0 limit of the Euler-Lagrange equations is singular because the derivation divides by σ. The conclusion section itself limits the claim to 'within our formulation of timelike and lightlike character based on the Riemannian line element.' The headline null result is thus an artifact of the causal classification.

  2. self definitional [Section IV.B (Kerr-Randers geodesics, after Eq. (64))]
    "As before we note that lightlike geodesics, found with σ = 0, are identical to their Riemannian counterparts, and so all associated quantities remain invariant in transitioning to the Finsler-Randers framework."

    The same reduction is repeated for the Kerr case: in Eq. (64) the Randers force is σ g^{μν}Φ^{(R)}_{μν}, so taking 'σ = 0' erases the Randers contribution by construction. The conclusion that Kerr lightlike geodesics are unchanged is therefore a restatement of the chosen lightlike convention, not a computed consequence of the Finsler-Randers geodesic equation.

full rationale

The timelike-sector results are not circular: given the assumed Randers one-form, Eqs. (50), (53), and (56) follow by direct algebra, and the Kerr components A4 and A7 are obtained by solving the linearized field equations (59) numerically. Importing the spherically symmetric Randers solution from the authors' earlier [12] is a citation of prior work rather than a circular step, and the numerical solution is an independent computation even though it is verified only for a subset of index choices. The circularity is confined to the null sector: defining lightlike curves via the Riemannian interval σ²=0 makes the Randers term vanish identically, so the paper's repeated claim that lightlike geodesics are unaffected is true by definition. The authors candidly flag the limitation with 'within our formulation of timelike and lightlike character based on the Riemannian line element.' Because the central null prediction reduces to this definitional choice, while the timelike predictions remain parametric in the free constant Ã0 and thus not forced by any fit, a partial circularity score of 6 is appropriate rather than a higher score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central derivation rests on the generalized field equations of Finsler-Randers gravity taken from earlier work by the same group, the smallness of the Randers one-form, and a definitional choice that classifies geodesics by the Riemannian interval rather than the full Finsler metric. The RN anisotropy field is borrowed from the spherically symmetric solution, and the Kerr field is a numerical construction with arbitrary normalization and an explicit lack of full verification. No new particles or forces are introduced; the free parameter \tilde A_0 controls all observable deviations.

free parameters (3)
  • \tilde A_0 (Randers one-form normalization)
    Introduced in Eq. (42) for RN and in the Kerr numerical solution; an unspecified constant satisfying |A|<<1. It sets the magnitude of all observable deviations and is not fixed by theory or observation.
  • Kerr A_4, A_7 normalization = arbitrary; maxima ~10^-5
    Section IV.A: the numerical solution is determined up to a normalizing factor chosen so maximum absolute values are of order 10^-5; observables depend on this choice.
  • Kerr integration domain and cutoff = r/Rs in (2,10), theta in (0,pi)
    Section IV.A: the PDE is integrated only in a restricted region and the solution is discarded beyond a radial cutoff because it diverges; the domain choice affects the fitted solution and the orbit illustration.
assumptions (5)
  • domain assumption The generalized field equations (21)-(23) derived from the Hilbert-like action on the tangent bundle are the correct equations of Finsler-Randers gravity.
    These equations are taken from prior work by the same group and are not derived in this paper; they define the framework.
  • domain assumption The Randers one-form is small, |A|<<1, and quadratic and higher terms in A are neglected.
    Stated in Section II.C and used to linearize the field equations into Eq. (59); the Kerr solution is only valid where this first-order scheme applies.
  • ad hoc to paper Geodesics are classified as timelike, lightlike, or spacelike by the sign of the Riemannian line element sigma, with sigma=0 for lightlike.
    Section III: this choice removes all Randers terms for lightlike curves by construction, so the no-imprint result is built into the definition.
  • domain assumption The matter energy-momentum tensor T_mu_nu for the RN spacetime is the same electromagnetic energy-momentum tensor as in GR.
    Appendix: assumed because variation of the electromagnetic Lagrangian with respect to g_mu_nu should not change in a Finslerian perspective.
  • domain assumption Torsion quadratic terms T^gamma_kappa_gamma T^beta_lambda_beta are neglected as first order in w^alpha_beta.
    Appendix: used to reduce Eq. (68) to the linear equation for the Randers one-form.
invented entities (1)
  • Randers one-form components A_4 and A_7 for Kerr spacetime
    purpose: Induce local anisotropy in the Kerr geometry and modify timelike particle orbits.
    These components are obtained only numerically in a finite region with arbitrary normalization and are not verified for the full field equations; the framework provides no independent handle such as a predicted mass or coupling.

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Pith. "Pith review of Reissner-Nordstr\"om and Kerr-like solutions in Finsler-Randers Gravity." pith.science (2026). https://pith.science/paper/ETUDG5Y7

@misc{pith2026250508009,
  author       = {Pith},
  title        = {Pith review of: Reissner-Nordstr\"om and Kerr-like solutions in Finsler-Randers Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ETUDG5Y7}},
  note         = {Machine review of arXiv:2505.08009}
}
read the original abstract

In a previous study we investigated the spherically symmetric Schwarzschild and Schwarzschild-de Sitter solutions within a Finsler-Randers-type geometry. In this work we extend our analysis to charged and rotating solutions, focusing on the Reissner-Nordstr\"om and Kerr-like metrics in the Finsler-Randers gravitational framework. In particular, we extract the modified gravitational field equations and we examine the geodesic equations, analyzing particle trajectories and quantifying the deviations from their standard counterparts. Moreover, we compare the results with the predictions of general relativity, and we discuss how potential deviations from Riemannian geometry could be reached observationally.

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Pith tools

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