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Gravitational fingerprints of a black hole solution in Palatini Kalb-Ramond gravity

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Carlos F. S. Pereira, Araújo Filho, Marcos V. de S. Silva, N. Heidari

General Relativity and Quantum Cosmology · Submitted 19 Oct 2026 · 91 references

Pith's verdict: A systematic and mostly sound catalogue of which Palatini Kalb-Ramond black hole observables stay Schwarzschild-degenerate at fixed orbital mass, held back by a localized lensing-convention mess and a load-bearing dependence on an unpublished companion paper.

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CitationAra\'ujo Filho, Carlos F. S. Pereira, Marcos V. de S. Silva, N. Heidari. "Gravitational fingerprints of a black hole solution in Palatini Kalb-Ramond gravity." arXiv:2610.26692 (2026). https://pith.science/paper/TMDE5YH2
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The paper

The paper claims

At fixed orbital mass, this black hole keeps Schwarzschild's horizon, photon sphere, and impact parameter; every Lorentz-violating effect lives in one radial factor $\sigma$ that rescales distances, tides, and axial ringdown.

The reading

This paper studies a static, spherically symmetric black hole in Palatini Kalb-Ramond gravity, a modified theory in which a background antisymmetric tensor field spontaneously breaks Lorentz symmetry and the metric and the affine connection are varied independently. Its central claim is that, once the orbital mass $\bar{M}=M/\rho$ that actually enters the Newtonian potential is held fixed, the horizon at $r_h=2\bar{M}$, the photon sphere at $r_{\mathrm{ph}}=3\bar{M}$, the critical impact parameter $b_c=3\sqrt{3}\,\bar{M}$, and the innermost stable circular orbit at $6\bar{M}$ keep their Schwarzschild values, while a single radial factor $\sigma^2=1-q/2$ (with $q=\ell_2/(1-\ell_1)$) carries the entire deformation. Every quantity that samples the radial metric — proper radial distances, infall times, accumulated azimuthal angles, tidal eigenvalues, and the photon-orbit instability exponent — rescales by an exact power of $\sigma$, and in the homogeneous axial gravitational sector the quasinormal frequencies obey the exact relation $\hat{\omega}_{\mathrm{T}}(q)=\hat{\omega}_{\mathrm{Sch}}/\sigma$. This matters because it separates observables that cannot distinguish the Lorentz-violating geometry from Schwarzschild, such as the shadow diameter once the dynamical mass-to-distance ratio is calibrated, from observables that can, such as tidal forces, strong-field lensing image separations, and gravitational ringdown rates.

What carries the argument

The load-bearing object is the line element $ds^2=-F(r)\,dt^2+(\sigma^2/F(r))\,dr^2+r^2\,d\Omega^2$ with $F(r)=1-2\bar{M}/r$, a one-parameter radial deformation of Schwarzschild in which $\sigma^2=1-q/2$ measures the Lorentz-violating deformation and $\bar{M}=M/\rho$ separates the conserved Iyer-Wald mass $M$ (the Noether-charge mass) from the orbital mass that enters the Newtonian potential. The argument is carried by the identity $\psi=\varphi/\sigma$: substituting it into the geodesic equation puts the orbit problem in Schwarzschild form, so any quantity fixed by the effective potential $F(r)(\epsilon+L^2/r^2)$ is shared with Schwarzschild, while any quantity that integrates the radial metric — proper length, proper time, azimuthal accumulation, the tortoise coordinate — acquires a power of $\sigma$. In the gravitational sector the same factorization is realized by an auxiliary metric whose background is exactly Schwarzschild under rescaled time, radius, and mass, which is what produces the exact axial spectral map $\hat{\omega}_{\mathrm{T}}=\hat{\omega}_{\mathrm{Sch}}/\sigma$.

Extended reading notes

Core claim

The paper's central claim, in its own normalization, is an exact factorization of the physics. With the orbital mass $\bar{M}=M/\rho$ defined so that the gravitational potential reads $F(r)=1-2\bar{M}/r$, and with $\sigma^2=1-q/2$, $q=\ell_2/(1-\ell_1)$, the quantities fixed by the circular-orbit effective potential — horizon, photon sphere, critical impact parameter, ISCO, marginally bound orbit, and the radial frequency ratios of the Doppler analysis — are exactly Schwarzschild, because that potential depends only on $\bar{M}$. The deformation sits entirely in the radial metric component $G(r)=\sigma^2/F(r)$, entering through the exact rescaling $\psi=\varphi/\sigma$, which maps the geodesic equation to Schwarzschild form: at fixed $\bar{M}$, energy, and angular momentum, $r(\varphi)=r_{\mathrm{Sch}}(\varphi/\sigma)$. Consequently proper distances, infall times, and accumulated azimuthal angles scale with $\sigma$; the radial and transverse tidal eigenvalues for free fall gain a factor $\sigma^{-2}$ while remaining traceless and in the ratio $-2$; and the photon-orbit instability exponent scales as $\sigma^{-1}$ while the orbital frequency stays fixed. In the wave sector, scalar, electromagnetic, and massless Dirac fields propagate on the physical metric, whereas a homogeneous axial gravitational sector with $\delta B_{\mu\nu}=0$ is carried by an auxiliary metric that becomes exactly Schwarzschild under the rescalings $T=\sqrt{1-\ell_1}\,t$, $R=\sqrt{1-\ell_1-\ell_2/2}\,r$; this yields the exact spectral relation $\hat{\omega}_{\mathrm{T}}(q)=\hat{\omega}_{\mathrm{Sch}}/\sigma$, so both oscillation and damping rates of these modes scale as $\sigma^{-1}$ at fixed $\bar{M}$, with WKB and time-domain evolution agreeing to within $0.7\%$ for the reported fundamental modes.

Load-bearing premise

The paper's scaling laws all rest on the imported claim, taken from a companion preprint and not re-derived here, that the line element with $F(r)=1-2\bar{M}/r$ is an exact solution of the Palatini Kalb-Ramond field equations and that $M$ is its Iyer-Wald conserved mass; if that solution, mass identification, or positivity domain $1-\ell_1>0$, $1-\ell_1-\ell_2/2>0$ fails, the factorization into $\sigma$ and $\bar{M}$ collapses.

Editorial extensions

If this is right

  • Shadow measurements of Sgr A* and M87* cannot by themselves pin down the Kalb-Ramond couplings: at fixed $\bar{M}/D_A$ the model predicts exactly zero deviation in both published angular statistics, so the couplings remain free at that level of inference.
  • Strong-field lensing breaks that degeneracy: the limiting relativistic-image position $\theta_\infty$ grows with $q$ while the angular separation of the outermost image from the asymptotic image position shrinks, so image-separation measurements would probe $\sigma$ directly.
  • The axial gravitational ringdown is fully predicted from Schwarzschild: at fixed $\bar{M}$ every multipole obeys $\hat{\omega}_{\mathrm{T}}=\hat{\omega}_{\mathrm{Sch}}/\sigma$, so measuring one axial mode together with an independent mass calibration determines $\sigma$, while at fixed conserved mass $M$ the spectrum is independent of $\ell_2$.
  • Test-field ringdown and photon-orbit instability respond differently: at leading eikonal order the scalar, electromagnetic, and Dirac oscillation frequencies track the physical photon orbit and stay fixed at fixed $\bar{M}$, while their damping rates scale as $\sigma^{-1}$, so the same shadow radius can accompany different relaxation times.
  • The choice of mass convention changes what is predicted: at fixed conserved mass $M$ rather than $\bar{M}$, the horizon, photon sphere, and impact parameter all grow like $1/\rho$, so astrophysical comparisons must state which mass is being calibrated.

Reading between the lines

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

  • The exact auxiliary-metric map suggests a stronger, unproven statement worth testing: the entire homogeneous axial sector may be physically indistinguishable from Schwarzschild with a rescaled time coordinate, which would make ringdown a clean null test of Lorentz violation in this theory class.
  • The paper's conversion formulas imply a concrete observational route it leaves implicit: a single independent measurement of proper radial distance (yielding $\delta_{\mathrm{prop}}=\sigma-1$) or of the conserved mass would convert the currently degenerate shadow intervals of its Table III into a strip bound on $q$ — at $\ell_1=0$, the Sgr A* VLTI interval alone would roughly bound $-0.9\lesssim q
  • If the imported solution is confirmed, the split between conserved mass and orbital mass is likely generic in Palatini theories with a second metric scale: mass estimates from the Newtonian potential and from Noether charges need not coincide, and conflating them could mimic either a shadow anomaly or an accidental agreement with general relativity.
Original abstract

We investigate particle motion, optical observables, and linear wave propagation around a static black hole in Palatini Kalb Ramond gravity with a vanishing cosmological constant. The analysis distinguishes the conserved mass $M$ from the orbital mass $\bar{M}=M/\rho$ and isolates the radial deformation $\sigma^2=1-q/2$, where $q=\ell_2/(1-\ell_1)$. At fixed $\bar{M}$, the photon sphere, critical impact parameter, and innermost stable circular orbit retain their Schwarzschild values. Radial frequency ratios also preserve their Schwarzschild profiles for the same areal release and particle radii, while proper distances, infall times, and accumulated orbital angles depend on $\sigma$. For radial fall, the tidal eigenvalues acquire a factor $\sigma^{-2}$, and the instability exponent of the photon orbit scales as $\sigma^{-1}$. In addition, we examine light deflection in the weak and strong regimes with an explicit account of the conical asymptotics. The shadow comparison with Sgr A* and M87* exhibits a coupling degeneracy when the orbital mass and areal distance are set. Scalar, electromagnetic, and massless Dirac perturbations probe the physical metric, whereas a homogeneous axial gravitational sector with $\delta B_{\mu\nu}=0$ admits an exact Schwarzschild spectral map. In this sector, both oscillation and attenuation rates scale as $\sigma^{-1}$ at fixed $\bar{M}$. Third order WKB estimates agree with frequencies extracted from time evolution to within $0.7\%$ for the reported fundamental modes.

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, 2026-10-20: 1 request, 7,741 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.68 · human participants 0.01 · controlled comparison 0.41 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We investigate particle motion, optical observables, and linear wave propagation around a static black hole in Palatini Kalb Ramond gravity with a vanishing cosmological constant. The analysis distinguishes the conserved mass $M$ from the orbital mass $\bar{M}=M/\rho$ and isolates the radial deformation $\sigma^2=1-q/2$, where $q=\ell_2/(1-\ell_1)$. At fixed $\bar{M}$, the photon sphere, critical impact parameter, and innermost stable circular orbit retain their Schwarzschild values. Radial frequency ratios also preserve their Schwarzschild profiles for the same areal release and particle radii, while proper distances, infall times, and accumulated orbital angles depend on $\sigma$. For radial fall, the tidal eigenvalues acquire a factor $\sigma^{-2}$, and the instability exponent of the photon orbit scales as $\sigma^{-1}$. In addition, we examine light deflection in the weak and strong regimes with an explicit account of the conical asymptotics. The shadow comparison with Sgr A* and M87* exhibits a coupling degeneracy when the orbital mass and areal distance are set. Scalar, electromagnetic, and massless Dirac perturbations probe the physical metric, whereas a homogeneous axial gravitational sector with $\delta B_{\mu\nu}=0$ admits an exact Schwarzschild spectral map. In this sector, both oscillation and attenuation rates scale as $\sigma^{-1}$ at fixed $\bar{M}$. Third order WKB estimates agree with frequencies extracted from time evolution to within $0.7\%$ for the reported fundamental modes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate particle motion, optical observables, and linear wave propagation around a static black hole in Palatini Kalb Ramond gravity with a vanishing cosmological constant. contribution 1.00 0.93 0.05 asserted 0.85
s1 The analysis distinguishes the conserved mass $M$ from the orbital mass $\bar{M}=M/\rho$ and isolates the radial deformation $\sigma^2=1-q/2$, where $q=\ell_2/(1-\ell_1)$. method 0.49 0.92 0.09 asserted 1.00
s2 At fixed $\bar{M}$, the photon sphere, critical impact parameter, and innermost stable circular orbit retain their Schwarzschild values. result 1.00 0.86 0.28 asserted 1.00
s3 Radial frequency ratios also preserve their Schwarzschild profiles for the same areal release and particle radii, while proper distances, infall times, and accumulated orbital angles depend on $\sigma$. result 1.00 0.90 0.21 asserted 1.00
s4 For radial fall, the tidal eigenvalues acquire a factor $\sigma^{-2}$, and the instability exponent of the photon orbit scales as $\sigma^{-1}$. result 1.00 0.89 0.10 asserted 1.00
s5 In addition, we examine light deflection in the weak and strong regimes with an explicit account of the conical asymptotics. contribution 0.98 0.91 0.03 asserted 0.97
s6 The shadow comparison with Sgr A* and M87* exhibits a coupling degeneracy when the orbital mass and areal distance are set. result 0.93 0.90 0.10 asserted 1.00
s7 Scalar, electromagnetic, and massless Dirac perturbations probe the physical metric, whereas a homogeneous axial gravitational sector with $\delta B_{\mu\nu}=0$ admits an exact Schwarzschild spectral map. contribution 0.56 0.87 0.08 asserted 1.00
s8 In this sector, both oscillation and attenuation rates scale as $\sigma^{-1}$ at fixed $\bar{M}$. result 1.00 0.89 0.09 asserted 1.00
s9 Third order WKB estimates agree with frequencies extracted from time evolution to within $0.7\%$ for the reported fundamental modes. result 0.88 0.93 0.05 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.91.

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
  • R cross-paper relations

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. Null trajectories for vr0 = −018, r0 = 77M¯ , 0 = 3π2, and M¯ 0 = 0022, at xed M¯ . The ray with q = −06 is captured after approaching the photon sphere; the remaining four rays scatter. The gray disk and dashed circle have radii 2M¯ and 3M¯ , respectively
Figure 2. Null trajectories for vr0 = −021 and the remaining launch data of
Figure 3. Captured null trajectories for vr0 = −025, with the same initial position and angular velocity. Every impact parameter lies below bc, while the angular accumulation before horizon arrival depends on q
Figure 4. Captured null trajectories for vr0 = −031. The increased inward coordinate velocity lowers the impact parameters in all ve cases. Each curve terminates at r = 2M¯ . launch angle at a common r0, and Eq. (23) supplies a common closest approach. The ingoing radial velocity becomes vr0 = − 1 σ √ 1 − b2 γ F(r0) r2 0  (27)
Figure 5. Scattering trajectories at xed E = 1 and bγ = 6M¯ , launched from r0 = 20M¯ and 0 = 3π2. The common closest approach is rmin ≃ 44534M¯ ; dierences between the curves arise from the angular factor σ. The same q values and color assignment as in Figs. 1–4 are used. sweep. An important remark which deserves being pointed out, at xed M, changing the couplings would also change M¯ = Mρ and rescale the horizon, the photon sphere, and the turning radii. C. Angular bending from weak to strong deection For a scattered photon arriving from and returning to large radius, the total…
Figure 6. Coordinate angular excess cone relative to the zero mass conical geometry, computed from Eq. (28). The distance from the critical impact parameter is shown on a logarithmic scale. The curves retain the exact dependence on σ; the divergence has the coecient given in Eq. (31). Near the capture, let us set  = bγbc − 1 > 0. The Schwarzschild strong deection expansion [? ] yields cone = σ [ − ln  + ln 216[7 − 4 √ 3] − π ] + O( ln ) (31) The coecient of the logarithm is σ, while the critical impact parameter stays xed at a given M¯
Figure 7. Dimensionless instability exponent M¯ λt and photon orbit frequency M¯ Ωph as functions of q, at xed M¯ . Increasing q lowers σ = 1 − q2 and increases the instability exponent. The orbital frequency remains constant. E. Bound timelike motion and orbital stability For a massive particle, the eective potential is V1 = F(1 + L2r2). A circular orbit at r = rc requires V1(rc) = E2 c and V ′ 1(rc) = 0, giving E2 c = (rc − 2M¯ )2 rc(rc − 3M¯ ) , L2 c = Mr ¯ 2 c rc − 3M¯  (35) Circular timelike trajectories exist for rc > 3M¯ . Their radial stability follows from V ′′ 1 (rc) = 2
Figure 8. Bound timelike trajectories over two radial periods, with
Figure 9. Outgoing frequency ratio for release from rest at
Figure 10. Ingoing frequency ratio for the same release radius. During the fall,
Figure 11. Outgoing frequency ratio as a function of the proper radial distance from the release point,
Figure 12. Ingoing frequency ratio versus proper radial distance for the same con
Figure 13. Proper time from release at B = 10M¯ to the horizon, in units of M¯ . The dependence on the KR couplings is ∆τh(q) = 1 − q2 ∆τh(0) at xed M¯ and BM¯ . σ: ∆τ (B → r) = σ  B3 2M¯  arccos  r B + r(B − r) B  , (54) ∆tγ(r → B) = σ  B − r + 2M¯ lnB − 2M¯ r − 2M¯   (55) The static observer at B measures a signal propagation interval F(B) ∆tγ. The particle crosses the horizon after a nite proper time, while the outgoing propagation interval grows logarithmically as the emission radius approaches rh
Figure 14. Dimensionless radial tidal eigenvalue M¯ 2λr versus rM¯ for representative q = −06, −03, 0, 03, 06. Each exterior prole begins at the common areal horizon rhM¯ = 2. Positive q increases the radial stretching at xed rM¯
Figure 15. Radial tidal strength at the horizon in units of
Figure 16. Dimensionless transverse eigenvalue M¯ 2λ⊥ for the congurations used in
Figure 17. Normalized null eective potential U = (1 − 2x)x2. The maximum at rM¯ = 3 determines the photon sphere. The horizon is at rM¯ = 2, where the potential vanishes. The mass scale M¯ is held xed. A. Null geodesics and the photon sphere Spherical symmetry allows the photon trajectory to be placed in the equatorial plane. For an ane parameter λ, the conserved photon energy and angular momentum are Eγ = F(r) dt dλ, Lγ = r2 d dλ , bγ = Lγ Eγ  (75) The null condition gives σ2  dr dλ 2 + Vγ(r) = E2 γ, Vγ(r) = L2 γ F(r) r2  (76) A circular photon orbit satises V ′ γ(rph) = 0 and…
Figure 18. Dimensionless radial Lyapunov exponent of the photon orbit. The KR dependence is
Figure 19. Angular radius of the shadow measured by a static observer outside the photon sphere. The angle is
Figure 20. Shadow contour on the normalized areal screen for a distant observer. The radius is
Figure 21. Sgr A* shadow-deviation intervals from the VLTI and Keck priors, shown with the KR prediction
Figure 22. M87* angular mass consistency interval and the static KR prediction. The 68% range is
Figure 23. Sgr A* and M87* statistics with their 68% intervals. The vertical line marks the static KR prediction. Sgr
Figure 24. Sensitivity strips in the KR coupling plane, obtained from Eq. (
Figure 25. Gaussian curvature of the optical geometry as a function of
Figure 26. Deection angle in the weak-eld limit as a function of Mb for dierent values of the Lorentz-symmetry violation parameter As can be seen in the expression above, Eq. (101), we calculate the deection of light in the weak-eld regime using the Gauss-Bonnet method, which is applicable to both asymptotically at and non-asymptotically at geometries. In this context, our primary objective is to establish a proper comparison between the results obtained via this formalism and those derived from the method traditionally employed in the literature, as presented in Section VIII. It is…
Figure 27. Light angular deection diagram. the lens L, and the observer O a scenario in which eects associated with the formation of relativistic images become particularly signicant [34, 35] we can establish the gravitational lens equation that relates the angular positions  and , expressed as:  =  − DLS DOS ∆n, (113) where ∆n is the deection angle with all the loops completed by the photons before reaching the observer subtracted out; that is, ∆n =  − 2nπ. In this approach, we use the following approximation for the impact parameter: b ≈ DOL. Thus, the contributions to the…
Figure 28. Dimensionless eective potentials for scalar, vector, tensor, and spinor perturbations as functions of x∗ = r∗M¯ , at xed M¯ . Each panel identies the angular eigenvalue and the values of q used in the calculation. The spinor panel displays U+. The tensor interpretation is specied in Sec. IX C. E. Geometric interpretation and numerical analysis The connection with the unstable null orbit gives an independent way to interpret the leading behavior at large angular momentum [86]. In the normalization used here the circular orbit and its radial instability
Figure 29. Real parts Re(M¯ ) of the quasinormal frequencies as functions of q, with the mass M¯ held xed. The panels correspond to scalar, vector, tensor, and spinor perturbations. The angular label and overtone number are indicated inside each panel. exponent satisfy rph = 3M, ¯ M¯ Ωph = 1 3 √3 , M¯ λt = 1 3 √3 σ  (148) For the scalar and electromagnetic elds the leading eikonal frequencies are ˆℓn = ℓ + 1 2 3 √3 − i n + 1 2 3 √3 σ + O(ℓ−1), ℓ → ∞, n xed, (149) while the Dirac eld has the corresponding form ˆκn = κ 3 √3 − i n + 1 2 3 √3 σ + O(κ−1), κ → ∞, n x
Figure 30. Attenuation rates − Im(M¯ ) as functions of q for the same congurations as
Figure 31. Internal diagnostics of the WKB calculation. The left panel displays the percentage order variation
Figure 32. Scalar time domain response for ℓ = 2 at xobs ∗ = 30. The left panel resolves the signed amplitude, and the right panel follows its absolute value on a logarithmic scale. The same pulse is used for every value of q, with M¯ held xed. Table XIII. Fundamental Maxwell frequencies for ℓ = 2 at xed M¯ . The columns follow Table XII, and the WKB values are those of Table VIII. q ˆTD ˆ(3) ∆WKB [%] ϵTD −06 0460781 − 0083573 i 0460505 − 0083604 i 0059 872 × 10−7 00 0457596 − 0095004 i 0457131 − 0095065 i 0100 539 × 10−7 06 0451681 − 0112920 i 0450737 − 0113073 i 0206…
Figure 33. Electromagnetic time domain response for
Figure 34. Time domain response of the homogeneous axial tensor sector for
Figure 35. Spinor time domain response for κ = 2 at xobs ∗ = 30, using U+. The signed amplitude and its absolute value follow from the same Gaussian prescribed in the bosonic evolutions. Table XVI. Temporal convergence at q = 0, evaluated on common samples in 40 ≤ τ ≤ 150. The ratio R and the relative estimate E0.05 are dened in Eq. (161). The observed order is p = log2 R. Perturbation R p E0.05 Scalar 4001548 2000558 164 × 10−5 Vector 4001451 2000523 160 × 10−5 Tensor 4001174 2000424 147 × 10−5 Spinor 4000987 2000356 952 × 10−6
Figure 36. Convergence of the temporal waveforms at
Figure 37. Temporal signals at q = 0 and their rank eight matrix pencil reconstructions. The t uses 90 ≤ τ ≤ 130; the vertical dotted line marks the start of the subsequent prediction, displayed through τ = 150. The four panels use the same angular labels and observation point as the preceding gures. XI. CONCLUSION We investigated the gravitational and optical properties of a static black hole in Palatini Kalb Ramond gravity with a vanishing cosmological constant. The analysis separated the conserved Iyer Wald mass M from the orbital mass M¯ = Mρ and used the latter to compare…

Figures reproduced from arXiv:2610.26692 by the authors.

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