Periodical orbits and waveforms with spontaneous Lorentz symmetry-breaking in Kalb-Ramond gravity

Diego Rubiera-Garcia; Ednaldo L. B. Junior; Francisco S. N. Lobo; Henrique A. Vieira; Jos\'e Tarciso S. S. Junior; Lu\'is F. Dias da Silva; Manuel E. Rodrigues

REVIEW 5 major objections 5 minor 6 cited by

Periodical orbits and waveforms with spontaneous Lorentz symmetry-breaking in Kalb-Ramond gravity

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Kalb-Ramond Lorentz-breaking parameter l shifts the phase, and slightly the amplitude, of gravitational waves from a body spiraling past a black hole.

desk verdict Clean geodesic and periodic-orbit analysis for the KR black hole, but the waveform claim rests on applying the GR quadrupole formula in proper time rather than detector time, so the phase shift is not yet an observable prediction. read the letter →

arxiv 2412.00769 v2 pith:JTR2B6RC submitted 2024-12-01 gr-qc astro-ph.HEhep-th

Ednaldo L. B. Junior , José Tarciso S. S. Junior , Francisco S. N. Lobo , Manuel E. Rodrigues , Diego Rubiera-Garcia , Luís F. Dias da Silva , Henrique A. Vieira This is my paper · ORCID

classification gr-qcastro-ph.HEhep-th PACS 04.30.-w04.70.-s11.30.Cp

keywords Kalb-RamondgravityLorentzsymmetrybreakingblackholegeodesicsperiodicorbitszoom-whirlextrememass-ratioinspiralgravitationalwaveformseffectivepotential

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

Kalb-Ramond gravity is a string-inspired extension of general relativity in which an antisymmetric tensor field takes on a background value, spontaneously breaking Lorentz symmetry and leaving a single dimensionless parameter l in a Schwarzschild-like black-hole metric. This paper derives the geodesic motion in that metric and asks whether l leaves an imprint on gravitational-wave signals. It shows that the parameter rescales the marginally bound orbit and the innermost stable circular orbit, shifts the energy and angular momentum of the zoom-whirl periodic orbits between them, and most importantly changes the phase of the two gravitational-wave polarizations emitted by an extreme mass-ratio inspiral, while changing the amplitude only slightly. If this is right, future space-based detectors could distinguish Kalb-Ramond black holes from Schwarzschild black holes by measuring the phase of these signals.

What carries the argument

The load-bearing object is the effective potential for timelike geodesics, $V_{\rm eff}(r)=\left(\frac{1}{1-l}-\frac{2M}{r}\right)\left(1+\frac{L^2}{r^2}\right)$, whose extrema fix the marginally bound orbit and the ISCO, and whose zeroes of $\dot r^2=E^2-V_{\rm eff}$ give the turning points that define bound orbits. The periodic-orbit selection rule is the rational-number condition $q\equiv \Delta\phi/2\pi-1=w+v/z$, where $z$ counts the zoom leaves, $w$ the whirl turns around periastron, and $v$ the vertex label of the successive apastron. Solving this condition numerically for the energy and angular momentum of each taxonomy $(z,w,v)$ produces the orbit families; the same $\dot r^2$ and turning-point data enter the quadrupole polarization formulas $h_+=-(2\beta M^2/(D_L r))(1+\cos^2\iota)\cos(2\phi+2\zeta)$ and $h_\times=-(4\beta M^2/(D_L r))\cos\iota\sin(2\phi+2\zeta)$, which convert the periodic orbits into waveforms. The l-dependence of the critical orbits enters through the combination $(1-l)$, which is why the waveform effect shows up as a phase shift rather than a change in orbit taxonomy.

What would settle it

Compute gravitational-wave emission from the full Kalb-Ramond field equations at linear order and compare the resulting phase evolution with the Fig. 10 waveforms; if the symmetry-breaking field contributes to the source or changes the wave speed, the predicted phase shift would change. Observationally, matched filtering of a long EMRI signal with Kalb-Ramond versus Schwarzschild templates would settle which phase evolution a detector actually sees.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the spontaneous Lorentz symmetry-breaking parameter l is not just a theoretical curiosity but produces a waveform signature. Using the effective potential for timelike geodesics, the authors obtain closed-form expressions for the marginally bound orbit, $r_{\rm MBO}=4M(1-l)$, and the innermost stable circular orbit, $r_{\rm ISCO}=6M(1-l)$, so the critical radii and angular momenta shrink uniformly with l, while the ISCO energy grows. They then classify orbits by the rational number $q=w+v/z$ from the $(z,w,v)$ taxonomy, solve the turning-point equation numerically for orbits between MBO and ISCO, and find that for fixed energy $l<0$ orbits carry higher angular momentum and higher eccentricity than $l>0$ orbits. Feeding those orbits through the quadrupole Kludge formula produces $h_+$ and $h_\times$ waveforms for an EMRI system with a Sgr A*-like massive black hole; compared with Schwarzschild, the waveforms for $l=-0.01285$ and $l=0.011746$ differ mainly in phase, with a smaller amplitude change. The paper interprets this phase difference as a promising observational signature for Lorentz symmetry-breaking.

Load-bearing premise

The predicted phase difference relies on the assumption that the symmetry-breaking field changes the orbit but leaves the emitted gravitational waves otherwise identical to general relativity's, and that the time coordinate used to plot the waves matches what a detector measures.

Editorial extensions

If this is right

In Kalb-Ramond gravity, the marginally bound orbit and the ISCO shrink uniformly as $(1-l)$, so the radii and angular momenta of these special orbits are rescaled compared to Schwarzschild while the ISCO energy rises with l.
For the same periodic-orbit taxonomy $(z,w,v)$ at fixed energy, negative l yields higher angular momentum and more eccentric orbits, whereas positive l yields lower angular momentum and less eccentric orbits; at fixed angular momentum the trend in energy is reversed.
The gravitational-wave polarizations $h_+$ and $h_\times$ from an EMRI in the KR spacetime carry a clear phase shift, and a mild amplitude change, relative to Schwarzschild; this phase imprint is uniform across the taxonomy, so it does not depend on the particular $(z,w,v)$ chosen.
A space-based detector sensitive to EMRI waveforms could in principle use this phase shift to distinguish a KR central black hole from a Schwarzschild one within the observationally allowed range of l, offering a new observational channel for Lorentz symmetry-breaking beyond precession and shadow measurements.

Reading between the lines

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

The paper does not stress that the $(1-l)$ scaling of all critical radii makes l degenerate with a mass rescaling in any single orbital measurement; the waveform phase is the observable that could break this degeneracy.
If the actual radiative sector of the theory differs from the Kludge quadrupole limit, the predicted phase shift would be modified; deriving that sector is the natural next step before building search templates.
Because the photon sphere also rescales with $(1-l)$, the same parameter connects these waveform predictions to shadow observations; combining an EMRI phase measurement with a shadow-radius measurement would give a consistency test of the theory.

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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

What you should know: this is a straightforward, conventional application of the Levin–Perez-Giz periodic orbit taxonomy and the Kludge waveform method to the spherically symmetric Kalb-Ramond black hole metric. The geodesic part is the solid core. The MBO and ISCO formulas—r_MBO = 4M(1−l), r_ISCO = 6M(1−l), E_ISCO ∝ 1/sqrt(1−l)—are simple consequences of the rescaled metric and correctly reduce to Schwarzschild at l=0. The periodic orbit plots (Figs. 6–9) are internally consistent and show how l shifts momentum and energy for fixed taxonomy. That part is worth having.

The genuinely new thing is the first application of this taxonomy and waveform pipeline to this spacetime. No prior work in the reference list computes these orbits or waveforms for the KR metric, so it is a new application rather than a restatement.

The soft spots are in the waveform section. The most concrete problem: Fig. 10 plots h+ and h× against the inspiraling body's proper time τ, while the quadrupole formula (22) is derived with coordinate-time derivatives and a phase argument in coordinate time t. For this metric, dt/dτ = E/A(r) with A(r) = 1/(1−l) − 2M/r, so the mapping is neither constant nor l-independent. The phase difference between l=0 and l≠0 therefore mixes the physical orbital effect with a time-reparametrization effect. LISA records strain versus detector time, so Fig. 10 is not yet a measurable prediction. This is fixable—output t(τ) and present h(t), or state that the plot is schematic—but as it stands the central claim of an observable phase shift is unsupported.

Relatedly, the paper applies the GR quadrupole formula to KR gravity without deriving or citing the radiative sector of the theory. That is a common kludge approximation, but it deserves an explicit caveat. Two minor points: Eq. (3) has the epsilon assignment reversed, and the ISCO energy formula in Eq. (15) is written ambiguously; the stated monotonicity implies the denominator form, which is what they intend.

The conclusion overreaches when it calls the deviations 'substantial.' Within the constrained interval (2), the effect is small, and with the proper-time issue, 'promising avenue' is too strong.

Bottom line: the geodesic and periodic-orbit analysis is solid and worth refereeing; the waveform claim needs a coordinate-time conversion and a statement about the radiative sector. A serious referee should engage with it. For my own work, I would cite the geodesic formulas if needed, but not the waveform claim.

Send it to review, conditionally.

Referee Report

5 major / 5 minor

Summary. The manuscript studies timelike geodesics in a static spherically symmetric Kalb-Ramond black-hole spacetime characterized by the Lorentz-violating parameter l. It derives the effective potential, computes the marginally bound orbit and innermost stable circular orbit, classifies periodic orbits using the (z, w, v) taxonomy, and constructs h+ and h× waveforms for an extreme mass ratio inspiral using the adiabatic approximation and a quadrupole kludge formula. The central claim is that l produces observable phase deviations in the gravitational waveforms relative to Schwarzschild, offering a possible LISA signature.

Significance. The geodesic, MBO, ISCO, and periodic-orbit analysis is internally consistent, reduces to Schwarzschild at l = 0, and uses the parameter interval from the external constraint of Ref. [102] without fitting predictions back to the input. The analytic MBO/ISCO expressions and the numerical classification of periodic orbits are useful additions. The observational claim, however, rests on a waveform computation that is not expressed in detector time and that assumes the general-relativistic radiative sector; these two issues are load-bearing for the claim that LISA could distinguish KR black holes from Schwarzschild black holes and must be resolved before the phase shifts in Fig. 10 can be considered observable.

major comments (5)

[V, Eq. (22), Fig. 10] The waveforms in Fig. 10 are plotted as functions of the smaller body's proper time, with no conversion to the coordinate time of the source frame or to detector time. In the quadrupole formula (22) and in the kludge method, the velocities v_i and the phase argument are defined with respect to coordinate time t. For the KR metric, dt/dτ = E/A(r) with A(r) = 1/(1 - l) - 2M/r, so the mapping between τ and t depends on l and on r. Consequently, the phase difference between l = 0 and l ≠ 0 shown in Fig. 10 mixes a genuine orbital effect with an l-dependent time reparametrization. The authors should recompute h+ and h× as functions of coordinate time, or provide the explicit conversion and replot, before drawing conclusions about observable phase shifts.
[V, Eq. (22)] Equation (22) is the general-relativistic quadrupole formula. The paper applies it to gravitational radiation from orbits in KR gravity without deriving or citing the radiative sector of the KR theory. If the KR field modifies wave generation, the polarization content, or wave propagation, the predicted phase and amplitude differences would not be the signals reaching LISA. This assumption is load-bearing for the central claim and should be stated and justified explicitly, or the observational claim should be correspondingly softened.
[V] Although the adiabatic approximation is invoked in Sec. V, the computation uses fixed E = 0.96 and a fixed periodic orbit; no radiation-reaction evolution of E and L is implemented. The resulting h+ and h× are periodic-orbit waveform snippets rather than full inspiral waveforms, so the comparison with EMRI signals in LISA is incomplete. The text should state this limitation clearly when presenting Fig. 10.
[II, Eq. (3)] The assignment ε = -1 for light-like and ε = 0 for time-like is reversed; for time-like geodesics one needs g_μν \dot{x}^μ \dot{x}^ν = -1. The subsequent equations (7) and (10) effectively use the timelike value ε = -1, so the error is typographical in nature, but it should be corrected to avoid confusion.
[III, Eq. (15)] The printed expression E_ISCO = 2√2 / 3√(1 - l) is ambiguous; the surrounding text and Fig. 1 indicate that E_ISCO increases with l, which requires the denominator form 2√2 / (3√(1 - l)). Please add parentheses to remove the ambiguity.

minor comments (5)

[V] The stated values ι = 4/π and ζ = 4/π are unusual; these are likely intended to be π/4, and the text should be corrected.
[IV] The text contains a typo: 'Schwarzsichild' should be 'Schwarzschild'.
[I] The Introduction contains a typo: 'obervations' should be 'observations'; the affiliation line also contains the typo 'Brazill'.
[V, Fig. 10] The horizontal axis of Fig. 10 is not labeled; the text says the waveforms are plotted against the eigentime of the lowest-mass object, but the figure should indicate the quantity and units explicitly.
[V] The term 'eigentime' should be replaced by 'proper time' for clarity, and the text should specify whether the plotted time is in units of the black-hole mass M.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: orbital and waveform results are computed from the externally sourced KR metric (1) and the observationally constrained l-interval (2), with l never fitted to the predicted outputs.

full rationale

This paper's derivation chain is self-contained given two external inputs: the KR line element (1), adopted from the independent solution paper [97] (Yang, Chen, Duan, Zhao), and the observationally constrained interval (2), taken from the authors' previous study [102] of S2-star periastron precession around Sgr A*. The interval (2) is a fit to external astrometric data, not a fit to the periodic orbits or waveforms computed here, so it is independent support rather than a load-bearing self-citation: the same orbit/waveform computation would go through for any l. From the metric the paper derives the effective potential (10), MBO/ISCO relations (13)-(15), bound-orbit regions (18), the periodic-orbit condition q = Δφ/(2π) − 1 (20)-(21), and the Kludge polarizations (24)-(25) by direct numerical integration of the geodesic equations. The parameter l is never adjusted to reproduce an orbit or a waveform; for each l, L or E is solved from the geodesic condition at fixed taxonomy and fixed E or L. The resulting l-dependence of the waveforms is therefore an output of the input metric, and the Schwarzschild limit l → 0 is recovered throughout (r_ISCO = 6M, E_ISCO = 2√2/3), providing an external check. The self-citations [101], [102] are contextual or provide that input range; no uniqueness theorem is invoked to forbid alternative metrics. Two limitations are flagged and weighed: Sec. V explicitly acknowledges truncation beyond quadrupole order ("we have excluded the multipole contribution beyond the quadratic order"), and the KR radiative sector is nowhere derived, with Eq. (22) cited from the GR Kludge literature; Fig. 10 is plotted against the inspiraling body's proper time rather than detector coordinate time. These are correctness/observability risks, not reductions of a prediction to its inputs by construction, so they do not constitute circularity under the hard rules.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the KR metric from [97], the constrained parameter interval from [102], and the assumption that the GR quadrupole formula describes radiation in this modified theory. No new particle or field is introduced; the free parameters are the theory parameter l and the hand-picked orbital references E=0.96 and L=3.7 used for the illustrations.

free parameters (3)

l (Kalb-Ramond Lorentz-violating parameter) = constrained to [-0.185022, 0.060938] by S2 star precession in [102]
Controls the metric; values selected via alpha in Eq. (11). Not fitted in this paper.
E_ref = 0.96 = 0.96
Hand-chosen reference energy used in Figs. 4, 6, 7 and 10 because it admits bound orbits for the sampled l values; the claimed l effects are demonstrated at this value.
L_ref = 3.7 = 3.7
Hand-chosen reference angular momentum used in Figs. 5, 8 and 9; together with E_ref selects illustrative periodic orbits.

assumptions (5)

domain assumption The metric (1) is a valid static, spherically symmetric vacuum solution of non-minimally coupled Einstein-Kalb-Ramond gravity.
The paper takes this solution from Yang et al. [97] and performs all calculations in it; if the solution is not physical, all results fail.
ad hoc to paper The general-relativistic quadrupole formula (Eq. 22) correctly gives the gravitational waveform emitted by an inspiral in KR gravity.
Used without deriving the radiative sector of KR gravity; this is the principal unverified premise for the waveform claim.
domain assumption The adiabatic approximation holds, so the inspiraling body follows geodesics with nearly constant E and L over many orbital periods.
Stated in Sec. V and supported by prior work [42-44], but not quantified for this metric.
standard math The effective-potential turning-point analysis defines MBO and ISCO via Eqs. (12)-(14).
Standard condition for circular and marginal orbits in static spherically symmetric spacetimes.
standard math Time-like geodesics satisfy g_mu_nu xdot^mu xdot^nu = -1; Eq. (3) mislabels epsilon.
The effective potential (10) is derived with epsilon = -1 for time-like motion, not the value 0 printed in Eq. (3).

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Cite this review

Pith. "Pith review of Periodical orbits and waveforms with spontaneous Lorentz symmetry-breaking in Kalb-Ramond gravity." pith.science (2026). https://pith.science/paper/JTR2B6RC

@misc{pith2026241200769,
  author       = {Pith},
  title        = {Pith review of: Periodical orbits and waveforms with spontaneous Lorentz symmetry-breaking in Kalb-Ramond gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JTR2B6RC}},
  note         = {Machine review of arXiv:2412.00769}
}

abstract

In this paper, we study time-like geodesics around a spherically symmetric black hole in Kalb-Ramond (KR) gravity, characterized by the parameter $l$, which induces spontaneous Lorentz symmetry breaking. The geodesic equations and effective potential are derived to investigate the influence of $l$. We calculate the marginally bound orbits and innermost stable circular orbits, analyzing the parameter's impact. Periodic orbits are computed numerically and classified within the standard taxonomy, revealing significant effects of $l$ on their momentum and energy. Additionally, we explore an extreme mass ratio inspiral system under the adiabatic approximation to derive gravitational waveforms emitted by an object orbiting a supermassive black hole in KR gravity. These waveforms reflect the distinctive characteristics of periodic orbits and highlight the influence of $l$. With advancements in gravitational wave detection, these results offer insights into black holes influenced by Lorentz symmetry-breaking fields.

Figures

Figures reproduced from arXiv: 2412.00769 by the authors.

**Figure 2.** Representation of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Representation of the allowed region for energy and momentum of the marginally bound orbits. In Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Graphical representation of ˙r 2 depending on r with E = 0.96 fixed and different values of l, with L varying for ϵ = 0, ϵ = 0.25, ϵ = 0.45, ϵ = 0.65, ϵ = 0.85, ϵ = 1. Bound orbits only exist when ˙r 2 = 0 has at least two roots. Periodic orbits are defined as those th…

**Figure 5.** Figure 5: Graphical representation of ˙r 2 depending on r with a fixed values of L = 3.7 and different values of l, with E varying for η = 0, η = 0.2, η = 0.4, η = 0.6, η = 0.8, η = 1 according to Eq.(17). Bound orbits only exist when ˙r 2 = 0 has at least two roots. ever, the d…

**Figure 6.** Figure 6: Periodic orbits for different values of L and (z, w, v) with E = 0.96 and α = 0.7 (l = −0.01285) in (11). Here x and y have units of meter [m]. the black hole (x, y, z) such that these adapted coordinates are given, as described in [104], by eX = [cos ζ, − sin ζ, 0] ,…

**Figure 7.** Figure 7: Periodic orbits for different values of L and (z, w, v) with E = 0.96 and α = 0.8 (l = 0.011746) in (11). Here x and y have units of meter [m]. wave is valid when the radiation reaction takes place over a time scale much longer than the orbital period [106], which occu…

**Figure 8.** Figure 8: Periodic orbits for different values of E and (z, w, v) with L = 3.7 and α = 0.7 (l = −0.01285) in Eq.(11). Here x and y have units of meter [m]. of the wave is largest when the orbiting object is rotating in the periastron, and as it moves away towards the apoastron …

**Figure 9.** Figure 9: Periodic orbits for different values of E and (z, w, v) with L = 3.7 and α = 0.8 (l = 0.011746) in (11). Here x and y have units of meter [m]. VI. SUMMARY AND CONCLUSION In this work, we have analyzed a static, spherically symmetric solution expressed in the form of a…

**Figure 10.** Figure 10: Graphical representation of the polarization modes [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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