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Periodic orbits and their gravitational wave radiations around the Schwarzschild-MOG black hole

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that the MOG parameter alpha shifts periodic-orbit energies around a Schwarzschild-MOG black hole relative to Schwarzschild: lower for alpha>0, higher for -1<alpha<0.

desk verdict A routine periodic-orbit extension to a metric that is Reissner-Nordstrom in disguise for α>0; the headline energy ordering is confounded by the Lav convention, and the IBCO formula as printed does not work. read the letter →

arxiv 2501.04367 v2 pith:ARE4SGXN submitted 2025-01-08 gr-qc astro-ph.GA

Oreeda Shabbir , Mubasher Jamil , Mustapha Azreg-Aïnou This is my paper · ORCID
classification gr-qcastro-ph.GA MSC 83C5783C1083C3583D05 PACS 04.70.-s04.30.-w04.25.-g
keywords Schwarzschild-MOGblackholemodifiedgravityperiodicorbitseffectivepotentialgravitationalwavesextrememass-ratioinspiralsgeodesicszoom-whirl
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 studies bound orbits of a test particle around a Schwarzschild-MOG black hole, a modified-gravity spacetime with a single parameter alpha that controls the effective gravitational strength. Using the Hamiltonian formalism and a rational classification of periodic orbits by a triplet of integers, it computes the orbit label q = w + v/z and the energy required for each label. The central result is that this required energy shifts systematically with alpha: for alpha > 0 the periodic orbits are energetically cheaper than in Schwarzschild, and for -1 < alpha < 0 they are more expensive. The paper also computes the gravitational waveforms emitted by these orbits, arguing that the zoom-whirl structure leaves a distinct quiet-loud signature in the plus and cross polarizations that could be used to constrain the MOG parameter with future detectors.

What carries the argument

The argument is carried by the rational orbit label $q = w + v/z$, defined from the azimuthal advance $\Delta\phi$ between successive apastra by $q = \Delta\phi/(2\pi) - 1$ and computed by an integral over the radial turning points (Eq. 25), together with the MOG-modified effective potential $V_{\rm eff}(r) = (1 - 2M(1+\alpha)/r + M^2\alpha(1+\alpha)/r^2)(L^2/r^2 + 1)$. The label and the chosen angular momentum $L_{av}$ select a unique periodic orbit, so the energy for each $(z,w,v)$ can be read off; the same potential determines the ISCO, IBCO and photon-sphere radii. For the waveforms the paper adopts the kludge quadrupole formula $h_{ij} = 4\eta M/D_L (V_i V_j - m/r\, n_i n_j)$ and its circular-orbit projection, which yields $h_+ \propto \cos(2\phi+2\zeta)/r$ and $h_\times \propto \sin(2\phi+2\zeta)/r$.

What would settle it

Recompute the (z,w,v) periodic-orbit energies for Schwarzschild-MOG with an independent high-precision integrator and without fixing L = L_av, or compute the emitted radiation with a full black-hole perturbation scheme; if the energy ordering E(alpha>0) < E_Schwarzschild reverses for any orbit label, or if the waveform phases differ by more than a radian over one inspiral timescale, the central claim fails.

Watch

Extended reading notes

Core claim

For a test particle moving between the ISCO and IBCO of a Schwarzschild-MOG black hole, the paper finds that the energy E(z,w,v) needed to realize a given periodic orbit, labeled by the integers (z,w,v), depends monotonically on the MOG parameter alpha when angular momentum is fixed to the average L_av = (L_ISCO + L_IBCO)/2. For alpha > 0 the energy is lower than the corresponding Schwarzschild value, while for -1 < alpha < 0 it is higher. The same parameter increases the horizon, photon-sphere, ISCO and IBCO radii, so the whole allowed band of bound orbits moves outward as alpha grows. In the extremal case alpha = -1 the event and Cauchy horizons coincide and the ISCO radius diverges, so the periodic-orbit analysis is restricted to alpha in (-1, infinity).

Load-bearing premise

The gravitational waveforms are computed with a circular-orbit formula (Eqs. 28-29), but the periodic orbits studied here are strongly eccentric zoom-whirl trajectories, and the paper does not test whether that formula remains valid in this regime.

Editorial extensions

If this is right

  • If alpha > 0, an extreme-mass-ratio inspiral around a Schwarzschild-MOG black hole will pass through the periodic-orbit sequence with less energy per rational label than in Schwarzschild, shifting the inspiral rate and the emitted frequency evolution.
  • The monotonic energy shift lets the rational orbit label serve as a diagnostic for the MOG parameter: identifying the (z,w,v) template in a measured waveform would place bounds on alpha.
  • Because the ISCO, IBCO, and horizon radii all grow with alpha, the allowed band of bound orbits moves outward, so an inspiral in MOG will end with a larger final circular orbit before plunge for alpha > 0.
  • In the extremal case alpha = -1 the horizon structure degenerates and the ISCO radius diverges, so the paper's periodic-orbit analysis, which requires a finite band between ISCO and IBCO, only applies for alpha in (-1, infinity).

Reading between the lines

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

  • An immediate test of the energy ordering is to repeat the tables with L fixed at a different representative value (not L_av); if the monotonic direction is an artifact of the averaging convention, the central claim would weaken.
  • The kludge generator likely underestimates the distinguishability of MOG from Schwarzschild because the circular-orbit projection discards the strong-field multipole structure; a full perturbation calculation could reveal larger phase differences and make alpha easier to constrain with future space-based detectors.
  • The same Hamiltonian-plus-taxonomy pipeline could be applied to rotating Kerr-MOG or MOG with a cosmological constant to check whether the energy shift direction persists across the family of modified spacetimes.
  • If the shift is monotone and known as a function of alpha, then an observed EMRI waveform's orbit-label sequence could be inverted to estimate alpha while simultaneously fitting the central mass, though systematic waveform-model error would dominate the error budget.
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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 paper studies timelike geodesics in the Schwarzschild-MOG (STVG) black-hole spacetime, parametrized by the MOG parameter α. It derives the geodesic equations and effective potential via a Hamiltonian formalism, identifies the ISCO, IBCO, and photon-sphere radii, and then applies the Levin-Perez-Giz taxonomy to construct periodic orbits characterized by rational numbers q = w + v/z. The central claim is an energy-ordering statement: for α>0 the periodic-orbit energies are lower than in Schwarzschild, while for -1<α<0 they are higher. The paper also studies precessing orbits near periodic orbits and computes gravitational-wave polarizations using a kludge-type formula for an EMRI configuration. The main quantitative results are presented in Tables 1-2 and in the waveform plots of Sections 6.

Significance. If the central energy-ordering claim is correct, the paper would provide a clean qualitative signature of the MOG parameter in periodic-orbit energies and waveform morphology. The geodesic setup and the use of the rational-orbit taxonomy are standard, and the α=0 limit of the ISCO formula checks out. However, the headline claim is currently read off from tables that vary both α and L, the IBCO formula is problematic as printed for α>0, and the gravitational-wave section applies a circular-orbit projection to highly eccentric zoom-whirl orbits without validation. These are load-bearing issues: until they are resolved, the paper's main conclusions are not established. The paper does not ship code or machine-checked derivations, but the numerical tables are reproducible in principle from Eq. (25).

major comments (4)
  1. [Eqs. (19)-(20)] The IBCO formula is not usable as printed for the positive-α branch that the paper emphasizes. The expression for Z contains the square root of -32α^3 -69α^4 -42α^5 -5α^6, whose argument is negative for every α>0; hence Z is complex or undefined over the reals for exactly the α values used in Table 2 and in the positive-α figures. In addition, if the prefactor in the second term of Eq. (19) is interpreted as 4/∛2 rather than 4∛2, the α=0 limit does not reduce to rIBCO=4M. The authors should state the unambiguous real algebraic form used to compute LIBCO, verify the α=0 limit, and report values for the α>0 cases explicitly.
  2. [Section 4, Tables 1-2] The central energy-ordering claim is confounded by the simultaneous variation of L and α. Each row of Tables 1-2 uses a different angular momentum L=Lav(α) defined by Eq. (21), and q in Eq. (25) depends on E, L, and α. Thus the comparison 'for α>0 energies are lower, for -1<α<0 energies are higher' does not isolate the effect of the MOG parameter; it also moves L across the allowed interval. The statement after Tables 1-2 and in the abstract therefore needs either a fixed-L comparison, a physical prescription for choosing L, or a demonstration that the ordering is robust under different choices of representative angular momentum.
  3. [Table 1] Table 1 contains an internal inconsistency: the rows for α=-0.2 and α=-0.1 have different Lav entries but identical entries in all four energy columns. This indicates that at least one of these rows was not produced by solving Eq. (25) for the stated Lav, and it undermines confidence in the numerical data feeding the energy-ordering comparison. The table should be regenerated and checked.
  4. [Section 6, Eqs. (27)-(31)] The gravitational-wave results are not supported by the formulas used. Equations (28)-(29) are the leading-order circular-orbit polarization projections, yet they are evaluated on the highly eccentric, zoom-whirl trajectories shown in Figs. 8-13. For a general eccentric geodesic, the radiation contains multiple harmonics and the amplitude and phase structure differ from cos(2ϕ+2ζ)/r and sin(2ϕ+2ζ)/r. No justification, derivation, or numerical cross-check is given for applying this circular-orbit kludge to these orbits. The waveform plots should either be derived from a genuinely general kludge/quadrupole integral, or the section should be explicitly restricted to a regime where the approximation is controlled.
minor comments (5)
  1. [Eq. (21) and Section 4] The definition of Lav as the arithmetic mean of LISCO and LIBCO is an arbitrary convention. Since the paper uses this convention as the basis for all energy tables, its role should be stated more transparently and, ideally, accompanied by a sensitivity test.
  2. [Figure 4] The left panel caption says q is plotted versus E for several α, but the right panel's description 'with the energy kept fixed for the (1,1,0) orbit' is not precise enough; please specify which energy value is held fixed for each α.
  3. [Figure 5] The first panel of Fig. 5 has the energy label E=0.957888 but no (z,w,v) label or α value; the reader cannot identify which periodic orbit it represents.
  4. [Reference [35]] Reference [35] appears to be an unrelated CMS search paper, not a Taiji gravitational-wave detector reference; the citation should be corrected.
  5. [Throughout] There are typographical inconsistencies such as 'processing orbits' in the abstract and 'Precession' in the section heading; these should be harmonized.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: orbit energies are computed from the geodesic integral, not fitted or defined from the target claim.

full rationale

The paper's central result, the alpha-dependent ordering of periodic-orbit energies, is obtained by solving Eq. (25) for E at fixed rational q and fixed L = Lav(alpha). The target ordering is not an input to any equation: E is neither fitted to the Schwarzschild comparison nor defined in terms of it. The Lav convention, while arbitrary and potentially confounding because it changes L with alpha, is an analysis choice rather than a circular derivation, since Lav is computed from LISCO and LIBCO and does not encode the final energy values. The metric, effective potential, ISCO, and IBCO are external theory inputs, and the ISCO/IBCO conditions are standard geodesic conditions, not restatements of the periodic-orbit energies. The gravitational-wave section applies an externally referenced kludge formula (Eqs. 27-29) to the already-computed orbits; the waveform plots are consequences of the orbital kinematics, not independent predictions fitted to those same orbits. There are no load-bearing self-citations, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via a self-citation. The duplicated Table 1 row for alpha = -0.2 and -0.1 and the questionable IBCO expression (Eqs. 19-20) are correctness and internal-consistency concerns, not circularity. Accordingly, the derivation chain is self-contained with respect to circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the MOG metric (external theory input), the periodic orbit taxonomy, and the quadrupole kludge waveform approximation. The angular momentum averaging Lav is a hand-chosen convention. No new entities are introduced.

free parameters (2)
  • Lav (average angular momentum) = varies with alpha; e.g., 3.732055 for alpha = 0
    Defined as (LISCO + LIBCO) / 2 to fix a single angular momentum for each alpha; this is an arbitrary modeling choice, not derived from physics, and it determines the energy tables.
  • Inclination angle iota and latitudinal angle zeta = pi/4
    Set to pi/4 for simplicity in the waveform plots; arbitrary, not varied.
assumptions (4)
  • domain assumption The Schwarzschild-MOG metric (Eq. 1) is the correct vacuum solution of MOG/STVG and the parameter alpha is a free theory parameter.
    Taken from Moffat's STVG theory [10,17]; not derived in this paper. The entire analysis proceeds within this metric.
  • domain assumption The Levin-Perez-Giz rational periodic orbit taxonomy applies to this static, spherically symmetric spacetime.
    The paper assumes that any bound orbit can be labelled by (z,w,v) and that q = omega_phi / omega_r is rational for periodic orbits, following [28].
  • ad hoc to paper The kludge gravitational waveform formula (27) and the projection (28)-(29) are valid for computing GWs from test particles on these periodic orbits.
    The formula is borrowed from EMRI kludge waveforms [49-51] and applied here to eccentric zoom-whirl orbits without justification; it is a circular-orbit quadrupole approximation.
  • standard math The Hamiltonian normalization and separation constant setup (Eqs. (6)-(12)) are standard.
    The geodesic equations are derived using standard Hamiltonian mechanics in GR.

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Cite this review

Pith. "Pith review of Periodic orbits and their gravitational wave radiations around the Schwarzschild-MOG black hole." pith.science (2026). https://pith.science/paper/ARE4SGXN

@misc{pith2026250104367,
  author       = {Pith},
  title        = {Pith review of: Periodic orbits and their gravitational wave radiations around the Schwarzschild-MOG black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARE4SGXN}},
  note         = {Machine review of arXiv:2501.04367}
}
abstract

This article explores the motion of massive particles in the gravitational field of a modified gravity (MOG) black hole (BH), characterized by the parameter $\alpha$. Using the Hamiltonian formalism, the geodesic equations and the effective potential governing particle trajectories are derived. Key features, including the innermost stable circular orbit (ISCO) and the innermost bound circular orbit (IBCO), are analyzed, revealing their dependence on the particle's energy, angular momentum, and the MOG parameter. In the extremal case, where $\alpha=-1$, the event horizon merges with the Cauchy horizon, forming a distinctive BH configuration. Numerical methods are employed to compute periodic orbits in this spacetime, with a comparison drawn to the Schwarzschild BH. The findings indicate that for $\alpha>0$, periodic orbits around Schwarzschild-MOG BH exhibit lower energy requirements than those in Schwarzschild spacetime, whereas for $-1<\alpha<0$, the energy requirements are higher. Precessing orbits near periodic trajectories are also examined, offering insights into their complex dynamical behavior. Finally, the gravitational wave (GW) radiation from the periodic orbits of a test particle around the Schwarzschild-MOG BH is examined, generating intricate waveforms that provide insights into the gravitational structure of the system.

Figures

Figures reproduced from arXiv: 2501.04367 by the authors.

Figure 1
Figure 1. Behavior of f(r)−r curves for different α values and M = 1, illustrating the shift between the event and Cauchy horizons [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The characteristic radii, namely, the horizon, photon sphere, IBCO and ISCO position as [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Behavior of effective potential with angular momentum for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Variation of q as a function of energy and angular momentum. For the left plot, α = 0.1, 0.2, 0.3, 0.4, 0.5, with their corresponding Lav. The right plot shows the same α values, with the energy kept fixed for the (1, 1, 0) orbit [PITH_FULL_IMAGE:figures/full_fig_p011…
Figure 5
Figure 5. Figure 5: Zoom-whirl periodic orbits characterized by ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Zoom-whirl periodic orbits characterized by [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Precessing orbits with varying energy values near periodic orbits for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: The case α = 0.05, L = 3.8939047 = Lav and E = 0.9672876. The periodic orbit and the GW of the Schwarzschild-MOG BH for the triplet of integers (3, 1, 1). In the right panels, the vertical axis represents cos(2ϕ + 2ζ)/r and sin(2ϕ + 2ζ)/r, which are proportional to h+ …
Figure 9
Figure 9. Figure 9: The case α = −0.05, L = 3.5697415 = Lav and E = 0.9680222. The periodic orbit and the GW of the Schwarzschild-MOG BH for the triplet of integers (3, 1, 1). In the right panels, the vertical axis represents cos(2ϕ + 2ζ)/r and sin(2ϕ + 2ζ)/r, which are proportional to h+…
Figure 10
Figure 10. Figure 10: Upper Panel: Combination of the plots of [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: The case α = 0.01, L = 3.7644566 = Lav and E = 0.9672595. The periodic orbit and the GW of the Schwarzschild-MOG BH for the triplet of integers (4, 1, 1). In the right panels, the vertical axis represents cos(2ϕ + 2ζ)/r and sin(2ϕ + 2ζ)/r, which are proportional to h+…
Figure 12
Figure 12. Figure 12: The case α = −0.01, L = 3.6996268 = Lav and E = 0.9674087. The periodic orbit and the GW of the Schwarzschild-MOG BH for the triplet of integers (4, 1, 1). In the right panels, the vertical axis represents cos(2ϕ + 2ζ)/r and sin(2ϕ + 2ζ)/r, which are proportional to h…
Figure 13
Figure 13. Figure 13: Upper Panel: Combination of the plots of [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

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Forward citations

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Reviewed August 10, 2026 · model on record in the stance chip above.