Pith. sign in

REVIEW 7 minor 1 cited by

Motion of spinning particles around a quantum-corrected black hole without Cauchy horizons

T0 review · 0 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Spin, not the quantum parameter, controls where particles can orbit this quantum-corrected black hole.

desk verdict Routine but clean MPD application to a third quantum-corrected black hole; the ISCO analysis survives the u^r/P^r concern, and the paper merits a serious referee. read the letter →

arxiv 2509.07682 v1 pith:TEMIC4BF submitted 2025-09-09 gr-qc

Jiawei Chen , Jinsong Yang This is my paper · ORCID
classification gr-qc
keywords SpinningparticlemotionQuantum-correctedblackholeEffectivequantumgravityPole-dipoleequationsSpinsupplementaryconditionpotentialInnermoststablecircularorbitBoundorbits
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 a specific quantum-corrected black hole spacetime—one without an inner (Cauchy) horizon—and asks how a small spinning test particle orbits it. It solves the standard pole-dipole equations for a spinning body and builds an effective radial potential. The central finding is an asymmetry: the quantum parameter $\zeta$ lowers the potential and weakly pushes the innermost stable circular orbit outward, while the particle's spin $S$ changes the potential's magnitude far more strongly. For small spin, bound trajectories look almost like Schwarzschild, but the same initial conditions around two other quantum-corrected black holes produce visibly different trajectories. If correct, this offers a concrete route to distinguishing quantum-corrected black hole models through the motion of spinning particles.

What carries the argument

The central object is the pole-dipole system of equations for a spinning body—the standard relativistic equations that replace geodesic motion once spin couples to spacetime curvature—closed by the spin supplementary condition $S^{ab}P_b=0$ that fixes the particle's center of mass. The argument is carried by the effective potential $V_{\rm eff}$, defined from the squared radial momentum $(P_r)^2=XE^2+YE+Z$ as the positive root $V_{\rm eff}=(-Y+\sqrt{Y^2-4XZ})/(2X)$; circular orbits, their stability, and the ISCO are read off from $V_{\rm eff}=E$, $dV_{\rm eff}/dr=0$, and $d^2V_{\rm eff}/dr^2=0$. The quantum parameter $\zeta$ enters through the metric functions $g_{tt}$ and $g_{rr}$, and the paper restricts it to $\zeta/M<2(\pi/2)^{3/2}\approx 3.94$.

What would settle it

At fixed $L=4.5$ and $S=0.5$, evaluate $V_{\rm eff}$ from Eq. (2.27) at $r=6$ (with $M=1$) for $\zeta=0,1,2,3,3.9$; the paper's claim requires $V_{\rm eff}$ to decrease monotonically as $\zeta$ grows. A numerical evaluation that does not show this ordering would refute the central claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that in the quantum-corrected black hole without a Cauchy horizon—the model it labels BH-III—the effective potential of a spinning test particle decreases as the quantum parameter $\zeta$ grows, while the spin parameter $S$ has a markedly stronger effect on the potential's size. As a consequence, the innermost stable circular orbit (ISCO) radius, its specific angular momentum, and its specific energy all grow weakly with $\zeta$ and more strongly with $S$. Bound orbits with small spin ($|S|=0.1$ in the examples) are almost indistinguishable from Schwarzschild orbits, whereas under the paper's chosen initial conditions the trajectories around BH-I and BH-II separate clearly from BH-III; this separation is the paper's proposed way of telling the models apart.

Load-bearing premise

The results assume one particular convention for attaching the particle's center of mass to its spin (one of several possible supplementary conditions), and a different convention could change the effective potential, the ISCO quantities, and the trajectories.

Editorial extensions

If this is right

  • At fixed spin, increasing $\zeta$ lowers the effective potential while raising the ISCO radius, angular momentum, and energy, so the quantum correction shows up mainly as a slight outward shift of the last stable orbit.
  • At fixed $\zeta$, changing $S$ moves the ISCO quantities much more, so spin-curvature coupling dominates the orbital structure in this model.
  • For small spin, bound orbits in BH-III are effectively Schwarzschild-like; distinguishing this model from Schwarzschild would require high spin or carefully chosen initial conditions.
  • Under chosen initial conditions ($E=0.984$, $L=4.5$, $\zeta=3$), bound orbits around BH-I, BH-II, and BH-III separate enough to identify which model is being observed.
  • The timelike condition restricts which $(S,\zeta)$ combinations allow a physical particle to sit at the ISCO; outside the allowed region the particle's would-be 4-velocity is spacelike.

Reading between the lines

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

  • Beyond the paper, a natural check is to repeat the calculation under a different spin supplementary condition; the paper itself notes the condition is not unique, and the sign or strength of the $\zeta$-dependence could change.
  • Because the paper restricts to equatorial orbits, inclined or precessing orbits—where the spin-curvature torque acts out of the plane—might make even low-spin particles reveal the quantum correction more clearly.
  • The same effective-potential machinery could be used to build extreme-mass-ratio inspiral waveforms; the weak $\zeta$-dependence might accumulate into a measurable phase shift over many orbits even if single orbits look Schwarzschild-like.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 7 minor

Summary. This paper studies the motion of spinning test particles in the covariant quantum-corrected Schwarzschild-like black hole without Cauchy horizons (BH-III), using the Mathisson-Papapetrou-Dixon (MPD) equations with the Tulczyjew spin supplementary condition. The authors derive the conserved four-momentum components and an effective potential for radial motion, then use it to obtain circular orbits, the ISCO, and a timelike-condition constraint at the ISCO. They also integrate the radial/azimuthal equations to produce bound-orbit trajectories for small spins and compare them with trajectories in two other covariant quantum-corrected black hole models. The main quantitative results are that the effective potential, ISCO radius, ISCO angular momentum, and ISCO energy all increase weakly with the quantum parameter ζ and more strongly with the spin parameter S, and that small-spin bound trajectories around BH-III are practically indistinguishable from Schwarzschild but differ visibly from BH-I and BH-II for the chosen initial data.

Significance. If correct, the paper extends the existing studies of spinning particles in covariant effective quantum gravity to the third solution (BH-III) and provides concrete, falsifiable predictions for distinguishing the three models through ISCO data and precessing bound orbits. The derivation follows the standard MPD route: no parameters are fitted to the target results; the quantum parameter is inherited from the metric of Ref. [29], while S, E, and L are initial conditions. The paper also honestly notes the non-uniqueness of the spin supplementary condition in Sec. II.B. I specifically checked the apparent tension between Eq. (2.24) and the use of (P^r)^2 as an effective potential. For the static, spherically symmetric metric (2.2), the spin-curvature term in the numerator of u^r is proportional to S^{tφ}, and S^{tφ} is itself proportional to P^r; hence P^r=0 implies u^r=0 provided the scalar prefactor does not vanish. The central worry therefore does not land. The remaining issues are mostly matters of derivation, notation, and reproducibility.

minor comments (7)
  1. [Sec. III.C, Eq. (3.3)] The velocity-momentum relation is asserted as following from Eqs. (2.4)-(2.7) but is not derived, and its index structure is not transparent. Since the numerical timelike-condition check uses Eqs. (2.24)-(2.25) rather than Eq. (3.3), please either derive Eq. (3.3) in the manuscript's notation or remove it, and state explicitly which expression is actually used to produce Fig. 4.
  2. [Sec. II.C, Eq. (2.26) and Fig. 1] The effective-potential coefficients are written in terms of J, while the text and figure captions fix L. Since L is defined as J-S, please state explicitly that J = L+S is used in all numerical calculations; without this, the curves cannot be reproduced and the comparison across different S values is ambiguous.
  3. [Sec. II.C, before Eq. (2.26)] The statement that 'the radial velocity u^r is proportional to the radial momentum P^r' is too strong; Eq. (2.24) shows that u^r is proportional to P^r only through a metric- and spin-dependent scalar factor. The effective-potential argument still works because the zero sets coincide, but the wording should be corrected.
  4. [Sec. IV, Fig. 6] The BH-I and BH-II trajectories are computed from formulas that are not reproduced in this manuscript, making the comparison non-self-contained. Please include the relevant metric functions or give explicit equations in an appendix so that the comparison can be verified.
  5. [Sec. II.B and Sec. V] The paper acknowledges the non-uniqueness of the spin supplementary condition, but all conclusions in Secs. III and IV are established only under the Tulczyjew SSC. Please add a sentence in Sec. V stating that the qualitative trends are not shown to be SSC-independent.
  6. [Sec. III.B] The text refers to 'Fig. 3.2'; this should be 'Fig. 3'. Please also ensure the description of the sub-panels matches the figure layout.
  7. [Throughout] There are several typographical and formatting issues: '4-monmentum' in Sec. II.B should be '4-momentum'; the DOI for Ref. [29] appears malformed; and the Riemann index order in expressions such as R^φ_{t μν}S^{μν} should be defined once to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper applies the MPD equations to an externally given quantum-corrected spacetime and does not fit any parameter to its target results.

full rationale

The derivation is self-contained against external inputs. The starting point is the BH-III metric in Eq. (2.2), taken from the prior solution paper [29], together with the standard MPD equations (2.4)-(2.5) and the Tulczyjew SSC (2.6). From these, Eqs. (2.19)-(2.21) give the 4-momentum components and Eq. (2.27) defines the effective potential; the circular-orbit and ISCO results in Sec. III and the trajectories in Sec. IV are computed directly from these equations. No parameter in this chain is fitted to the target outputs: the quantum parameter zeta is an input of the spacetime, while S, E, and L are chosen initial conditions, and the reported r_ISCO, L_ISCO, E_ISCO, and bound-orbit plots are functions of these inputs, not quantities imposed by construction. Refs. [28-30] share an author with this paper, but they supply the spacetime background as prior parameter-free derivations and do not contain the spinning-particle predictions made here. The possible inconsistency between the P^r-based effective-potential conditions and the actual radial velocity u^r in Eq. (2.24) is a physical-correctness concern, not circularity: it does not make any output equivalent to an input by definition.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claims are parametric in the quantum parameter and the spin, both chosen by hand within ranges inherited from the model and figures. Energy and angular momentum are fixed initial conditions for the plotted trajectories, not fitted parameters. All other ingredients, such as the MPD equations, conserved quantities, effective potential, and timelike condition, are standard or cited from prior work. No new entities are introduced; BH-III is inherited from [29].

free parameters (2)
  • Quantum parameter = varied over [0, 3.9]
    Input parameter of the BH-III metric inherited from [29]; not fitted here, but every central result is a parametric function of it.
  • Spin parameter S = varied over [-1, 1] in figures
    Test-particle spin-to-mass ratio; chosen by hand for each calculation, and the paper's central claim is that S dominates the orbit changes.
assumptions (6)
  • domain assumption The metric (2.1)-(2.2) with n=0 is a covariant quantum-corrected black hole solution without Cauchy horizons, as derived in [29].
    The paper uses this metric as its spacetime and does not re-derive the covariance conditions or the no-Cauchy-horizon property.
  • domain assumption The MPD equations (2.4)-(2.5) under the pole-dipole approximation describe the motion of a spinning test particle.
    Standard formalism adopted in Sec. II.B; limits the analysis to test particles with negligible back-reaction.
  • domain assumption The Tulczyjew spin supplementary condition (2.6) is a valid and sufficient supplementary condition.
    The paper notes other SSCs exist; results may depend on this choice.
  • standard math The Killing vectors give conserved quantities through Eq. (2.16).
    Standard conserved-charge construction for MPD, invoked in Eqs. (2.17)-(2.18).
  • domain assumption The relation (3.3) between four-velocity and four-momentum, cited to [61], is correct in the adopted conventions.
    Used to build the timelike condition; not derived in the text, and the displayed index structure is unclear.
  • domain assumption Setting u^t=1 is a legitimate parameterization of the worldline.
    Adopted before Eq. (2.22); changes the affine parameter but not the orbit geometry, and the timelike condition is imposed afterward in Eq. (3.4).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Motion of spinning particles around a quantum-corrected black hole without Cauchy horizons." pith.science (2026). https://pith.science/paper/TEMIC4BF

@misc{pith2026250907682,
  author       = {Pith},
  title        = {Pith review of: Motion of spinning particles around a quantum-corrected black hole without Cauchy horizons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TEMIC4BF}},
  note         = {Machine review of arXiv:2509.07682}
}
abstract

In this paper, we investigate the motion of spinning particles around a covariant quantum-corrected black hole without a Cauchy horizon within the framework of effective quantum gravity, and examine the influence of quantum gravitational effects on the motion of these spinning particles. First, we employ the Mathisson-Papapetrou-Dixon equations to derive the 4-momentum and 4-velocity of spinning particles, and introduce the effective potential for radial motion using the components of the 4-momentum. We find that an increase in the quantum parameter $\zeta$ leads to a decrease in the effective potential, while the spin $S$ significantly affects the magnitude of the effective potential. Then, through the effective potential, we investigate the properties of circular orbits and the innermost stable circular orbit, and discuss the timelike condition that spinning particles must satisfy when moving around the black hole. Finally, we study the trajectories of spinning particles on bound orbits around the quantum-corrected black hole and compare them with those around other covariant quantum-corrected black holes. The results show that the trajectories of spinning particles in this quantum-corrected black hole model are weakly influenced by $\zeta$, making them almost indistinguishable from those in the Schwarzschild black hole, but they can be distinguished from other covariant quantum-corrected models under certain initial conditions. These results contribute to our understanding of black hole properties under quantum corrections.

Figures

Figures reproduced from arXiv: 2509.07682 by the authors.

Figure 1
Figure 1. FIG. 1: Variation of the e [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Behavior of the circular orbit radius [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Variation of ISCO physical quantities ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Constraints on [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Trajectories of spinning particles with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Trajectories of spinning particles with [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Evaporation and fate of covariant quantum black holes

    gr-qc 2026-07 conditional novelty 5.0 of 10

    For a covariant LQG black-hole metric, Hawking mass-loss rates depend on emitted-particle spin and deviate from Schwarzschild rates only for sub-Planckian masses.

Reference graph

Works this paper leans on

66 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [61]

    Rakhimova, F

    G. Rakhimova, F. Atamurotov, N. Juraeva, A. Abdujabbarov, and G. Mustafa, Spinning particle motion around charged de- coupled hairy black hole. Phys. Dark Univ.47, 101721 (2025). https://doi.org/10.1016/j.dark.2024.101721

    arXiv 2025

  2. [29]

    Zhang, J

    C. Zhang, J. Lewandowski, Y . Ma, and J. Yang, Black holes and covariance in effective quantum gravity. Phys. Rev. D 111, L081504 (2025). https://doi.org/10.1103/PhysRevD.111. L081504. arXiv:2407.10168

    arXiv 2025

  3. [1]

    Penrose, Gravitational Collapse and Space-time Singular- ities

    R. Penrose, Gravitational Collapse and Space-time Singular- ities. Phys. Rev. Lett.14, 57 (1965). https://doi.org/10.1103/ PhysRevLett.14.57

    work page 1965

  4. [2]

    Hawking and R

    S.W. Hawking and R. Penrose, The Singularities of gravita- tional collapse and cosmology. Proc. Roy. Soc. Lond. A314, 529 (1970). https://doi.org/10.1098/rspa.1970.0021

    arXiv 1970

  5. [3]

    Surya, The causal set approach to quantum grav- ity

    S. Surya, The causal set approach to quantum grav- ity. Living Rev. Rel.22, 5 (2019). https://doi.org/10.1007/ s41114-019-0023-1. arXiv:1903.11544

    arXiv 2019

  6. [4]

    Rovelli, Zakopane lectures on loop gravity

    C. Rovelli, Zakopane lectures on loop gravity. PoS QGQGS2011, 003 (2011). https://doi.org/10.22323/1.140

    work page doi:10.22323/1.140 2011

  7. [5]

    5: Trajectories of spinning particles withE=0.976 andL=4.5 in the equatorial plane around BH-III for different values of spinSand quantum parameterζ

    arXiv:1102.3660 8 S=-0.1 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 x y ζ=0 (a)S=−0.1 andζ=0 S=0 -20 -10 0 10 20 -20 -10 0 10 20 x y ζ=0 (b)S=0 andζ=0 S=0.1 -20 -10 0 10 20 -20 -10 0 10 20 x y ζ=0 (c)S=0.1 andζ=0 S=-0.1 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 x y ζ=1 (d)S=−0.1 andζ=1 S=0 -20 -10 0 10 20 -20 -10 0 10 20 x y ζ=1 (e)S=0 andζ=1 S=0.1...

    arXiv

  8. [6]

    Rovelli, Loop quantum gravity

    C. Rovelli, Loop quantum gravity. Living Rev. Rel.1, 1 (1998). https://doi.org/10.12942/lrr-1998-1. arXiv:gr-qc/9710008

    arXiv 1998

Show all 66 references
  1. [7]

    Ashtekar and M

    A. Ashtekar and M. Bojowald, Quantum geometry and the Schwarzschild singularity. Class. Quant. Grav.23, 391 (2006). https://doi.org/10.1088/0264-9381/23/2/008. arXiv:gr- qc/0509075

    2006

  2. [8]

    Ashtekar and J

    A. Ashtekar and J. Lewandowski, Background indepen- dent quantum gravity: A Status report. Class. Quant. Grav. 21, R53 (2004). https://doi.org/10.1088/0264-9381/21/15/R01. arXiv:gr-qc/0404018

    2004 arXiv

  3. [9]

    Ashtekar, Introduction to loop quantum gravity and cosmol- ogy

    A. Ashtekar, Introduction to loop quantum gravity and cosmol- ogy. Lect. Notes Phys.863, 31 (2013). https://doi.org/10.1007/ 978-3-642-33036-0 2. arXiv:1201.4598

    2013 arXiv

  4. [10]

    M. Han, W. Huang, and Y . Ma, Fundamental structure of loop quantum gravity. Int. J. Mod. Phys. D16, 1397 (2007). https: //doi.org/10.1142/S0218271807010894. arXiv:gr-qc/0509064

    2007 arXiv

  5. [11]

    Modesto, Semiclassical loop quantum black hole

    L. Modesto, Semiclassical loop quantum black hole. Int. J. Theor. Phys.49, 1649 (2010). https://doi.org/10.1007/ s10773-010-0346-x. arXiv:0811.2196

    2010 arXiv

  6. [12]

    Perez, Black Holes in Loop Quantum Gravity

    A. Perez, Black Holes in Loop Quantum Gravity. Rept. Prog. Phys.80, 126901 (2017). https://doi.org/10.1088/1361-6633/ aa7e14. arXiv:1703.09149

    2017 arXiv

  7. [13]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, Quantum Transfigu- ration of Kruskal Black Holes. Phys. Rev. Lett.121, 241301 (2018). https://doi.org/10.1103/PhysRevLett.121.241301. arXiv:1806.00648

    2018 arXiv

  8. [14]

    Bodendorfer, F.M

    N. Bodendorfer, F.M. Mele, and J. M ¨unch, Effective Quantum Extended Spacetime of Polymer Schwarzschild Black Hole. Class. Quant. Grav.36, 195015 (2019). https://doi.org/10.1088/ 1361-6382/ab3f16. arXiv:1902.04542

    2019 arXiv

  9. [15]

    Kelly, R

    J.G. Kelly, R. Santacruz, and E. Wilson-Ewing, Effective loop quantum gravity framework for vacuum spherically symmetric spacetimes. Phys. Rev. D102, 106024 (2020). https://doi.org/ 10.1103/PhysRevD.102.106024. arXiv:2006.09302

    2020 arXiv

  10. [16]

    Gan, N.O

    W.-C. Gan, N.O. Santos, F.-W. Shu, and A. Wang, Properties of the spherically symmetric polymer black holes. Phys. Rev. D102, 124030 (2020). https://doi.org/10.1103/PhysRevD.102. 124030. arXiv:2008.09664

    2020 arXiv

  11. [17]

    Sartini and M

    F. Sartini and M. Geiller, Quantum dynamics of the black hole interior in loop quantum cosmology. Phys. Rev. D 103, 066014 (2021). https://doi.org/10.1103/PhysRevD.103. 066014. arXiv:2010.07056

    2021 arXiv

  12. [18]

    S. Song, H. Li, Y . Ma, and C. Zhang, Entropy of black holes with arbitrary shapes in loop quantum gravity. Sci. China Phys. Mech. Astron.64, 120411 (2021). https://doi.org/10. 1007/s11433-021-1770-3. arXiv:2002.08869

    2021 arXiv

  13. [19]

    Zhang, Y

    C. Zhang, Y . Ma, S. Song, and X. Zhang, Loop quantum Schwarzschild interior and black hole remnant. Phys. Rev. D102, 041502 (2020). https://doi.org/10.1103/PhysRevD.102. 041502. arXiv:2006.08313

    2020 arXiv

  14. [20]

    Zhang, Y

    C. Zhang, Y . Ma, S. Song, and X. Zhang, Loop quantum de- parametrized Schwarzschild interior and discrete black hole mass. Phys. Rev. D105, 024069 (2022). https://doi.org/10. 1103/PhysRevD.105.024069. arXiv:2107.10579

    2022 arXiv

  15. [21]

    Lewandowski, Y

    J. Lewandowski, Y . Ma, J. Yang, and C. Zhang, Quantum Oppenheimer-Snyder and Swiss Cheese Models. Phys. Rev. Lett.130, 101501 (2023). https://doi.org/10.1103/PhysRevLett. 130.101501. arXiv:2210.02253

    2023 arXiv

  16. [22]

    Bojowald, S

    M. Bojowald, S. Brahma, and J.D. Reyes, Covariance in models of loop quantum gravity: Spherical symmetry. Phys. Rev. D92, 045043 (2015). https://doi.org/10.1103/PhysRevD.92.045043. arXiv:1507.00329

    2015 arXiv

  17. [23]

    Bojowald and S

    M. Bojowald and S. Brahma, Covariance in models of loop quantum gravity: Gowdy systems. Phys. Rev. D92, 065002 (2015). https://doi.org/10.1103/PhysRevD.92.065002. arXiv:1507.00679

    2015 arXiv

  18. [24]

    Ben Achour and S

    J. Ben Achour and S. Brahma, Covariance in self dual in- homogeneous models of effective quantum geometry: Spher- ical symmetry and Gowdy systems. Phys. Rev. D97, 126003 (2018). https://doi.org/10.1103/PhysRevD.97.126003. arXiv:1712.03677

    2018 arXiv

  19. [25]

    Bojowald, No-go result for covariance in models of loop quantum gravity

    M. Bojowald, No-go result for covariance in models of loop quantum gravity. Phys. Rev. D102, 046006 (2020). https://doi. org/10.1103/PhysRevD.102.046006. arXiv:2007.16066

    2020 arXiv

  20. [26]

    Gambini, J

    R. Gambini, J. Olmedo, and J. Pullin, Towards a quantum no- tion of covariance in spherically symmetric loop quantum grav- ity. Phys. Rev. D105, 026017 (2022). https://doi.org/10.1103/ PhysRevD.105.026017. arXiv:2201.01616

    2022 arXiv

  21. [27]

    Han and H

    M. Han and H. Liu, Covariant ¯µ-scheme effective dynam- ics, mimetic gravity, and nonsingular black holes: Applica- tions to spherically symmetric quantum gravity. Phys. Rev. D109, 084033 (2024). https://doi.org/10.1103/PhysRevD.109. 084033. arXiv:2212.04605

    2024 arXiv

  22. [28]

    Li and J.-P

    S. Li and J.-P. Wu, Gravitational waves with generalized holon- omy corrections. Eur. Phys. J. C84, 629 (2024). https://doi.org/ 10.1140/epjc/s10052-024-13010-2. arXiv:2309.05535

    2024 arXiv

  23. [30]

    Zhang, J

    C. Zhang, J. Lewandowski, Y . Ma, and J. Yang, Black holes and covariance in effective quantum gravity: A solution without Cauchy horizons. Phys. Rev. D112, 044054 (2025). https://doi. org/10.1103/d6ks-d576. arXiv:2412.02487

    2025

  24. [31]

    J. Yang, C. Zhang, and Y . Ma, Covariant effective spacetimes of spherically symmetric electro-vacuum with a cosmological constant. arXiv:2503.15157

  25. [32]

    Konoplya and O.S

    R.A. Konoplya and O.S. Stashko, Probing the effective quan- tum gravity via quasinormal modes and shadows of black holes. Phys. Rev. D111, 104055 (2025). https://doi.org/10. 1103/PhysRevD.111.104055. arXiv:2408.02578

    2025 arXiv

  26. [33]

    W. Liu, D. Wu, and J. Wang, Light rings and shadows of static black holes in effective quantum gravity. Phys. Lett. B858, 139052 (2024). https://doi.org/10.1016/j.physletb.2024. 139052. arXiv:2408.05569

    2024 arXiv

  27. [34]

    Liu, M.-Y

    H. Liu, M.-Y . Lai, X.-Y . Pan, H. Huang, and D.-C. Zou, Gravi- tational lensing effect of black holes in effective quantum grav- ity. Phys. Rev. D110, 104039 (2024). https://doi.org/10.1103/ PhysRevD.110.104039. arXiv:2408.11603

    2024 arXiv

  28. [35]

    L.-G. Zhu, G. Fu, S. Li, D. Zhang, and J.-P. Wu, Quasinor- mal modes of a charged loop quantum black hole. Phys. Rev. D111, 104008 (2025). https://doi.org/10.1103/PhysRevD.111. 104008. arXiv:2410.00543

    2025 arXiv

  29. [36]

    Y . Wang, A. Vachher, Q. Wu, T. Zhu, and S.G. Ghosh, Strong gravitational lensing by static black holes in effective quantum gravity. Eur. Phys. J. C85, 302 (2025). https://doi.org/10.1140/ epjc/s10052-025-13970-z. arXiv:2410.12382

    2025 arXiv

  30. [37]

    Z. Ban, J. Chen, and J. Yang, Shadows of rotating black holes in effective quantum gravity. Eur. Phys. J. C85, 878 (2025). https://doi.org/10.1140/epjc/s10052-025-14614-y. arXiv:2411.09374

    2025 arXiv

  31. [38]

    J. Lin, X. Zhang, and M. Bravo-Gaete, Mass inflation and strong cosmic censorship conjecture in the covariant quantum black hole. Phys. Rev. D111, 106025 (2025). https://doi.org/ 10.1103/n7jv-crs9. arXiv:2412.01448

    2025 arXiv

  32. [39]

    Shu and J.-H

    Y .-H. Shu and J.-H. Huang, Circular orbits and thin accre- tion disk around a quantum corrected black hole. Phys. Lett. B864, 139411 (2025). https://doi.org/10.1016/j.physletb.2025. 11 139411. arXiv:2412.05670

    2025

  33. [40]

    W. Liu, D. Wu, and J. Wang, Light rings and shad- ows of static black holes in effective quantum gravity II: A new solution without Cauchy horizons. Phys. Lett. B 868, 139742 (2025). https://doi.org/10.1016/j.physletb.2025. 139742. arXiv:2412.18083

    2025

  34. [41]

    Bojowald, E.I

    M. Bojowald, E.I. Duque, and D. Hartmann, Covariant Lemaˆıtre-Tolman-Bondi collapse in models of loop quantum gravity. Phys. Rev. D111, 064002 (2025). https://doi.org/10. 1103/PhysRevD.111.064002. arXiv:2412.18054

    2025 arXiv

  35. [42]

    Konoplya and O.S

    R.A. Konoplya and O.S. Stashko, Transition from regular black holes to wormholes in covariant effective quantum gravity: Scattering, quasinormal modes, and Hawking radiation. Phys. Rev. D111, 084031 (2025). https://doi.org/10.1103/PhysRevD. 111.084031. arXiv:2502.05689

    2025 arXiv

  36. [43]

    Chen and J

    J. Chen and J. Yang, Shadows and optical appearance of quantum-corrected black holes illuminated by static thin accre- tions. Eur. Phys. J. C85, 512 (2025). https://doi.org/10.1140/ epjc/s10052-025-14230-w. arXiv:2503.06215

    2025 arXiv

  37. [44]

    L ¨utf¨uo˘glu, Long-lived quasinormal modes around regu- lar black holes and wormholes in Covariant Effective Quan- tum Gravity

    B.C. L ¨utf¨uo˘glu, Long-lived quasinormal modes around regu- lar black holes and wormholes in Covariant Effective Quan- tum Gravity. JCAP06, 057 (2025). https://doi.org/10.1088/ 1475-7516/2025/06/057. arXiv:2504.09323

    2025 arXiv

  38. [45]

    Chen and J

    J. Chen and J. Yang, Periodic orbits and gravitational wave- forms in quantum-corrected black hole spacetimes. Eur. Phys. J. C85, 726 (2025). https://doi.org/10.1140/epjc/ s10052-025-14457-7. arXiv:2505.02660

    2025 arXiv

  39. [46]

    Al-Badawi, F

    A. Al-Badawi, F. Ahmed, and ˙I. Sakallı, Effective quan- tum gravity black hole with cloud of strings surrounded by quintessence field. Nucl. Phys. B1017, 116961 (2025). https: //doi.org/10.1016/j.nuclphysb.2025.116961

    2025

  40. [47]

    Zhang and Z

    C. Zhang and Z. Cao, Covariant dynamics from static spheri- cally symmetric geometries. arXiv:2506.09540

  41. [48]

    Sekhmani, H

    Y . Sekhmani, H. Ali, S.G. Ghosh, and K. Boshkayev, Rotat- ing charged nonsingular black holes in loop quantum gravity and their observational imprints from EHT. JHEAp49, 100425 (2026). https://doi.org/10.1016/j.jheap.2025.100425

    2026

  42. [49]

    B.P. Abbottet al.(LIGO Scientific Collaboration and Virgo Collaboration), Observation of Gravitational Waves from a Binary Black Hole Merger, inCentennial of General Relativity: A Celebration, edited by C.A.Z. Vasconcellos (World Scientific, Singapore, 2017). https://doi.org/1...

    2017

  43. [50]

    Akiyamaet al.(Event Horizon Telescope Collabora- tion), First M87 Event Horizon Telescope Results

    K. Akiyamaet al.(Event Horizon Telescope Collabora- tion), First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett. 875, L1 (2019). https://doi.org/10.3847/2041-8213/ab0ec7. arXiv:1906.11238

    2019 arXiv

  44. [51]

    Hughes, Gravitational waves from extreme mass ratio inspirals: Challenges in mapping the space-time of massive, compact objects

    S.A. Hughes, Gravitational waves from extreme mass ratio inspirals: Challenges in mapping the space-time of massive, compact objects. Class. Quant. Grav.18, 4067 (2001). https: //doi.org/10.1088/0264-9381/18/19/314. arXiv:gr-qc/0008058

    2001 arXiv

  45. [52]

    Jefremov, O.Y

    P.I. Jefremov, O.Y . Tsupko, and G.S. Bisnovatyi-Kogan, Innermost stable circular orbits of spinning test particles in Schwarzschild and Kerr space-times. Phys. Rev. D91, 124030 (2015). https://doi.org/10.1103/PhysRevD.91.124030. arXiv:1503.07060

    2015 arXiv

  46. [53]

    Mathisson, Neue mechanik materieller systemes

    M. Mathisson, Neue mechanik materieller systemes. Acta Phys. Polon.6, 163 (1937)

    1937

  47. [54]

    Papapetrou, Spinning test particles in general relativity

    A. Papapetrou, Spinning test particles in general relativity. 1. Proc. Roy. Soc. Lond. A209, 248 (1951). https://doi.org/10. 1098/rspa.1951.0200

    1951

  48. [55]

    Dixon, Dynamics of extended bodies in general relativity

    W.G. Dixon, Dynamics of extended bodies in general relativity. I. Momentum and angular momentum. Proc. Roy. Soc. Lond. A 314, 499 (1970). https://doi.org/10.1098/rspa.1970.0020

    1970

  49. [56]

    Dixon, Dynamics of extended bodies in general relativity

    W.G. Dixon, Dynamics of extended bodies in general relativity. II. Moments of the charge-current vector. Proc. Roy. Soc. Lond. A319, 509 (1970). https://doi.org/10.1098/rspa.1970.0191

    1970

  50. [57]

    Dixon, Dynamics of extended bodies in general relativity III

    W.G. Dixon, Dynamics of extended bodies in general relativity III. Equations of motion. Phil. Trans. Roy. Soc. Lond. A277, 59 (1974). https://doi.org/10.1098/rsta.1974.0046

    1974

  51. [58]

    Zhang, S.-W

    Y .-P. Zhang, S.-W. Wei, W.-D. Guo, T.-T. Sui, and Y .- X. Liu, Innermost stable circular orbit of spinning particle in charged spinning black hole background. Phys. Rev. D97, 084056 (2018). https://doi.org/10.1103/PhysRevD.97.084056. arXiv:1711.09361

    2018 arXiv

  52. [59]

    Toshmatov and D

    B. Toshmatov and D. Malafarina, Spinning test particles in the γspacetime. Phys. Rev. D100, 104052 (2019). https://doi.org/ 10.1103/PhysRevD.100.104052. arXiv:1910.11565

    2019 arXiv

  53. [60]

    Zhang, Y .-B

    Y .-P. Zhang, Y .-B. Zeng, Y .-Q. Wang, S.-W. Wei, and Y .-X. Liu, Equatorial orbits of spinning test particles in rotating boson stars. Eur. Phys. J. C82, 809 (2022). https://doi.org/10.1140/ epjc/s10052-022-10743-w. arXiv:2201.01498

    2022 arXiv

  54. [62]

    Q. Tan, W. Deng, S. Long, and J. Jing, Motion of spin- ning particles around black hole in a dark matter halo. JCAP 05, 044 (2025). https://doi.org/10.1088/1475-7516/2025/05/

    2025 doi

  55. [63]

    Skoup ´y and V

    V . Skoup ´y and V . Witzany, Analytic Solution for the Mo- tion of Spinning Particles in Kerr Spacetime. Phys. Rev. Lett. 134, 171401 (2025). https://doi.org/10.1103/PhysRevLett.134. 171401. arXiv:2411.16855

    2025 arXiv

  56. [64]

    Alimova, F

    A. Alimova, F. Atamurotov, A. Abdujabbarov, G. Mustafa, and P. Channuie, Impact of quantum-corrected parameter on spin- ning particle motion around a black hole. Eur. Phys. J. C85, 646 (2025). https://doi.org/10.1140/epjc/s10052-025-14385-6

    2025 doi

  57. [65]

    Y . Du, Y . Liu, and X. Zhang, Spinning particle dynamics and the innermost stable circular orbit in covariant loop quan- tum gravity. JCAP05, 045 (2025). https://doi.org/10.1088/ 1475-7516/2025/05/045. arXiv:2411.13316

    2025 arXiv

  58. [66]

    Umarov, F

    D. Umarov, F. Atamurotov, S.G. Ghosh, A. Abdujabbarov, and G. Mustafa, Dynamics of spinning particles around static black holes in effective quantum gravity. Eur. Phys. J. C85, 800 (2025). https://doi.org/10.1140/epjc/s10052-025-14541-y

    2025 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.