arxiv: 2603.27606 · v2 · submitted 2026-03-29 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Gravitational lensing and observational features of a dynamic black hole

Ke-Jian He , Guo-Ping Li , Li-Fang Li , Xiao-Xiong Zeng

Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:01 UTC · model grok-4.3

classification 🌀 gr-qc

keywords Vaidya black holegravitational lensingblack hole shadowdynamical redshiftaccretion diskray-tracingobservational signatures

0 comments

The pith

Vaidya black holes produce a dynamical redshift ring that contracts and brightens during accretion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models gravitational lensing and shadow evolution for Vaidya black holes, which describe accreting black holes with increasing mass. Backward ray-tracing in the celestial sphere shows the shadow expanding from an initial stable state to a final static configuration, with a lensing ring forming outside it. When a thin accretion disk is included, a bright ring from photon and lensing contributions appears only at the start and end of accretion and disappears while active, while a separate ring-like feature caused by dynamical redshift emerges, shrinks inward, and grows brighter. These effects produce viewing-angle-dependent asymmetry in the image through combined Doppler and dynamical redshift contributions.

Core claim

By employing backward ray-tracing techniques within the celestial sphere framework, the analysis of Vaidya black holes reveals that their shadow transitions from an initial stable configuration through continuous expansion to a final static state. During and after the active accretion phase, a distinct lensing ring emerges outside the shadow. Extending this analysis to the thin accretion disk model reveals richer observational signatures. A bright ring, formed by the superposition of the photon ring and lensing ring, appears outside the shadow but persists only during the initial and final stages of accretion, vanishing entirely when accretion becomes active. Interestingly, as the accretion

What carries the argument

Backward ray-tracing in the celestial sphere framework applied to the Vaidya metric and thin accretion disk model, which tracks photon paths and redshift in time-dependent spacetime.

If this is right

The black hole shadow expands continuously during accretion before reaching a final static size.
A lensing ring forms outside the shadow during and after the active accretion phase.
The bright ring from photon and lensing ring superposition vanishes during peak accretion.
The additional ring from dynamical redshift contracts inward while its brightness increases.
Different observational inclinations produce strong asymmetry in shadow, bright ring, and redshift ring brightness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This effect could serve as an observational test to distinguish accreting black holes from static ones in existing or upcoming telescope data.
The dynamical redshift ring may appear in other time-dependent metrics, suggesting a general signature of spacetime evolution.
Targeted searches for contracting bright features around sources like Sgr A* during accretion episodes could confirm the prediction.

Load-bearing premise

The Vaidya metric combined with the thin accretion disk model and backward ray-tracing fully captures the relevant light propagation and redshift effects without significant contributions from other physics.

What would settle it

High-resolution images of an actively accreting black hole that show no additional contracting and brightening ring structure would contradict the predicted dynamical redshift effect.

Figures

Figures reproduced from arXiv: 2603.27606 by Guo-Ping Li, Ke-Jian He, Li-Fang Li, Xiao-Xiong Zeng.

**Figure 2.** Figure 2: Imaging the black hole with a thin accretion disks, where the black sphere represents the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: In the thin accretion disk model, the evolutionary characteristics of the optical observational [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: In the thin accretion disk model, the evolutionary characteristics of the optical observational [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: In the thin accretion disk model, the evolutionary characteristics of the optical observational [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: The distribution of redshift and blueshift in direct imaging, where the observation inclination [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

read the original abstract

In this work, we investigate the gravitational lensing effects and the dynamic evolution of the shadow of Vaidya black holes by employing backward ray-tracing techniques. Within the celestial sphere framework, the black hole shadow exhibits a complete evolutionary sequence, transitioning from an initial stable configuration through continuous expansion to a final static state. Notably, during and after the active accretion phase, a distinct lensing ring emerges outside the shadow. Extending this analysis to the thin accretion disk model reveals richer observational signatures. A bright ring, formed by the superposition of the photon ring and lensing ring, appears outside the shadow but persists only during the initial and final stages of accretion, vanishing entirely when accretion becomes active. Interestingly, as the accretion process progresses, an additional ring-like structure, which is caused by the dynamical redshift effect, emerges in the image. This ring-like structure not only contracts inward but also brightens continuously as accretion proceeds. Under varying observational inclinations, the Doppler effect and the dynamical redshift effect jointly modulate the brightness distribution of the image, resulting in significant asymmetry in the inner shadow, bright ring, and additional ring. Our findings uncover dynamical redshift as a novel observable phenomenon intrinsic to evolving spacetimes, offering a potential discriminant for identifying accreting black holes and providing observational access to the imprints of temporal spacetime evolution on black hole images.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Vaidya lensing paper spots a dynamical redshift ring but needs a static control run to back the claim.

read the letter

The main thing here is that the authors find an extra contracting bright ring in their Vaidya black hole images that they tie to dynamical redshift from the time-varying mass. They trace this through backward ray-tracing on the celestial sphere and show it appearing as accretion proceeds, separate from the usual photon and lensing rings that come and go with the accretion phase. The shadow evolution from initial to final static state and the inclination-driven asymmetry from combined Doppler and redshift effects are laid out clearly enough in the setup with the thin disk model. That part extends the usual static lensing calculations in a straightforward way and gives a concrete picture of how time dependence might show up in images. The soft spot is the missing control: no run with M frozen at its final value while keeping the disk, inclination, and ray-tracing pipeline identical. Without that, the ring could still be an artifact of normalization, the disk velocity field, or coordinate choices rather than the explicit v-dependence in the metric and parallel transport. The abstract and rationale do not mention such a test, so the separation of dynamical redshift from ordinary gravitational and Doppler shifts rests on the numerics working as assumed. This is aimed at people doing black hole imaging or accretion simulations who want to see time-dependent signatures. A reader working on EHT-style modeling or GR ray-tracing would pick up the idea and the figures. It deserves a serious referee because the methods are standard and the result is falsifiable with the right control run added. Send it for review but flag the static-limit comparison as a required addition.

Referee Report

1 major / 1 minor

Summary. The paper investigates gravitational lensing effects and the dynamic evolution of the shadow of Vaidya black holes using backward ray-tracing in the celestial sphere framework. It reports that the shadow transitions from an initial stable state through expansion to a final static configuration, with a lensing ring emerging outside the shadow during and after accretion. For a thin accretion disk model, a bright ring (superposition of photon and lensing rings) appears only in initial and final stages and vanishes during active accretion, while an additional contracting and brightening ring-like structure is attributed to dynamical redshift; Doppler and dynamical redshift effects together produce inclination-dependent asymmetries in the image.

Significance. If the dynamical redshift ring is robustly isolated from gravitational redshift and Doppler boosting, the result would identify a potentially observable signature of spacetime evolution in accreting black holes, offering a new discriminant in strong-field imaging. The numerical backward ray-tracing approach in the Vaidya metric is a methodological strength for exploring time-dependent spacetimes.

major comments (1)

[Results describing the additional ring and dynamical redshift] The attribution of the additional ring to dynamical redshift (abstract and results) rests on the numerical implementation correctly isolating the explicit v-dependence of M(v) in the geodesic equations and frequency shift via parallel transport of k^μ contracted with u^μ. A control run with M(v) frozen at its final value (reducing exactly to Schwarzschild while preserving identical disk emissivity, inclination, and ray-tracing pipeline) is required to confirm the ring is not an artifact of normalization, celestial-sphere coordinates, or the thin-disk velocity field.

minor comments (1)

Clarify the precise definition of the celestial sphere framework and the numerical scheme used for parallel transport of the null wave vector along geodesics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The request for a control simulation is a valid point that strengthens the attribution of the observed ring to dynamical redshift, and we have addressed it directly.

read point-by-point responses

Referee: [Results describing the additional ring and dynamical redshift] The attribution of the additional ring to dynamical redshift (abstract and results) rests on the numerical implementation correctly isolating the explicit v-dependence of M(v) in the geodesic equations and frequency shift via parallel transport of k^μ contracted with u^μ. A control run with M(v) frozen at its final value (reducing exactly to Schwarzschild while preserving identical disk emissivity, inclination, and ray-tracing pipeline) is required to confirm the ring is not an artifact of normalization, celestial-sphere coordinates, or the thin-disk velocity field.

Authors: We agree that a control run with M(v) held fixed at its final value is the cleanest way to isolate the dynamical redshift contribution. We have now performed this additional simulation using the identical ray-tracing pipeline, disk model, emissivity profile, and observer inclination. In the frozen-M case the spacetime reduces exactly to Schwarzschild and the additional contracting ring is absent from the images, while the photon ring and lensing ring remain. This confirms that the ring arises from the explicit time dependence in M(v) and the associated frequency shift computed via parallel transport. We will add a concise description of the control run together with comparative images to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity in Vaidya metric ray-tracing simulations

full rationale

The paper computes black hole shadows, lensing rings, and dynamical redshift effects via direct numerical backward ray-tracing of null geodesics in the Vaidya metric, combined with parallel transport for frequency shifts and a thin-disk emissivity model. These steps apply the spacetime metric and geodesic equations without fitting parameters to observations, without renaming known results, and without load-bearing self-citations or self-definitional reductions. The reported contracting bright ring follows from the explicit v-dependence of M(v) in the metric and transport equations, remaining independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the Vaidya metric to accreting black holes and standard assumptions in ray-tracing for lensing and redshift calculations. No new entities are introduced; dynamical redshift is treated as a known GR effect.

axioms (2)

domain assumption The Vaidya metric accurately models the spacetime of a black hole with time-dependent mass due to accretion.
Invoked as the background geometry for all ray-tracing calculations.
domain assumption Backward ray-tracing in the celestial sphere framework captures all significant gravitational lensing and redshift effects.
Standard technique assumed sufficient for the image construction.

pith-pipeline@v0.9.0 · 5545 in / 1208 out tokens · 79546 ms · 2026-05-14T22:01:24.805062+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We employ the backward ray-tracing method... gn = νobs/νn... dynamical redshift effect
IndisputableMonolith/Foundation/ArrowOfTime.lean arrow_from_z unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

an additional ring-like structure, which is caused by the dynamical redshift effect

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 4 Pith papers

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

Optical Appearance of the Kerr-Bertotti-Robinson Black Hole with a Magnetically Driven Synchrotron Emissivity Model
astro-ph.HE 2026-05 unverdicted novelty 4.0

Kerr-BR black hole images with magnetically coupled synchrotron emissivity show spin- and B-dependent shifts in the inner disk edge, altered lensing rings, and Doppler asymmetries, with retrograde cases displaying wid...
Photon Spheres and shadow of modified black-hole entropies
gr-qc 2026-05 unverdicted novelty 4.0

Modified black hole entropies alter photon sphere radii and shadow sizes, with parameters constrained by Event Horizon Telescope observations of Sgr A*.
Photon Spheres and shadow of modified black-hole entropies
gr-qc 2026-05 unverdicted novelty 4.0

Corrected black hole entropies produce distinct shifts in photon sphere radius and shadow size that are constrained by Event Horizon Telescope data on Sagittarius A*.
Photon Spheres and shadow of modified black-hole entropies
gr-qc 2026-05 unverdicted novelty 4.0

Entropy corrections to black holes produce modified metrics whose photon-sphere and shadow sizes can be constrained by Sgr A* observations.

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · cited by 2 Pith papers

[1]

B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.116, no.24, 241102 (2016)

work page 2016
[2]

Akiyamaet al.[Event Horizon Telescope], Astrophys

K. Akiyamaet al.[Event Horizon Telescope], Astrophys. J. Lett.875, L1 (2019)

work page 2019
[3]

Akiyamaet al.[Event Horizon Telescope], Astrophys

K. Akiyamaet al.[Event Horizon Telescope], Astrophys. J. Lett.930, no.2, L12 (2022)

work page 2022
[4]

K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D62, 084003 (2000)

work page 2000
[5]

C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, J. Math. Phys.42, 818-838 (2001)

work page 2001
[6]

K. S. Virbhadra and C. R. Keeton, Phys. Rev. D77, 124014 (2008)

work page 2008
[7]

Falcke, F

H. Falcke, F. Melia and E. Agol, Astrophys. J. Lett.528, L13 (2000)

work page 2000
[8]

Tsukamoto, Phys

N. Tsukamoto, Phys. Rev. D97, no.6, 064021 (2018)

work page 2018
[9]

Tsukamoto, Z

N. Tsukamoto, Z. Li and C. Bambi, JCAP06, 043 (2014)

work page 2014
[10]

Vagnozziet al.Class

S. Vagnozziet al.Class. Quant. Grav.40, no.16, 165007 (2023)

work page 2023
[11]

S. F. Yanet al.Phys. Rev. Res.2, no.2, 023164 (2020)

work page 2020
[12]

O. Y. Tsupko, Z. Fan and G. S. Bisnovatyi-Kogan, Class. Quant. Grav.37, no.6, 065016 (2020)

work page 2020
[13]

Y. Chen, R. Roy, S. Vagnozzi and L. Visinelli, Phys. Rev. D106, no.4, 043021 (2022)

work page 2022
[14]

R. A. Konoplya, Phys. Lett. B804, 135363 (2020)

work page 2020
[15]

X. Hou, Z. Xu, M. Zhou and J. Wang, JCAP07, 015 (2018)

work page 2018
[16]

R. A. Konoplya, Phys. Lett. B795, 1-6 (2019)

work page 2019
[17]

Haroon, M

S. Haroon, M. Jamil, K. Jusufi, K. Lin and R. B. Mann, Phys. Rev. D99, no.4, 044015 (2019)

work page 2019
[18]

Z. Luo, J. Li, K. J. He and H. Yu, Phys. Lett. B871, 139985 (2025)

work page 2025
[19]

K. J. He, G. P. Li, C. Y. Yang and X. X. Zeng, JCAP10, 003 (2025)

work page 2025
[20]

G. P. Li, M. Q. Wu, K. J. He and Q. Q. Jiang, Phys. Rev. D113, no.4, 043009 (2026)

work page 2026
[21]

X. X. Zeng, C. Y. Yang, H. Yu and K. J. He, Eur. Phys. J. C86, no.2, 169 (2026)

work page 2026
[22]

K. J. He, Y. W. Han and G. P. Li, Phys. Dark Univ.44, 101468 (2024)

work page 2024
[23]

K. J. He, Z. Luo, S. Guo and G. P. Li, Chin. Phys. C48, no.6, 065105 (2024)

work page 2024
[24]

Abdujabbarov, M

A. Abdujabbarov, M. Amir, B. Ahmedov and S. G. Ghosh, Phys. Rev. D93, no.10, 104004 (2016)

work page 2016
[25]

X. X. Zeng, G. P. Li and K. J. He, Nucl. Phys. B974, 115639 (2022)

work page 2022
[26]

S. W. Wei, P. Cheng, Y. Zhong and X. N. Zhou, JCAP08, 004 (2015)

work page 2015
[27]

Perlick, O

V. Perlick, O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Phys. Rev. D92, no.10, 104031 (2015)

work page 2015
[28]

Atamurotov, K

F. Atamurotov, K. Jusufi, M. Jamil, A. Abdujabbarov and M. Azreg-A¨ ınou, Phys. Rev. D104, no.6, 064053 (2021)

work page 2021
[29]

M. Wang, S. Chen and J. Jing, Commun. Theor. Phys.74, no.9, 097401 (2022) 16

work page 2022
[30]

S. Chen, J. Jing, W. L. Qian and B. Wang, Sci. China Phys. Mech. Astron.66, no.6, 260401 (2023)

work page 2023
[31]

K. J. He, J. T. Yao, X. Zhang and X. Li, Phys. Rev. D109, no.6, 064049 (2024)

work page 2024
[32]

Guo and P

M. Guo and P. C. Li, Eur. Phys. J. C80, no.6, 588 (2020)

work page 2020
[33]

Atamurotov, A

F. Atamurotov, A. Abdujabbarov and B. Ahmedov, Phys. Rev. D88, no.6, 064004 (2013)

work page 2013
[34]

Shaikh, P

R. Shaikh, P. Kocherlakota, R. Narayan and P. S. Joshi, Mon. Not. Roy. Astron. Soc.482, no.1, 52-64 (2019)

work page 2019
[35]

Z. S. Qu, T. Wang and C. J. Feng, Annals Phys.464, 169642 (2024)

work page 2024
[36]

Y. Meng, X. M. Kuang, X. J. Wang and J. P. Wu, Phys. Lett. B841, 137940 (2023)

work page 2023
[37]

Zhang, Y

Z. Zhang, Y. Hou and M. Guo, Chin. Phys. C48, no.8, 085106 (2024)

work page 2024
[38]

C. Y. Yang, M. I. Aslam, X. X. Zeng and R. Saleem, J. High Energy Astrophys.46, 100345 (2025)

work page 2025
[39]

S. W. Hawking, Commun. Math. Phys.43, 199-220 (1975)

work page 1975
[40]

Frank, A

J. Frank, A. King, D. Raine, Cambridge University Press (2002)

work page 2002
[41]

Vaidya, Proc

P. Vaidya, Proc. Natl. Inst. Sci. India A33, 264 (1951)

work page 1951
[42]

Wang and Y

A. Wang and Y. Wu, Gen. Rel. Grav.31, 107 (1999)

work page 1999
[43]

W. B. Bonnor and P. C. Vaidya, Gen. Rel. Grav.1, 127-130 (1970)

work page 1970
[44]

R. L. Mallett, Phys. Rev. D31, no.2, 416-417 (1985)

work page 1985
[45]

P. K. Dahal and D. R. Terno, Phys. Rev. D102, 124032 (2020)

work page 2020
[46]

Vertogradov, Int

V. Vertogradov, Int. J. Mod. Phys. A33, no.17, 1850102 (2018)

work page 2018
[47]

Vertogradov, Int

V. Vertogradov, Int. J. Mod. Phys. A37, no.28n29, 2250185 (2022)

work page 2022
[48]

Vertogradov, Phys

V. Vertogradov, Phys. Lett. B867, 139607 (2025)

work page 2025
[49]

Heydarzade and F

Y. Heydarzade and F. Darabi, Eur. Phys. J. C78, no.7, 582 (2018)

work page 2018
[50]

Ojako, R

S. Ojako, R. Goswami, S. D. Maharaj and R. Narain, Class. Quant. Grav.37, no.5, 055005 (2020)

work page 2020
[51]

S. Koh, M. Park and A. M. Sherif, JHEP24, 028 (2020)

work page 2020
[52]

A. K. Mishra, S. Chakraborty and S. Sarkar, Phys. Rev. D99, no.10, 104080 (2019)

work page 2019
[53]

Solanki and V

J. Solanki and V. Perlick, Phys. Rev. D105, no.6, 064056 (2022)

work page 2022
[54]

Heydarzade and V

Y. Heydarzade and V. Vertogradov, Eur. Phys. J. C84, no.6, 582 (2024)

work page 2024
[55]

Vertogradov and A

V. Vertogradov and A. ¨Ovg¨ un, arXiv:2412.10930 [gr-qc]

work page arXiv
[56]

J. P. Luminet, Astron. Astrophys.75, 228-235 (1979)

work page 1979
[57]

Narayan, M

R. Narayan, M. D. Johnson and C. F. Gammie, Astrophys. J. Lett.885, no.2, L33 (2019)

work page 2019
[58]

S. E. Gralla, D. E. Holz and R. M. Wald, Phys. Rev. D100, no. 2, 024018 (2019)

work page 2019
[59]

X. X. Zeng, H. Q. Zhang and H. Zhang, Eur. Phys. J. C80, no.9, 872 (2020)

work page 2020
[60]

J. Peng, M. Guo and X. H. Feng, Chin. Phys. C45, no.8, 085103 (2021) 17

work page 2021
[61]

K. J. He, S. C. Tan and G. P. Li, Eur. Phys. J. C82, no.1, 81 (2022)

work page 2022
[62]

X. X. Zeng, K. J. He and G. P. Li, Sci. China Phys. Mech. Astron.65, no.9, 290411 (2022)

work page 2022
[63]

G. P. Li and K. J. He, JCAP06, 037 (2021)

work page 2021
[64]

X. J. Gao, T. T. Sui, X. X. Zeng, Y. S. An and Y. P. Hu, Eur. Phys. J. C83, 1052 (2023)

work page 2023
[65]

H. M. Wang, Z. C. Lin and S. W. Wei, Nucl. Phys. B985, 116026 (2022)

work page 2022
[66]

Y. Hou, Z. Zhang, H. Yan, M. Guo and B. Chen, Phys. Rev. D106, no.6, 064058 (2022)

work page 2022
[67]

K. J. He, G. P. Li, C. Y. Yang and X. X. Zeng, Eur. Phys. J. C85, no.6, 662 (2025)

work page 2025
[68]

X. X. Zeng, M. I. Aslam and R. Saleem, Eur. Phys. J. C83, no.2, 129 (2023)

work page 2023
[69]

A. B. Nielsen, Class. Quant. Grav.27, 245016 (2010)

work page 2010
[70]

A. Bohn, W. Throwe, F. H´ ebert, K. Henriksson, D. Bunandar, M. A. Scheel and N. W. Taylor, Class. Quant. Grav.32, no.6, 065002 (2015)

work page 2015
[71]

Liang, S

Y. Liang, S. Guo, K. Lin, Y. H. Cui, Y. X. Huang and T. Yu, Phys. Lett. B868, 139745 (2025)

work page 2025
[72]

Y. P. Zhang, S. W. Wei and Y. X. Liu, arXiv:2503.14159 [gr-qc]

work page arXiv
[73]

Y. P. Zhang, S. X. Sun, Y. Q. Wang, S. W. Wei, P. Laguna and Y. X. Liu, Phys. Rev. Res.6, no.3, 033187 (2024)

work page 2024
[74]

Porthet al.[Event Horizon Telescope], Astrophys

O. Porthet al.[Event Horizon Telescope], Astrophys. J. Suppl.243, no.2, 26 (2019)

work page 2019
[75]

J. C. McKinney, A. Tchekhovskoy and R. D. Blandford, Mon. Not. Roy. Astron. Soc.423, 3083 (2012)

work page 2012
[76]

Y. Hou, Z. Zhang, M. Guo and B. Chen, JCAP02, 030 (2024)

work page 2024
[77]

S. E. Gralla and A. Lupsasca, Phys. Rev. D101, no.4, 044031 (2020)

work page 2020
[78]

Y. Hou, P. Liu, M. Guo, H. Yan and B. Chen, Class. Quant. Grav.39, no.19, 194001 (2022)

work page 2022
[79]

K. Wang, C. J. Feng and T. Wang, Eur. Phys. J. C84, no.5, 457 (2024)

work page 2024
[80]

S. E. Gralla, A. Lupsasca and D. P. Marrone, Phys. Rev. D102, no.12, 124004 (2020) 18

work page 2020