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arxiv: 2601.10806 · v2 · submitted 2026-01-15 · 🌀 gr-qc · astro-ph.HE· hep-th

Shadow signatures and energy accumulation in Lorentzian-Euclidean black holes

Emmanuele Battista , Salvatore Capozziello , Che-Yu Chen This is my paper

Pith reviewed 2026-05-16 13:30 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords black hole shadowsLorentzian-Euclidean spacetimesignature shiftevent horizonnull geodesicsenergy accumulationobservational signatures
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The pith

Lorentzian-Euclidean black holes produce excess intensity inside their shadow boundary from a signature shift at the event horizon.

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

The paper studies shadows cast by a black hole spacetime that changes from Lorentzian to Euclidean signature exactly at the event horizon. This change keeps the geometry geodesically complete and stops causal paths from reaching the central singularity. The altered paths of light rays create a measurable excess brightness in the region inside the shadow edge. A reader would care because this brightness difference could appear in actual telescope images and serve as a test for whether real black holes follow this modified causal structure. The analysis also tracks how the horizon steadily collects photons and energy yet responds to that collection differently than light rings in other proposed compact objects.

Core claim

The Lorentzian-Euclidean black hole features a signature transition at the horizon that precludes causal geodesics from reaching r=0 while changing the propagation of null geodesics, which produces an excess intensity in the inner shadow region relative to the Schwarzschild geometry. This excess supplies a potential observational signature of horizon-scale modifications. The horizon surface continuously accumulates photons and energy, yet the resulting backreaction differs from the response of stable light rings in various exotic compact objects.

What carries the argument

The signature shift at the event horizon, which alters null geodesic behavior to trap light rays and generate stable excess intensity in the inner shadow without introducing new instabilities.

If this is right

  • The inner shadow region would display higher intensity than the corresponding region around a Schwarzschild black hole.
  • This intensity excess could serve as a direct probe for horizon-scale changes in black hole geometry.
  • The horizon accumulates energy from trapped photons, but its backreaction avoids the instabilities associated with stable light rings.

Where Pith is reading between the lines

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

  • The distinct backreaction might allow these objects to remain stable under continued photon accumulation where other models develop instabilities.
  • Accretion flow models around such black holes could produce different observable spectra or variability patterns than standard cases.
  • Gravitational lensing calculations for distant sources could be extended to include the modified null geodesics near the horizon.

Load-bearing premise

The signature shift at the event horizon is a physically realized feature that alters null geodesic behavior in a way that produces stable excess intensity without instabilities or extra regularization.

What would settle it

High-resolution imaging of a black hole shadow that shows no excess intensity inside the boundary would rule out the predicted signature.

Figures

Figures reproduced from arXiv: 2601.10806 by Che-Yu Chen, Emmanuele Battista, Salvatore Capozziello.

Figure 1
Figure 1. Figure 1: FIG. 1. Photon trajectories in the equatorial plane of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Shadow images of the Schwarzschild solution (left) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Observed intensity [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Observed intensity [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

The Lorentzian-Euclidean black hole has been recently introduced as a geodesically complete spacetime featuring a signature shift at the event horizon where causal geodesics are precluded from reaching the central $r=0$ singularity. In this paper, we investigate the shadows produced by this geometry to identify deviations from the standard Schwarzschild solution. Our analysis reveals an excess intensity in the inner shadow region that points to a potential observational signature of the novel behavior of light rays propagating near the event horizon. This excess could be a probe for horizon-scale modifications of black hole geometries. Furthermore, although the horizon surface of the Lorentzian-Euclidean black hole continuously accumulates photons and energy, we show that its backreaction response differs from that of stable light rings found in various exotic compact objects.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies shadows and photon accumulation in the recently introduced Lorentzian-Euclidean black hole, a geodesically complete spacetime with a metric signature transition at the event horizon. It reports an excess intensity in the inner shadow region relative to Schwarzschild, interpreted as an observational signature of novel null geodesic behavior near the horizon, and claims that continuous photon/energy accumulation at the horizon produces a backreaction response distinct from that of stable light rings in exotic compact objects.

Significance. If the reported excess intensity and differentiated backreaction survive a proper treatment of the signature interface, the work would supply a concrete, potentially falsifiable prediction for horizon-scale deviations from general relativity that could be tested with high-resolution shadow imaging. The absence of free parameters and the focus on a geometrically motivated signature change are strengths that would make the result noteworthy in modified-gravity and quantum-gravity phenomenology.

major comments (2)
  1. [shadow calculation section] The central claim of excess inner-shadow intensity rests on the assumption that the standard Lorentzian effective potential V_eff(r) = f(r)(1 + L^2/r^2) and the associated critical impact parameter remain valid across the signature transition at r = r_h. No re-derivation of the null condition g_μν k^μ k^ν = 0 or of the conserved Killing quantities is supplied once the metric becomes Euclidean inside the horizon, nor are explicit junction conditions or numerical ray-tracing across the interface presented. This omission directly undermines the reported excess intensity (abstract and §3).
  2. [energy accumulation and backreaction discussion] The statement that the horizon “continuously accumulates photons and energy” while exhibiting a backreaction distinct from stable light rings is asserted without quantitative comparison. No explicit energy-momentum flux calculation, no time-dependent metric perturbation, and no comparison metric (e.g., against a known light-ring spacetime) are given to substantiate the difference in backreaction response.
minor comments (2)
  1. [metric definition] Notation for the metric functions before and after the signature flip should be introduced with explicit coordinate ranges and continuity conditions at r = r_h.
  2. [figures] Figure captions for the shadow images should state the numerical resolution, impact-parameter sampling, and whether rays are terminated at the signature interface or continued.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's thorough review and constructive feedback on our manuscript. We address the major comments point by point below, agreeing where revisions are necessary to strengthen the presentation.

read point-by-point responses

  1. Referee: [shadow calculation section] The central claim of excess inner-shadow intensity rests on the assumption that the standard Lorentzian effective potential V_eff(r) = f(r)(1 + L^2/r^2) and the associated critical impact parameter remain valid across the signature transition at r = r_h. No re-derivation of the null condition g_μν k^μ k^ν = 0 or of the conserved Killing quantities is supplied once the metric becomes Euclidean inside the horizon, nor are explicit junction conditions or numerical ray-tracing across the interface presented. This omission directly undermines the reported excess intensity (abstract and §3).

    Authors: We thank the referee for highlighting this important point. While the Lorentzian-Euclidean black hole is constructed to be geodesically complete with continuous matching at the horizon, the manuscript indeed relies on the exterior Lorentzian effective potential for the shadow calculation without explicitly re-deriving the null condition in the interior Euclidean region. To address this, we will revise the shadow calculation section to include a derivation of the geodesic equations across the signature transition, specify the junction conditions used, and provide numerical examples of ray-tracing that demonstrate the excess intensity persists. This will clarify that the inner shadow excess arises from the trapping behavior at the horizon interface. revision: yes

  2. Referee: [energy accumulation and backreaction discussion] The statement that the horizon “continuously accumulates photons and energy” while exhibiting a backreaction distinct from stable light rings is asserted without quantitative comparison. No explicit energy-momentum flux calculation, no time-dependent metric perturbation, and no comparison metric (e.g., against a known light-ring spacetime) are given to substantiate the difference in backreaction response.

    Authors: We agree that the backreaction discussion is currently qualitative. The distinction we draw is that, unlike stable light rings in exotic compact objects where photons orbit at a fixed radius, in the Lorentzian-Euclidean geometry the signature change at the horizon prevents photons from escaping or forming closed orbits, leading to accumulation directly at the horizon surface. However, we acknowledge the lack of quantitative support. In the revision, we will add a brief comparison to a standard light-ring spacetime (e.g., a thin-shell gravastar) and estimate the difference in energy accumulation rate using the conserved quantities, though a full time-dependent perturbation analysis is beyond the current scope and will be noted as future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the Lorentzian-Euclidean black hole geometry (recently defined elsewhere) and computes shadow intensity and photon accumulation directly from the metric's null geodesics and effective potential under the given signature shift. No equation reduces a reported excess intensity or accumulation result to a quantity defined by fitting the same data or by renaming the input geometry. The central claims follow from explicit integration of the geodesic equations in the provided spacetime rather than from self-referential normalization or unverified self-citation chains. The derivation remains self-contained against the external benchmark of standard Schwarzschild shadow calculations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the recently introduced Lorentzian-Euclidean metric is a valid, geodesically complete spacetime whose signature change at the horizon produces the stated light-ray and energy behaviors; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption The Lorentzian-Euclidean black hole is a geodesically complete spacetime with a signature shift at the event horizon that prevents causal geodesics from reaching r=0.
    Invoked in the first sentence of the abstract as the defining property of the geometry under study.

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Forward citations

Cited by 6 Pith papers

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

  1. Gravity/thermodynamics correspondence via black hole shadows

    gr-qc 2026-04 unverdicted novelty 6.0

    Cuspy black hole shadows correspond to swallowtail thermodynamic free energy, with boundary self-intersections marking geometric phase transitions whose critical exponents fall in the mean-field class.

  2. Quasinormal Modes and Neutrino Energy Deposition for a Magnetically Charged Black Hole in a Hernquist Dark Matter Halo

    gr-qc 2026-04 unverdicted novelty 5.0

    Computations for a new black hole metric with magnetic charge and Hernquist halo show that charge raises QNM frequencies while the halo lowers them, with similar opposing effects on shadow size and neutrino annihilati...

  3. Energy conditions in static, spherically symmetric spacetimes and effective geometries

    gr-qc 2026-04 unverdicted novelty 5.0

    A logarithmic correction to Schwarzschild in static spherical symmetry obeys all classical energy conditions and serves as an effective exterior for horizon-bearing and horizonless compact objects.

  4. 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*.

  5. 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*.

  6. 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

96 extracted references · 96 canonical work pages · cited by 4 Pith papers · 17 internal anchors

  1. [1]

    A. G. Abacet al.(LIGO Scientific, KAGRA, VIRGO), Astrophys. J. Lett.995, L18 (2025), arXiv:2508.18080 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  2. [2]

    First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  3. [3]

    First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J.Lett.930,L12(2022),arXiv:2311.08680[astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  4. [4]

    Bambi,Black Holes: A Laboratory for Testing Strong Gravity(Springer, 2017)

    C. Bambi,Black Holes: A Laboratory for Testing Strong Gravity(Springer, 2017)

    work page 2017

  5. [5]

    Black holes, gravitational waves and fundamental physics: a roadmap

    L. Baracket al., Class. Quant. Grav.36, 143001 (2019), arXiv:1806.05195 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  6. [6]

    Testing the nature of dark compact objects: a status report

    V. Cardoso and P. Pani, Living Rev. Rel.22, 4 (2019), arXiv:1904.05363 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  7. [7]

    N. U. Molla, H. Chaudhary, S. Capozziello, F. Atamuro- tov, G. Mustafa, and U. Debnath, Phys. Dark Univ.47, 101804 (2025), arXiv:2501.09439 [gr-qc]

    work page arXiv 2025

  8. [8]

    Capozziello, S

    S.Capozziello, S.Zare, L.M.Nieto, andH.Hassanabadi, Phys. Dark Univ.50, 102065 (2025), arXiv:2311.12896 [gr-qc]

    work page arXiv 2025

  9. [9]

    Calmet, R

    X. Calmet, R. Casadio, and F. Kuipers, Phys. Rev. D 100, 086010 (2019), arXiv:1909.13277 [hep-th]

    work page arXiv 2019

  10. [10]

    Quantum General Relativity and Effec- tive Field Theory,

    J. F. Donoghue, “Quantum General Relativity and Effec- tive Field Theory,” (2023) arXiv:2211.09902 [hep-th]

    work page arXiv 2023

  11. [11]

    C.-M. Chen, Y. Chen, A. Ishibashi, and N. Ohta, Chin. J. Phys.92, 766 (2024), arXiv:2308.16356 [hep-th]

    work page arXiv 2024

  12. [12]

    Battista, Phys

    E. Battista, Phys. Rev. D109, 026004 (2024), arXiv:2312.00450 [gr-qc]

    work page arXiv 2024

  13. [13]

    Zhang, Universe9, 313 (2023), arXiv:2308.10184 [gr- qc]

    X. Zhang, Universe9, 313 (2023), arXiv:2308.10184 [gr- qc]

    work page arXiv 2023

  14. [14]

    Del Piano, S

    M. Del Piano, S. Hohenegger, and F. Sannino, Phys. Rev. D109, 024045 (2024), arXiv:2307.13489 [gr-qc]

    work page arXiv 2024

  15. [15]

    M.DelPiano, S.Hohenegger, andF.Sannino,Eur.Phys. J. C84, 1273 (2024), arXiv:2403.12679 [gr-qc]

    work page arXiv 2024

  16. [16]

    Hohenegger, Eur

    S. Hohenegger, Eur. Phys. J. C85, 1413 (2025), arXiv:2508.17781 [gr-qc]

    work page arXiv 2025

  17. [17]

    Singh, B

    B. Singh, B. K. Singh, and D. V. Singh, Int. J. Geom. Meth. Mod. Phys.20, 2350125 (2023)

    work page 2023

  18. [18]

    Zeng and Y

    H. Zeng and Y. Meng, (2025), arXiv:2512.05147 [gr-qc]

    work page arXiv 2025

  19. [19]

    M.-S. Ma, Y. He, X.-M. Wang, and H.-F. Li, Phys. Lett. B870, 139961 (2025), arXiv:2510.06576 [hep-th]

    work page arXiv 2025

  20. [20]

    Vertogradov and A

    V. Vertogradov and A. Rincon, Phys. Dark Univ.50, 102066 (2025), arXiv:2508.14489 [gr-qc]

    work page arXiv 2025

  21. [21]

    R. A. Konoplya and A. Zhidenko, Phys. Lett. B856, 138945 (2024), arXiv:2404.09063 [gr-qc]

    work page arXiv 2024

  22. [22]

    Alshammari, S

    M. Alshammari, S. Alshammari, S. Khan, and M. M. Al-sawalha, Eur. Phys. J. C85, 1402 (2025)

    work page 2025

  23. [23]

    Vertogradov, Eur

    V. Vertogradov, Eur. Phys. J. C85, 839 (2025), arXiv:2504.19292 [gr-qc]

    work page arXiv 2025

  24. [24]

    Fathi, M

    M. Fathi, M. Molina, and J. R. Villanueva, Phys. Lett. B820, 136548 (2021), arXiv:2101.12253 [gr-qc]

    work page arXiv 2021

  25. [25]

    M. M. Gohain, K. Bhuyan, R. Borgohain, T. Gogoi, K. Bhuyan, and P. Phukon, Nucl. Phys. B1018, 117073 (2025), arXiv:2412.06252 [gr-qc]

    work page arXiv 2025

  26. [26]

    S. Bora, D. J. Gogoi, and P. K. Karmakar, (2025), arXiv:2510.04208 [gr-qc]

    work page arXiv 2025

  27. [27]

    Waseem, F

    A. Waseem, F. Javed, G. Mustafa, S. K. Maurya, F. Ata- murotov, and M. Shrahili, Annals Phys.480, 170087 (2025)

    work page 2025

  28. [28]

    Liang, Z

    Q.-Q. Liang, Z. Cai, D. Liu, and Z.-W. Long, (2025), arXiv:2511.02396 [gr-qc]

    work page arXiv 2025

  29. [29]

    Naseer, J

    T. Naseer, J. Levi Said, R. Altuijri, M. R. Eid, and A.-H. Abdel-Aty, Int. J. Geom. Meth. Mod. Phys.22, 2540055 (2025)

    work page 2025

  30. [30]

    Capozziello, S

    S. Capozziello, S. Gambino, and O. Luongo, Phys. Dark Univ.48, 101950 (2025), arXiv:2503.21987 [gr-qc]

    work page arXiv 2025

  31. [31]

    On a regular charged black hole with a nonlinear electric source

    H. Culetu, Int. J. Theor. Phys.54, 2855 (2015), arXiv:1408.3334 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  32. [32]

    Miao and S.-J

    W.-J. Miao and S.-J. Yang, JCAP05, 022 (2025), arXiv:2409.07305 [gr-qc]

    work page arXiv 2025

  33. [33]

    M. D. Sultan, S. Chaudhary, T. Anwar, A. Ashraf, A. M. Mubaraki, F. Atamurotov, and A. Abidi, Phys. Dark Univ.50, 102077 (2025)

    work page 2025

  34. [34]

    E. L. B. Junior, J. T. S. S. Junior, F. S. N. Lobo, M. E. Rodrigues, L. F. D. da Silva, and H. A. Vieira, Eur. Phys. J. C85, 724 (2025), arXiv:2502.13327 [gr-qc]

    work page arXiv 2025

  35. [35]

    Huang and X.-P

    H. Huang and X.-P. Rao, Phys. Rev. D111, 104040 (2025), arXiv:2503.13133 [gr-qc]

    work page arXiv 2025

  36. [36]

    Contreras, M

    E. Contreras, M. Carrasco-Hidalgo, P. Bargueño, and A. G. Suvorov, Phys. Rev. D112, 124053 (2025), arXiv:2511.01544 [gr-qc]

    work page arXiv 2025

  37. [37]

    Black-bounce to traversable wormhole

    A. Simpson and M. Visser, JCAP02, 042 (2019), arXiv:1812.07114 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  38. [38]

    Ditta, T

    A. Ditta, T. Xia, R. Ali, G. Mustafa, G. Mustafa, and A. Mahmood, Phys. Dark Univ.43, 101418 (2024)

    work page 2024

  39. [39]

    Singularities and the Finale of Black Hole Evaporation

    L. Xiang, Y. Ling, and Y. G. Shen, Int. J. Mod. Phys. D22, 1342016 (2013), arXiv:1305.3851 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  40. [40]

    Brahma, C.-Y

    S. Brahma, C.-Y. Chen, and D.-h. Yeom, Phys. Rev. Lett.126, 181301 (2021), arXiv:2012.08785 [gr-qc]

    work page arXiv 2021

  41. [41]

    Bhandari, S

    G. Bhandari, S. D. Pathak, M. Sharma, and M. Y. Khlopov, Eur. Phys. J. Plus140, 663 (2025), arXiv:2407.19268 [gr-qc]

    work page arXiv 2025

  42. [42]

    I.H.Belfaqih, M.Bojowald, S.Brahma, andE.I.Duque, Phys. Rev. D112, 046022 (2025), arXiv:2407.12087 [gr- qc]

    work page arXiv 2025

  43. [43]

    Calz` a, D

    M. Calzà, D. Pedrotti, and S. Vagnozzi, Phys. Rev. D 111, 024009 (2025), arXiv:2409.02804 [gr-qc]

    work page arXiv 2025

  44. [44]

    Calz` a, D

    M. Calzà, D. Pedrotti, and S. Vagnozzi, Phys. Rev. D 111, 024010 (2025), arXiv:2409.02807 [gr-qc]

    work page arXiv 2025

  45. [45]

    Calzà, D

    M. Calzà, D. Pedrotti, G.-W. Yuan, and S. Vagnozzi, (2025), 10.1103/4x1f-vctx, arXiv:2507.02396 [gr-qc]

    work page doi:10.1103/4x1f-vctx 2025

  46. [46]

    Calz` a, M

    M. Calzà, M. Rinaldi, and S. Vagnozzi, Phys. Rev. D 112, 104055 (2025), arXiv:2510.12257 [gr-qc]

    work page arXiv 2025

  47. [47]

    Li, J.-P

    S. Li, J.-P. Wu, and X.-H. Ge, (2025), arXiv:2512.00926 [gr-qc]. 8

    work page arXiv 2025

  48. [48]

    Capozziello, S

    S. Capozziello, S. De Bianchi, and E. Battista, Phys. Rev. D109, 104060 (2024), arXiv:2404.17267 [gr-qc]

    work page arXiv 2024

  49. [49]

    Capozziello, E

    S. Capozziello, E. Battista, and S. De Bianchi, Phys. Rev. D112, 044009 (2025), arXiv:2507.08431 [gr-qc]

    work page arXiv 2025

  50. [50]

    De Bianchi, S

    S. De Bianchi, S. Capozziello, and E. Battista, Found. Phys.55, 36 (2025), arXiv:2504.17570 [gr-qc]

    work page arXiv 2025

  51. [51]

    Slowly decaying waves on spherically symmetric spacetimes and ultracompact neu- tron stars,

    J. Keir, Class. Quant. Grav.33, 135009 (2016), arXiv:1404.7036 [gr-qc]

    work page arXiv 2016

  52. [52]

    Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects

    V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Phys. Rev. D90, 044069 (2014), arXiv:1406.5510 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  53. [53]

    P. V. P. Cunha, C. Herdeiro, E. Radu, and N. Sanchis-Gual, Phys. Rev. Lett.130, 061401 (2023), arXiv:2207.13713 [gr-qc]

    work page arXiv 2023

  54. [54]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D15, 2752 (1977)

    work page 1977

  55. [55]

    Esposito,Quantum gravity, quantum cosmology and Lorentzian geometries, Vol

    G. Esposito,Quantum gravity, quantum cosmology and Lorentzian geometries, Vol. 12 (1992)

    work page 1992

  56. [56]

    Battista and G

    E. Battista and G. Esposito, Eur. Phys. J. C82, 1088 (2022), arXiv:2202.03763 [gr-qc]

    work page arXiv 2022

  57. [57]

    Garnier, Eur

    A. Garnier, Eur. Phys. J. C84, 374 (2024), arXiv:2401.15809 [gr-qc]

    work page arXiv 2024

  58. [58]

    Garnier and E

    A. Garnier and E. Battista, Eur. Phys. J. C85, 284 (2025), arXiv:2502.10053 [gr-qc]

    work page arXiv 2025

  59. [59]

    Poisson and C

    E. Poisson and C. M. Will,Gravity: Newtonian, Post- Newtonian, Relativistic(Cambridge University Press, 2014)

    work page 2014

  60. [60]

    Hadamard Regularization

    L. Blanchet and G. Faye, J. Math. Phys.41, 7675 (2000), arXiv:gr-qc/0004008

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  61. [61]

    Dymnikova, Gen

    I. Dymnikova, Gen. Rel. Grav.24, 235 (1992)

    work page 1992

  62. [62]

    S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006), arXiv:gr-qc/0506126

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  63. [63]

    Noncommutative geometry inspired Schwarzschild black hole

    P. Nicolini, A. Smailagic, and E. Spallucci, Phys. Lett. B632, 547 (2006), arXiv:gr-qc/0510112

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  64. [64]

    Simpson and M

    A. Simpson and M. Visser, Universe6, 8 (2019), arXiv:1911.01020 [gr-qc]

    work page arXiv 2019

  65. [65]

    Eichhorn, A

    A. Eichhorn, A. Held, and P.-V. Johannsen, JCAP01, 043 (2023), arXiv:2204.02429 [gr-qc]

    work page arXiv 2023

  66. [66]

    Chen, JCAP05, 040 (2020), arXiv:2004.01440 [gr- qc]

    C.-Y. Chen, JCAP05, 040 (2020), arXiv:2004.01440 [gr- qc]

    work page arXiv 2020

  67. [67]

    Chen, Phys

    C.-Y. Chen, Phys. Rev. D106, 044009 (2022), arXiv:2205.06962 [gr-qc]

    work page arXiv 2022

  68. [68]

    Pedrotti and S

    D. Pedrotti and S. Vagnozzi, Phys. Rev. D110, 084075 (2024), arXiv:2404.07589 [gr-qc]

    work page arXiv 2024

  69. [69]

    Chen and Y

    C.-Y. Chen and Y. Yokokura, Phys. Rev. D109, 104058 (2024), arXiv:2403.09388 [gr-qc]

    work page arXiv 2024

  70. [70]

    Wang, Phys

    Z.-L. Wang, Phys. Rev. D112, 064052 (2025), arXiv:2506.21148 [gr-qc]

    work page arXiv 2025

  71. [71]

    R. C. Pantig, S. Kala, A. Övgün, and N. J. L. S. Lobos, (2024), 10.1142/S0219887825502408, arXiv:2410.13661 [gr-qc]

    work page doi:10.1142/s0219887825502408 2024

  72. [72]

    R. Ali, X. Tiecheng, M. Awais, and R. Babar, Int. J. Geom. Meth. Mod. Phys.21, 2450180 (2024)

    work page 2024

  73. [73]

    C.-Y. Yang, H. Ye, and X.-X. Zeng, (2025), arXiv:2510.21229 [gr-qc]

    work page arXiv 2025

  74. [74]

    Chen, C.-M

    C.-Y. Chen, C.-M. Chen, and N. Ohta, (2025), arXiv:2510.00708 [gr-qc]

    work page arXiv 2025

  75. [75]

    Al-Badawi, EPL152, 19002 (2025), arXiv:2510.22704 [gr-qc]

    A. Al-Badawi, EPL152, 19002 (2025), arXiv:2510.22704 [gr-qc]

    work page arXiv 2025

  76. [76]

    Kala and J

    S. Kala and J. Singh, Eur. Phys. J. C85, 1047 (2025), arXiv:2507.17280 [gr-qc]

    work page arXiv 2025

  77. [77]

    Y. Chen, G. Guo, B. Mu, and P. Wang, (2025), arXiv:2506.19581 [gr-qc]

    work page arXiv 2025

  78. [78]

    R. C. Pantig and A. Övgün, Annals Phys.480, 170104 (2025), arXiv:2503.18585 [gr-qc]

    work page arXiv 2025

  79. [79]

    Fathi and Y

    M. Fathi and Y. Sekhmani, Eur. Phys. J. C85, 477 (2025), arXiv:2503.02179 [gr-qc]

    work page arXiv 2025

  80. [80]

    Wang and E

    Z.-L. Wang and E. Battista, Eur. Phys. J. C85, 304 (2025), arXiv:2501.14516 [gr-qc]

    work page arXiv 2025

Showing first 80 references.