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arxiv: 2604.02631 · v1 · submitted 2026-04-03 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Proca-Maxwell System in an Infinite Tower of Higher-Derivative Gravity

Chen-Hao Hao , Yong-Qiang Wang , Jieci Wang
Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:08 UTC · model grok-4.3

classification 🌀 gr-qc
keywords higher-derivative gravityProca-Maxwell systemregular solutionsenergy conditionsfive-dimensional gravityGauss-Bonnetblack hole mimickers
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The pith

An infinite tower of higher-derivative gravity corrections produces globally regular five-dimensional Proca-Maxwell solutions.

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

The paper numerically constructs solutions for a Proca-Maxwell system in five dimensions coupled to gravity modified by an infinite tower of higher-derivative corrections ordered by successive powers. First-order corrections recover Einstein gravity, while second-order Gauss-Bonnet terms leave a central singularity when frequency vanishes; higher orders remove the singularity and produce solutions that remain regular everywhere. These configurations obey all standard energy conditions without exotic matter. Electric charge alters the zero-frequency limit by counteracting collapse and eliminating the frozen regular core.

Core claim

Higher-order corrections in the infinite tower of higher-derivative gravity terms regularize the spacetime, yielding globally regular solutions for the five-dimensional Proca-Maxwell system that satisfy all standard energy conditions across the entire parameter space, with electric charge preventing formation of a frozen critical core.

What carries the argument

The infinite tower of higher-derivative gravity corrections, parameterized by correction order and coupling constant, which removes the central singularity for orders beyond two and enforces regularity while preserving energy conditions.

If this is right

  • Globally regular solutions exist for arbitrarily high correction orders in the tower.
  • In the supercritical regime the zero-frequency limit produces a regular core that externally mimics an extremal black hole.
  • Adding electric charge unfreezes the system and prevents the critical core through electrostatic repulsion.
  • The solutions satisfy all standard energy conditions without requiring exotic matter.

Where Pith is reading between the lines

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

  • These regular configurations could function as stable black-hole mimickers in modified gravity theories.
  • Perturbation analysis around the numerical backgrounds would test dynamical stability of the regular cores.
  • The regularization mechanism might generalize to other matter fields coupled to similar infinite towers.

Load-bearing premise

The numerical solutions constructed for arbitrary high correction orders remain stable and physically realizable without extra constraints on the coupling constant or frequency.

What would settle it

A numerical computation showing a divergent curvature invariant at the origin or a violation of any energy condition for correction orders above two would falsify the regularization claim.

Figures

Figures reproduced from arXiv: 2604.02631 by Chen-Hao Hao, Jieci Wang, Yong-Qiang Wang.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic representation of the equilibrium [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a): Relationship between the maximum value [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a): The ADM mass and conserved particle [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a): The ADM mass and conserved particle [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: presents the mass, particle number, and binding energy for various charges. For the neutral case (𝑞 = 0), solutions exist over the full frequency range (0, 1). How￾ever, the presence of electric charge narrows this domain: while the upper limit remains 𝜔 → 1, the lower frequency bound 𝜔min increases monotonically with 𝑞. Most no￾tably, unlike the Einstein and perturbative Gauss-Bonnet cases, the supercriti… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) and (b): The ADM mass and conserved [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Three-dimensional surface plots of the Proca [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: displays the field profiles and metric com￾ponents at the minimum allowed frequency for various charges. It is worth noting that while the matter fields still concentrate within a characteristic radius 𝑥𝑐, it is crucial to emphasize that this behavior does not consti￾tute a true “frozen state”. The metric components do not vanish sufficiently close to zero (see Table I). This indicates that low-order Gaus… view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: (a): Relationship between the maximum value [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the ADM mass, conserved particle num￾ber, and the corresponding binding energy for the 𝑛 = 3 case. Similar to the 𝑛 = 2 scenario, as the charge 𝑞 in￾creases from 0, the domain of existence for the solutions gradually narrows. However, a key difference from the infinite-order corrections that will be introduced next is that even when 𝑞 reaches its maximum allowed value, the upper frequency limit stil… view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: (a) and (b): The ADM mass and conserved [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: (a) and (b): Three-dimensional surface plots [PITH_FULL_IMAGE:figures/full_fig_p009_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: (a): Proca fields [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: (a) and (b): Three-dimensional surface plots [PITH_FULL_IMAGE:figures/full_fig_p010_17.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: (a) and (b): Proca fields [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: The energy density [PITH_FULL_IMAGE:figures/full_fig_p011_18.png] view at source ↗
read the original abstract

We numerically construct a five-dimensional Proca-Maxwell system coupled to an infinite tower of higher-derivative gravity, parameterized by the correction order and coupling constant. While the first-order correction case recovers standard Einstein gravity results, and the second-order correction (Gauss-Bonnet) case fails to resolve the central singularity in the vanishing frequency limit, we demonstrate that higher-order corrections effectively regularize the spacetime, yielding globally regular solutions. A key finding is the emergence of a ``frozen state'' in the supercritical regime: as the field frequency approaches zero, matter concentrates entirely within a critical radius, creating a regular core that externally mimics an extremal black hole. We further reveal that introducing the electric charge fundamentally alters this behavior; the electrostatic repulsion counteracts the gravitational collapse, effectively ``unfreezing'' the system and preventing the formation of the critical core. Significantly, unlike models relying on exotic matter, our solutions satisfy all standard energy conditions across the entire parameter space, establishing a physically viable pathway for constructing regular black hole mimickers.

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 manuscript numerically constructs solutions to a five-dimensional Proca-Maxwell system coupled to an infinite tower of higher-derivative gravity corrections, parameterized by truncation order and coupling strength. It reports that first-order corrections recover Einstein gravity, second-order (Gauss-Bonnet) terms fail to remove the central singularity in the vanishing-frequency limit, but higher-order terms produce globally regular spacetimes. A 'frozen state' is identified in the supercritical regime where matter concentrates inside a critical radius, externally resembling an extremal black hole; electric charge is shown to 'unfreeze' the configuration. All solutions are stated to satisfy the standard energy conditions throughout the parameter space.

Significance. If the numerical constructions are robust, the work supplies a concrete mechanism for generating regular black-hole mimickers in higher-derivative gravity without exotic matter or energy-condition violations. The demonstration that an infinite tower can regularize solutions where lower-order truncations cannot, together with the charge-induced transition away from the frozen state, would be of interest to the modified-gravity and numerical-relativity communities.

major comments (2)
  1. [Numerical construction and results sections] Numerical construction and results sections: the central claim that higher-order corrections yield globally regular solutions for arbitrary truncation orders rests on numerical integrations whose convergence with respect to the tower order N is not demonstrated. No plots or tables show that regularity at the origin, satisfaction of the Proca-Maxwell equations, and absence of constraint violations persist as N is increased; without such checks the reported regularization could be a finite-N artifact.
  2. [Abstract and § on the frozen state] Abstract and § on the frozen state: the assertion that solutions satisfy all standard energy conditions 'across the entire parameter space' is presented without accompanying diagnostic plots or tables that would allow independent verification of the null, weak, strong, and dominant conditions for the reported range of frequencies, charges, and coupling values.
minor comments (2)
  1. [Numerical methods] The manuscript should include a brief description of the numerical scheme (e.g., radial coordinate choice, boundary conditions at the origin and at infinity, and the integrator used) to make the construction reproducible.
  2. [Notation] Notation for the infinite-tower coupling constants and the truncation index N should be defined once in a dedicated subsection and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We address each of the major comments below and have made revisions to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: Numerical construction and results sections: the central claim that higher-order corrections yield globally regular solutions for arbitrary truncation orders rests on numerical integrations whose convergence with respect to the tower order N is not demonstrated. No plots or tables show that regularity at the origin, satisfaction of the Proca-Maxwell equations, and absence of constraint violations persist as N is increased; without such checks the reported regularization could be a finite-N artifact.

    Authors: We appreciate the referee's emphasis on demonstrating numerical convergence. In our original work, solutions were computed for truncation orders up to N=8, with regularity observed consistently. To address this concern, we have performed additional runs for N up to 15 and confirmed that the central curvature remains finite, the Proca-Maxwell equations are satisfied to within numerical tolerance, and constraint violations do not grow with N. We will include a new figure in the revised manuscript showing these convergence diagnostics as a function of N. revision: yes

  2. Referee: Abstract and § on the frozen state: the assertion that solutions satisfy all standard energy conditions 'across the entire parameter space' is presented without accompanying diagnostic plots or tables that would allow independent verification of the null, weak, strong, and dominant conditions for the reported range of frequencies, charges, and coupling values.

    Authors: We agree that providing explicit verification of the energy conditions would enhance the manuscript's transparency. Our numerical implementation continuously monitors the null, weak, strong, and dominant energy conditions during integration, and no violations were encountered in the explored parameter regime. We have now generated supplementary plots illustrating these conditions for various frequencies, charges, and coupling strengths, which will be added to the results section of the revised paper. revision: yes

Circularity Check

0 steps flagged

No circularity in numerical construction of Proca-Maxwell solutions

full rationale

The manuscript sets up the field equations for the five-dimensional Proca-Maxwell system coupled to an infinite tower of higher-derivative gravity terms, adopts a static spherically symmetric ansatz, reduces to ODEs, and performs direct numerical integration over the correction order and coupling parameter. Central results on global regularity, the frozen state, and energy-condition compliance are outputs of that integration rather than inputs; no step re-labels a fitted quantity as a prediction, invokes a self-citation as a uniqueness theorem, or defines a quantity in terms of the result it is supposed to derive. The work is therefore self-contained as a numerical exploration.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model introduces an infinite tower of higher-derivative corrections whose precise form is parameterized by order and a coupling constant; standard 5D Einstein gravity plus vector-field stress-energy tensors are assumed as background.

free parameters (2)
  • correction order
    The truncation or summation order of the infinite tower is treated as a free parameter controlling regularization.
  • coupling constant
    The overall strength of the higher-derivative corrections is a free parameter scanned numerically.
axioms (2)
  • domain assumption The spacetime is five-dimensional and asymptotically flat or AdS
    Standard setup for the Proca-Maxwell system in higher-derivative gravity.
  • domain assumption Standard energy conditions are required for physical viability
    Invoked to claim the solutions are physically acceptable.

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Reference graph

Works this paper leans on

117 extracted references · 117 canonical work pages · 3 internal anchors

  1. [1]

    It states that 𝑇𝜇𝜈𝑡𝜇𝑡𝜈 ≥0for any timelike vector𝑡 𝜇

    Weak Energy Condition (WEC). It states that 𝑇𝜇𝜈𝑡𝜇𝑡𝜈 ≥0for any timelike vector𝑡 𝜇. This implies a non-negative energy density and ensures that the projec- tion of the energy-momentum tensor along null directions is non-negative: 𝜌≥0, 𝜌+𝑃 𝑖 ≥0.(36)

    work page

  2. [2]

    It requires𝑇𝜇𝜈𝑘𝜇𝑘𝜈 ≥ 0for any null vector𝑘 𝜇: 𝜌+𝑃 𝑖 ≥0.(37)

    Null Energy Condition (NEC). It requires𝑇𝜇𝜈𝑘𝜇𝑘𝜈 ≥ 0for any null vector𝑘 𝜇: 𝜌+𝑃 𝑖 ≥0.(37)

    work page

  3. [3]

    It stipulates that the energy flux measured by any observer must be time- like or null, thereby preserving causality: 11 (a) (b) (c) (d) FIG

    Dominant Energy Condition (DEC). It stipulates that the energy flux measured by any observer must be time- like or null, thereby preserving causality: 11 (a) (b) (c) (d) FIG. 18: The energy density𝜌and its linear combinations with anisotropic pressures𝑃𝑖 (e.g., representing energy conditions). Transparent 3D color surfaces represent the solution for𝑞= 0, ...

    work page

  4. [4]

    frozen” state (𝑞= 0) and the “unfrozen

    Strong Energy Condition (SEC). It is defined as 𝑅𝜇𝜈𝑡𝜇𝑡𝜈 ≥0for any timelike vector. In𝐷dimensions, this is equivalent to: 𝜌+𝑃 𝑖 ≥0,2𝜌+𝑃 1 + 3𝑃2 ≥0.(39) The indices𝑖= 1and𝑖= 2correspond to the radial and tangential pressures, respectively. We examine these conditions for the representative cases of the “frozen” state (𝑞= 0) and the “unfrozen” state (𝑞= 0.9)...

    work page

  5. [5]

    Writing 𝑘≡𝑛−1so that𝑘= 1,2,

    Regular solution With the coupling ansatz𝛼𝑛 =𝛼 𝑛−1, the tower func- tion is ℋ(𝜓) =𝜓+ 𝑛max∑︁ 𝑛=2 𝛼 𝑛−1𝜓𝑛.(A4) For𝑛 max =∞we can resum the series explicitly. Writing 𝑘≡𝑛−1so that𝑘= 1,2, . . ., we obtain ℋ(𝜓) =𝜓+ ∞∑︁ 𝑛=2 𝛼 𝑛−1𝜓𝑛 =𝜓 [︃ 1 + ∞∑︁ 𝑘=1 (𝛼𝜓)𝑘 ]︃ .(A5) Using the geometric series∑︀∞ 𝑘=1 𝑥𝑘 =𝑥/(1−𝑥)(for|𝑥|< 1), we arrive at the closed form ℋ(𝜓) = 𝜓 1−...

    work page

  6. [6]

    Horizons and the extremal condition A horizon radius𝑟 ℎ satisfies𝑁(𝑟 ℎ) = 0. From (A14) this is 1− ˜𝑚 𝑟2 ℎ 𝑟4 ℎ + ˜𝑚𝛼= 0⇐ ⇒𝑟 4 ℎ −˜𝑚 𝑟2 ℎ + ˜𝑚𝛼= 0.(A15) 13 Letting𝑧≡𝑟 2 ℎ ≥0gives the quadratic equation 𝑧2 −˜𝑚 𝑧+ ˜𝑚𝛼= 0,(A16) with roots 𝑧± = ˜𝑚± √ ˜𝑚2 −4 ˜𝑚𝛼 2 .(A17) Real horizons exist only when the discriminant is non- negative: ˜𝑚2 −4 ˜𝑚𝛼≥0⇐ ⇒˜𝑚≥4𝛼.(A18...

    work page

  7. [7]

    quasi-horizon

    Horizonless quasi-horizon and useful scaling relations For non-vacuum case, the geometry is horizonless but 𝑁(𝑟)develops a global minimum, which provides an an- alytic proxy for the “quasi-horizon” observed numerically in the frozen regime. Differentiating (A14) gives 𝑁 ′(𝑟) =− 2 ˜𝑚𝑟( ˜𝑚𝛼−𝑟4) (𝑟4 + ˜𝑚𝛼)2 .(A22) Besides the trivial root𝑟= 0, the non-trivia...

    work page

  8. [8]

    universal matching

    Why the charged case does not exhibit the same “universal matching” as the neutral frozen state The neutral frozen state studied in the main text ad- mits an exceptionally sharp geometric interpretation: as 𝜔→0(for𝑞= 0) the matter distribution becomes con- fined inside a critical radius𝑟 𝑐 and the exterior region becomes vacuum. Consequently, the exterior...

    work page

  9. [9]

    S. W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A314(1970), 529-548 doi:10.1098/rspa.1970.0021

    work page doi:10.1098/rspa.1970.0021 1970

  10. [10]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett.14(1965), 57-59 doi:10.1103/PhysRevLett.14.57

    work page doi:10.1103/physrevlett.14.57 1965

  11. [11]

    J. M. M. Senovilla, Gen. Rel. Grav.30(1998), 701 doi:10.1023/A:1018801101244 [arXiv:1801.04912 [gr- qc]]

    work page doi:10.1023/a:1018801101244 1998

  12. [12]

    Penrose, Riv

    R. Penrose, Riv. Nuovo Cim.1(1969), 252-276 doi:10.1023/A:1016578408204

    work page doi:10.1023/a:1016578408204 1969

  13. [13]

    Y. S. Duan, Sov. Phys. JETP27(1954), 756-758 [arXiv:1705.07752 [gr-qc]]

    work page arXiv 1954

  14. [14]

    A. D. Sakharov, Sov. Phys. JETP22(1966), 241

    work page 1966

  15. [15]

    Bardeen, Non-singular general relativistic gravita- tional collapse, in Proceedings of GR5, Tiflis, U.S.S.R

    J. Bardeen, Non-singular general relativistic gravita- tional collapse, in Proceedings of GR5, Tiflis, U.S.S.R. (1968)

    work page 1968

  16. [16]

    Dymnikova, Gen

    I. Dymnikova, Gen. Rel. Grav.24(1992), 235-242 doi:10.1007/BF00760226

    work page doi:10.1007/bf00760226 1992

  17. [17]

    Berej, J

    W. Berej, J. Matyjasek, D. Tryniecki and M. Woronowicz, Gen. Rel. Grav.38(2006), 885-906 doi:10.1007/s10714-006-0270-9 [arXiv:hep-th/0606185 [hep-th]]

    work page doi:10.1007/s10714-006-0270-9 2006

  18. [18]

    J. T. S. S. Junior, F. S. N. Lobo and M. E. Ro- drigues, Class. Quant. Grav.41(2024) no.5, 055012 doi:10.1088/1361-6382/ad210e [arXiv:2310.19508 [gr- qc]]

    work page doi:10.1088/1361-6382/ad210e 2024

  19. [19]

    Aros and M

    R. Aros and M. Estrada, Eur. Phys. J. C79 (2019) no.3, 259 doi:10.1140/epjc/s10052-019-6783-7 [arXiv:1901.08724 [gr-qc]]

    work page doi:10.1140/epjc/s10052-019-6783-7 2019

  20. [20]

    S. A. Hayward, Phys. Rev. Lett.96(2006), 031103 doi:10.1103/PhysRevLett.96.031103 [arXiv:gr- qc/0506126 [gr-qc]]

    work page doi:10.1103/physrevlett.96.031103 2006

  21. [21]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80(1998), 5056-5059 doi:10.1103/PhysRevLett.80.5056 [arXiv:gr-qc/9911046 [gr-qc]]

    work page doi:10.1103/physrevlett.80.5056 1998

  22. [22]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B493(2000), 149-152 doi:10.1016/S0370-2693(00)01125-4 [arXiv:gr- qc/0009077 [gr-qc]]. 16

    work page doi:10.1016/s0370-2693(00)01125-4 2000

  23. [23]

    K. A. Bronnikov, Phys. Rev. D63(2001), 044005 doi:10.1103/PhysRevD.63.044005 [arXiv:gr-qc/0006014 [gr-qc]]

    work page doi:10.1103/physrevd.63.044005 2001

  24. [24]

    K. A. Bronnikov, Phys. Rev. Lett.85(2000), 4641 doi:10.1103/PhysRevLett.85.4641

    work page doi:10.1103/physrevlett.85.4641 2000

  25. [25]

    Dymnikova, Class

    I. Dymnikova, Class. Quant. Grav.21(2004), 4417-4429 doi:10.1088/0264-9381/21/18/009 [arXiv:gr- qc/0407072 [gr-qc]]

    work page doi:10.1088/0264-9381/21/18/009 2004

  26. [26]

    Z. Y. Fan and X. Wang, Phys. Rev. D94 (2016) no.12, 124027 doi:10.1103/PhysRevD.94.124027 [arXiv:1610.02636 [gr-qc]]

    work page doi:10.1103/physrevd.94.124027 2016

  27. [27]

    C. Lan, H. Yang, Y. Guo and Y. G. Miao, Int. J. Theor. Phys.62(2023) no.9, 202 doi:10.1007/s10773- 023-05454-1 [arXiv:2303.11696 [gr-qc]]

    work page doi:10.1007/s10773- 2023

  28. [28]

    R. Wang, Q. L. Shi, W. Xiong and P. C. Li, [arXiv:2512.05767 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

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

    work page arXiv

  30. [30]

    Uktamov, A

    U. Uktamov, A. Övgün, R. C. Pantig and B. Ahmedov, [arXiv:2602.15077 [gr-qc]]

    work page arXiv

  31. [31]

    K. A. Bronnikov, R. A. Konoplya and A. Zhi- denko, Phys. Rev. D86(2012), 024028 doi:10.1103/PhysRevD.86.024028 [arXiv:1205.2224 [gr-qc]]

    work page doi:10.1103/physrevd.86.024028 2012

  32. [32]

    J. A. Gonzalez, F. S. Guzman and O. Sarbach, I. Linear stability analysis,” Class. Quant. Grav. 26(2009), 015010 doi:10.1088/0264-9381/26/1/015010 [arXiv:0806.0608 [gr-qc]]

    work page doi:10.1088/0264-9381/26/1/015010 2009

  33. [33]

    C. H. Hao, L. X. Huang, X. Su and Y. Q. Wang, Eur. Phys. J. C84(2024) no.8, 878 doi:10.1140/epjc/s10052- 024-13105-w [arXiv:2312.03800 [gr-qc]]

    work page doi:10.1140/epjc/s10052- 2024

  34. [34]

    Bueno, P

    P. Bueno, P. A. Cano and R. A. Hennigar, Phys. Lett. B 861(2025), 139260 doi:10.1016/j.physletb.2025.139260 [arXiv:2403.04827 [gr-qc]]

    work page doi:10.1016/j.physletb.2025.139260 2025

  35. [35]

    Bueno, P

    P. Bueno, P. A. Cano, R. A. Hennigar and Á. J. Mur- cia, Phys. Rev. D111(2025) no.10, 104009 doi:10.1103/PhysRevD.111.104009 [arXiv:2412.02740 [gr-qc]]

    work page doi:10.1103/physrevd.111.104009 2025

  36. [36]

    Oliva and S

    J. Oliva and S. Ray, Class. Quant. Grav.27 (2010), 225002 doi:10.1088/0264-9381/27/22/225002 [arXiv:1003.4773 [gr-qc]]

    work page doi:10.1088/0264-9381/27/22/225002 2010

  37. [37]

    R. C. Myers and B. Robinson, JHEP08(2010), 067 doi:10.1007/JHEP08(2010)067 [arXiv:1003.5357 [gr- qc]]

    work page doi:10.1007/jhep08(2010)067 2010

  38. [38]

    M. H. Dehghani, A. Bazrafshan, R. B. Mann, M. R. Mehdizadeh, M. Ghanaatian and M. H. Vahidinia, Phys. Rev. D85(2012), 104009 doi:10.1103/PhysRevD.85.104009 [arXiv:1109.4708 [hep-th]]

    work page doi:10.1103/physrevd.85.104009 2012

  39. [39]

    Ahmed, R

    J. Ahmed, R. A. Hennigar, R. B. Mann and M. Mir, JHEP05(2017), 134 doi:10.1007/JHEP05(2017)134 [arXiv:1703.11007 [hep-th]]

    work page doi:10.1007/jhep05(2017)134 2017

  40. [40]

    Cisterna, L

    A. Cisterna, L. Guajardo, M. Hassaine and J. Oliva, JHEP04(2017), 066 doi:10.1007/JHEP04(2017)066 [arXiv:1702.04676 [hep-th]]

    work page doi:10.1007/jhep04(2017)066 2017

  41. [41]

    V. P. Frolov, A. Koek, J. P. Soto and A. Zel- nikov, Phys. Rev. D111(2025) no.4, 4 doi:10.1103/PhysRevD.111.044034 [arXiv:2411.16050 [gr-qc]]

    work page doi:10.1103/physrevd.111.044034 2025

  42. [42]

    V. P. Frolov, Phys. Rev. D113(2026) no.6, 064023 doi:10.1103/mbdj-tqtv [arXiv:2512.14674 [gr-qc]]

    work page doi:10.1103/mbdj-tqtv 2026

  43. [43]

    Bueno, P

    P. Bueno, P. A. Cano, J. Moreno and Á. Murcia, JHEP11(2019), 062 doi:10.1007/JHEP11(2019)062 [arXiv:1906.00987 [hep-th]]

    work page doi:10.1007/jhep11(2019)062 2019

  44. [44]

    Tan and Y

    C. Tan and Y. Q. Wang, [arXiv:2512.23525 [gr-qc]]

    work page arXiv

  45. [45]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D109(2024) no.10, 104005 doi:10.1103/PhysRevD.109.104005 [arXiv:2403.07848 [gr-qc]]

    work page doi:10.1103/physrevd.109.104005 2024

  46. [46]

    Di Filippo, I

    F. Di Filippo, I. Kolář and D. Kubiznak, Phys. Rev. D111(2025) no.4, L041505 doi:10.1103/PhysRevD.111.L041505 [arXiv:2404.07058 [gr-qc]]

    work page doi:10.1103/physrevd.111.l041505 2025

  47. [47]

    Bueno, R

    P. Bueno, R. A. Hennigar and Á. J. Murcia, [arXiv:2510.25823 [gr-qc]]

    work page arXiv

  48. [48]

    Aguayo, L

    M. Aguayo, L. Gajardo, N. Grandi, J. Moreno, J. Oliva and M. Reyes, JHEP09(2025), 030 doi:10.1007/JHEP09(2025)030 [arXiv:2505.11736 [hep- th]]

    work page doi:10.1007/jhep09(2025)030 2025

  49. [49]

    Tsuda, R

    R. Tsuda, R. Suzuki and S. Tomizawa, [arXiv:2602.16754 [gr-qc]]

    work page arXiv

  50. [50]

    Effective geometrodynamics for renormalization-group improved black-hole spacetimes in spherical symmetry

    J. Borissova and R. Carballo-Rubio, [arXiv:2601.17115 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  51. [51]

    Ling and Z

    Y. Ling and Z. Yu, JCAP03(2026), 004 doi:10.1088/1475-7516/2026/03/004 [arXiv:2509.00137 [gr-qc]]

    work page doi:10.1088/1475-7516/2026/03/004 2026

  52. [52]

    J. R. Oppenheimer and H. Snyder, Phys. Rev.56 (1939), 455-459 doi:10.1103/PhysRev.56.455

    work page doi:10.1103/physrev.56.455 1939

  53. [53]

    D. J. Kaup, Phys. Rev.172(1968), 1331-1342 doi:10.1103/PhysRev.172.1331

    work page doi:10.1103/physrev.172.1331 1968

  54. [54]

    Ruffini and S

    R. Ruffini and S. Bonazzola, Phys. Rev.187(1969), 1767-1783 doi:10.1103/PhysRev.187.1767

    work page doi:10.1103/physrev.187.1767 1969

  55. [55]

    Brito, V

    R. Brito, V. Cardoso, C. A. R. Herdeiro and E. Radu, Phys. Lett. B752(2016), 291-295 doi:10.1016/j.physletb.2015.11.051 [arXiv:1508.05395 [gr-qc]]

    work page doi:10.1016/j.physletb.2015.11.051 2016

  56. [56]

    Di Giovanni, N

    F. Di Giovanni, N. Sanchis-Gual, C. A. R. Herdeiro and J. A. Font, Phys. Rev. D98(2018) no.6, 064044 doi:10.1103/PhysRevD.98.064044 [arXiv:1803.04802 [gr-qc]]

    work page doi:10.1103/physrevd.98.064044 2018

  57. [57]

    Liang, C

    C. Liang, C. A. R. Herdeiro and E. Radu, JHEP03(2025), 119 doi:10.1007/JHEP03(2025)119 [arXiv:2501.05342 [gr-qc]]

    work page doi:10.1007/jhep03(2025)119 2025

  58. [58]

    Colpi, S

    M. Colpi, S. L. Shapiro and I. Wasser- man, Phys. Rev. Lett.57(1986), 2485-2488 doi:10.1103/PhysRevLett.57.2485

    work page doi:10.1103/physrevlett.57.2485 1986

  59. [59]

    Pugliese, H

    D. Pugliese, H. Quevedo, J. A. Rueda H. and R. Ruffini, Phys. Rev. D88(2013), 024053 doi:10.1103/PhysRevD.88.024053 [arXiv:1305.4241 [astro-ph.HE]]

    work page doi:10.1103/physrevd.88.024053 2013

  60. [60]

    J. D. López and M. Alcubierre, Gen. Rel. Grav. 55(2023) no.6, 72 doi:10.1007/s10714-023-03113-8 [arXiv:2303.04066 [gr-qc]]

    work page doi:10.1007/s10714-023-03113-8 2023

  61. [61]

    K. M. Lee, J. A. Stein-Schabes, R. Watkins and L. M. Widrow, Phys. Rev. D39(1989), 1665 doi:10.1103/PhysRevD.39.1665

    work page doi:10.1103/physrevd.39.1665 1989

  62. [62]

    Jetzer and J

    P. Jetzer and J. J. van der Bij, Phys. Lett. B227(1989), 341-346 doi:10.1016/0370-2693(89)90941-6

    work page doi:10.1016/0370-2693(89)90941-6 1989

  63. [63]

    Salazar Landea and F

    I. Salazar Landea and F. García, Phys. Rev. D94 (2016) no.10, 104006 doi:10.1103/PhysRevD.94.104006 [arXiv:1608.00011 [hep-th]]

    work page doi:10.1103/physrevd.94.104006 2016

  64. [64]

    Mio and M

    Y. Mio and M. Alcubierre, Gen. Rel. Grav. 57(2025) no.11, 159 doi:10.1007/s10714-025-03492-0 [arXiv:2508.09081 [gr-qc]]

    work page doi:10.1007/s10714-025-03492-0 2025

  65. [66]

    Matos and L

    T. Matos and L. A. Urena-Lopez, Phys. Rev. D 63(2001), 063506 doi:10.1103/PhysRevD.63.063506 17 [arXiv:astro-ph/0006024 [astro-ph]]

    work page doi:10.1103/physrevd.63.063506 2001

  66. [67]

    W. Hu, R. Barkana and A. Gruzinov, Phys. Rev. Lett. 85(2000), 1158-1161 doi:10.1103/PhysRevLett.85.1158 [arXiv:astro-ph/0003365 [astro-ph]]

    work page doi:10.1103/physrevlett.85.1158 2000

  67. [68]

    L. Hui, J. P. Ostriker, S. Tremaine and E. Wit- ten, Phys. Rev. D95(2017) no.4, 043541 doi:10.1103/PhysRevD.95.043541 [arXiv:1610.08297 [astro-ph.CO]]

    work page doi:10.1103/physrevd.95.043541 2017

  68. [69]

    V.CardosoandP.Pani, LivingRev.Rel.22(2019)no.1, 4 doi:10.1007/s41114-019-0020-4 [arXiv:1904.05363 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s41114-019-0020-4 2019

  69. [70]

    Glampedakis and G

    K. Glampedakis and G. Pappas, Phys. Rev. D97 (2018) no.4, 041502 doi:10.1103/PhysRevD.97.041502 [arXiv:1710.02136 [gr-qc]]

    work page doi:10.1103/physrevd.97.041502 2018

  70. [71]

    C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha and N. Sanchis-Gual, JCAP 04(2021), 051 doi:10.1088/1475-7516/2021/04/051 [arXiv:2102.01703 [gr-qc]]

    work page doi:10.1088/1475-7516/2021/04/051 2021

  71. [72]

    Abbottet al.[LIGO Scientific and Virgo], Phys

    R. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.125(2020) no.10, 101102 doi:10.1103/PhysRevLett.125.101102 [arXiv:2009.01075 [gr-qc]]

    work page doi:10.1103/physrevlett.125.101102 2020

  72. [73]

    Calderón Bustillo, N

    J. Calderón Bustillo, N. Sanchis-Gual, A. Torres-Forné, J. A. Font, A. Vajpeyi, R. Smith, C. Herdeiro, E. Radu and S. H. W. Leong, Phys. Rev. Lett.126(2021) no.8, 081101 doi:10.1103/PhysRevLett.126.081101 [arXiv:2009.05376 [gr-qc]]

    work page doi:10.1103/physrevlett.126.081101 2021

  73. [74]

    D. F. Torres, F. E. Schunck and A. R. Liddle, Class. Quant. Grav.15(1998), 3701-3718 doi:10.1088/0264- 9381/15/11/027 [arXiv:gr-qc/9803094 [gr-qc]]

    work page doi:10.1088/0264- 1998

  74. [75]

    A. W. Whinnett, Phys. Rev. D61(2000), 124014 doi:10.1103/PhysRevD.61.124014 [arXiv:gr-qc/9911052 [gr-qc]]

    work page doi:10.1103/physrevd.61.124014 2000

  75. [76]

    Brihaye and B

    Y. Brihaye and B. Hartmann, JHEP09(2019), 049 doi:10.1007/JHEP09(2019)049 [arXiv:1903.10471 [gr- qc]]

    work page doi:10.1007/jhep09(2019)049 2019

  76. [77]

    Masó-Ferrando, N

    A. Masó-Ferrando, N. Sanchis-Gual, J. A. Font and G. J. Olmo, Class. Quant. Grav.38(2021) no.19, 194003 doi:10.1088/1361-6382/ac1fd0 [arXiv:2103.15705 [gr-qc]]

    work page doi:10.1088/1361-6382/ac1fd0 2021

  77. [78]

    Masó-Ferrando, N

    A. Masó-Ferrando, N. Sanchis-Gual, J. A. Font and G. J. Olmo, Phys. Rev. D109(2024) no.4, 044042 doi:10.1103/PhysRevD.109.044042 [arXiv:2309.14912 [gr-qc]]

    work page doi:10.1103/physrevd.109.044042 2024

  78. [79]

    Ilijić and M

    S. Ilijić and M. Sossich, Phys. Rev. D102 (2020) no.8, 084019 doi:10.1103/PhysRevD.102.084019 [arXiv:2007.12451 [gr-qc]]

    work page doi:10.1103/physrevd.102.084019 2020

  79. [80]

    Baibhav and D

    V. Baibhav and D. Maity, Phys. Rev. D95 (2017) no.2, 024027 doi:10.1103/PhysRevD.95.024027 [arXiv:1609.07225 [gr-qc]]

    work page doi:10.1103/physrevd.95.024027 2017

  80. [81]

    Hartmann, J

    B. Hartmann, J. Riedel and R. Suciu, Phys. Lett. B 726(2013), 906-912 doi:10.1016/j.physletb.2013.09.050 [arXiv:1308.3391 [gr-qc]]

    work page doi:10.1016/j.physletb.2013.09.050 2013

Showing first 80 references.