Pith. sign in

REVIEW 3 major objections 4 minor 33 references

Stable scalarized multi-horizon black holes exist only inside a window of primary scalar charge.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-14 10:27 UTC pith:XMKUDZRK

load-bearing objection Solid multi-horizon extension of EEHS scalarization that produces a clean intermediate qs window for L/C branches, but the window rests on a radial s-mode time-domain proxy and an ADM-mass cut whose necessity is not fully argued. the 3 major comments →

arxiv 2607.10614 v1 pith:XMKUDZRK submitted 2026-07-12 gr-qc

Scalarization of Einstein-Euler-Heisenberg black hole with multiple horizons

Hong Guo , Yun Soo Myung This is my paper
classification gr-qc
keywords scalarizationEinstein-Euler-Heisenbergmultiple horizonsprimary scalar chargetime-domain stabilitytachyonic instabilitynonlinear electrodynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies how a quadratic coupling of a scalar field to the Maxwell term can grow hair on magnetically charged Einstein-Euler-Heisenberg black holes that possess multiple horizons. For fixed mass and a small Euler-Heisenberg parameter, four horizon families appear (low, cold, negative, hot), and triple horizons occupy a narrow charge band. Onset analysis around the low, cold, and hot horizons produces infinite branches of scalarized solutions; the three fundamental branches are constructed explicitly. Positivity of the ADM mass then imposes upper bounds on the primary scalar charge for the low and cold families, which in turn restrict the allowed Hawking temperature and entropy. Time-domain evolution of radial scalar perturbations shows that instabilities occur only at small scalar charge. The net result is that dynamically stable and mass-positive scalarized black holes occupy an intermediate window of scalar charge for the low and cold horizons, and a lower bound for the hot horizon.

Core claim

In the multi-horizon regime of the Einstein-Euler-Heisenberg-scalar theory with quadratic Maxwell coupling, the fundamental branches of scalarized low-, cold-, and hot-horizon black holes are dynamically stable only above a critical primary scalar charge, while the low and cold families further require the scalar charge to remain below an upper bound set by positivity of the ADM mass; thus viable solutions live in an intermediate window (or a lower bound for the hot family).

What carries the argument

The time-domain evolution of the radial s-mode scalar perturbation on the scalarized background, together with the positivity cut on ADM mass as a function of primary scalar charge qs, which together carve out the intermediate window of viability.

Load-bearing premise

That radial s-mode time-domain decay plus positivity of ADM mass are enough to certify a black hole as physically viable, without controlling non-radial modes or full quasinormal spectra.

What would settle it

A computation of the full quasinormal spectrum (or a non-radial mode analysis) on the same scalarized low- or cold-horizon solutions that finds an unstable mode inside the claimed intermediate window of primary scalar charge.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies spontaneous scalarization of magnetically charged Einstein-Euler-Heisenberg black holes that possess multiple horizons (low, cold, negative, hot) for M=1 and μ=0.03. A quadratic coupling g(φ)=1-αφ^{2} is introduced only to the Maxwell term. Onset analysis (sufficient condition I_i<0, WKB bounds, and static scalar clouds) indicates infinite branches for the low, cold and hot horizons; the three fundamental (n=0) branches are then constructed numerically. Positivity of the ADM mass M_i(q_s) supplies upper bounds on primary scalar charge for the low and cold families, while time-domain evolution of a radial s-mode Gaussian packet under the Schrödinger-type equation with potential V_{s,i} shows that instabilities appear only at small q_s. The authors conclude that stable, physically viable scalarized solutions occupy an intermediate window of q_s for low/cold horizons and a lower bound for the hot horizon.

Significance. The multi-horizon EEHBH geometry (μ≤0.08) is a natural but previously unexplored arena for charge-induced scalarization. The numerical construction of three distinct fundamental branches, the mass-positivity cut that truncates the large-q_s tails of the L and C families, and the time-domain maps that reveal an unstable-to-stable transition within each fundamental branch are new results relative to the authors’ earlier single-horizon studies. If the intermediate-window claim survives more complete stability tests, it supplies a concrete, falsifiable prediction for the allowed range of primary scalar charge in nonlinear-electrodynamics scalarization models.

major comments (3)
  1. §6, Eqs. (41)–(43) and Figs. 10–15: Dynamical stability of the fundamental branches is inferred solely from late-time decay of a radial (l=0) s-mode Gaussian packet. No complex QNM frequencies are extracted, non-radial multipoles are omitted, and metric/Maxwell perturbations are not coupled. The authors themselves note that the frequencies appear purely imaginary at small q_s and that full QNM extraction is “numerically challenging” (§6.3). Because the intermediate-window claim rests on this proxy, at least a partial QNM spectrum (or an explicit statement of its limitations) is required before the stability conclusion can be regarded as robust.
  2. §5, Fig. 8 and asymptotic expansion (37): The upper bounds q_s < q_s^u for the L and C branches are imposed by hand once the ADM mass M_i becomes negative. While M_i>0 is a natural physical requirement, the manuscript supplies no thermodynamic or asymptotic argument that negative-mass solutions must be discarded once the field equations and asymptotic flatness are already satisfied. Clarifying the status of this cut (or showing that other viability criteria independently exclude the same tail) is needed to keep the intermediate-window claim well-defined.
  3. §3 and Fig. 6: The WKB integrals that define α_in for the hot and negative horizons are declared “not properly defined” because f_i(r)<0 immediately outside those horizons. Consequently the cold horizon is adopted as the sole representative of the triple-horizon band. A more quantitative discussion of whether scalar clouds can still be constructed on the hot/negative backgrounds (or an explicit demonstration that they cannot) would strengthen the claim that only three families admit infinite branches.
minor comments (4)
  1. Notation for the four horizon families (r_L, r_C, r_N, r_H) is introduced in §2 but the thermodynamic labels “low/cold/negative/hot” are used interchangeably with the subscripts; a single consistent glossary would help the reader.
  2. Figs. 10–15 display many overlapping curves; a clearer legend or a tabular summary of the critical q_s values that separate unstable from stable regimes would improve readability.
  3. The coupling is restricted to the Maxwell term only; a brief remark on why the NED term F^{2} is left uncoupled (beyond the statement that it is the choice made) would be useful for comparison with Refs. [18,19].
  4. Several self-citations to the authors’ single-horizon EEHS papers appear; ensuring that the multi-horizon novelty is stated explicitly in the introduction would help the reader locate the new contribution.

Circularity Check

0 steps flagged

No significant circularity: multi-horizon scalarized solutions, mass bounds, and time-domain stability windows are independent numerical outputs from the EEHS equations, not forced by redefinition or self-citation chains.

full rationale

The derivation chain begins from the EEHS action (1) with quadratic coupling g(φ)=1-αφ^{2}, obtains the bald multi-horizon EEHBH metric (8) for µ=0.03, performs linearized onset analysis via the effective potential (20)–(22) and WKB/static eigenvalue problems to locate bifurcation points α_th, then constructs the n=0 scalarized branches by shooting the full nonlinear system (30)–(32) subject to asymptotic flatness (37). Upper bounds q_s^u on primary scalar charge follow directly from the numerical observation that ADM mass M_i(q_s) extracted from (37) becomes negative (Fig. 8); dynamical windows follow from time-domain integration of the radial s-mode Schrödinger equation (41) with potential (43). These are free-parameter numerical outputs (M=1, µ=0.03, q=0.5/1/2, α free). Self-citations to the authors’ single-horizon papers supply background methods and contrast, but the intermediate q_s windows themselves are not imported, fitted, or definitionally equivalent to any prior result. No uniqueness theorem, ansatz, or fitted constant is smuggled in as a prediction. The analysis is therefore self-contained against its own equations.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The central claim rests on the EEHS action with a chosen quadratic Maxwell coupling, fixed background parameters (M, μ, q), the spontaneous-scalarization onset machinery, numerical shooting for n=0 branches, mass positivity as a physical cut, and s-mode time-domain stability as a proxy for dynamical viability. No new particles or forces are postulated; the scalar hair and horizon taxonomy are solution features within an existing theory class.

free parameters (4)
  • Euler-Heisenberg parameter μ
    Fixed by hand to μ=0.03 to place the system in the multi-horizon regime (μ≤0.08); all horizon families and windows depend on this choice.
  • Black-hole mass M
    Set to M=1 throughout; scales the charge bands and mass-positivity bounds.
  • Magnetic charges q (0.5, 1, 2)
    Representative values chosen for L, C, H families; results are not scanned continuously over all allowed q.
  • Coupling constants α_i and horizon scalars ϕ0,i
    α and ϕ0 are free inputs used to generate branches and profiles (e.g. ϕ0=0.5, α_L=50, α_C=5, α_H=0.7 for thermo plots); windows are reported in qs but depend on these choices.
axioms (5)
  • ad hoc to paper Einstein-Euler-Heisenberg-scalar action with g(ϕ)=1−αϕ² coupling only to the Maxwell term F, not to the NED F² term.
    Action (1); choice of quadratic Maxwell-only coupling defines the model and differs from exponential or dual-coupling cases in cited prior work.
  • domain assumption Spontaneous scalarization via tachyonic effective mass m²_eff=−α q²/r⁴ near the horizon with asymptotically vanishing scalar.
    Standard EMS-type mechanism used in §3 linearized analysis; assumed to generate infinite branches from scalar clouds.
  • domain assumption ADM mass positivity is required for physical viability of scalarized solutions.
    §5 uses Mi(qs)<0 to define upper bounds qu_s,i for L and C; this cut is load-bearing for the intermediate window.
  • domain assumption Radial s-mode time-domain stability is a sufficient proxy for dynamical viability of the fundamental branches.
    §6; QNMs not computed; non-radial modes not analyzed.
  • standard math Asymptotically flat magnetically charged EEHBH metric (8) and standard thermodynamic definitions of T and horizon area.
    Background solution from Yajima-Tamaki and subsequent literature; used throughout.
invented entities (1)
  • Four named horizon families (low, cold, negative, hot) of the multi-horizon EEHBH independent evidence
    purpose: Taxonomy of roots of f(r)=0 used to organize onset, solutions, and stability windows.
    Naming is interpretive (from temperature/heat capacity signs) rather than a new physical object; independent evidence is the metric roots themselves.

pith-pipeline@v1.1.0-grok45 · 27009 in / 3217 out tokens · 34376 ms · 2026-07-14T10:27:26.377304+00:00 · methodology

0 comments
read the original abstract

Scalarizations of the Einstein-Euler-Heisenberg (EEH) black hole (EEHBH) with multiple horizons are investigated in the EEH-scalar theory by introducing a quadratic scalar coupling to the Maxwell term. For mass $M=1$ and Euler-Heisenberg parameter $\mu=0.03$, the magnetically charged EEHBH admits four horizon families (low, cold, negative, and hot), with triple horizons appearing in the narrow band of magnetic charge $q\in[0.95,1.0065]$. The onset scalarization around the low, cold, and high horizons is then analyzed for the magnetic charge $q=0.5,\,1,\,2$, implying infinite branches of scalarized black holes for each case. We construct the three fundamental branches of scalarized black holes. From the positivity condition of their mass, we find the upper bounds on primary scalar charges $q_{s}$ for scalarized low and cold horizons. These bounds determine the allowable regions for the Hawking temperature and entropy. Furthermore, we perform a time-domain stability analysis and find that the instabilities arise only at small scalar charge regime. Therefore, stable and physically viable scalarized black holes exist in an intermediate window of the primary scalar charge for low and cold horizon solutions and a lower bound for hot horizon solution.

Figures

Figures reproduced from arXiv: 2607.10614 by Hong Guo, Yun Soo Myung.

Figure 1
Figure 1. Figure 1: (Left) Solution to f(r, M = 1, q, 0.03) = 0 with q = 0.5, 1, 2. For q = 1, one finds three real roots, implying three horizons. (Right) Four horizons rL(M = 1, qL, µ), rC(1, qC, µ), rN (1, qN , µ), rH(1, qH, µ) as functions of q with µ = 0.03. rRN (M = 1, q ∈ [0, 1]) represents the outer horizon for RN black hole, while rRN−(1, q) denotes its inner horizon. The shaded region denotes a narrow region of q ∈ … view at source ↗
Figure 2
Figure 2. Figure 2: (Left) Three reduced temperatures ti(M = 1, qi , µ = 0.03) for i = L, C, H with tRN (1, q). A dotted line denotes a line at q = 0.95 which three horizons meet. (Right) Three reduced heat capacities ci(M = 1, qi , µ = 0.03) for i = L, C, H with cRN (1, q). It includes two Davies points [dashed lines at q = 0.866(RN), 0871(L)] and two extremal points with tC(1, 1.0065) = cC(1, 1.0065) = 0 and tRN (1, 1) = cR… view at source ↗
Figure 3
Figure 3. Figure 3: Scalar potential ViEEH(r, M = 1, q, α) and its integration Ii for l = 0-scalar mode. (Left) qL = 0.5 for r ∈ [rL = 1.87, 10]. Here, α L takes 18.4, 19.51, 21. (Right) qH = 2 for r ∈ [rH = 0.493, 10]. Here, α H is chosen as 0.3, 0.4635, 0.6. VCEEH(r,1,1,1.2)→IC=0.02 VCEEH(r,1,1,1.3)→IC=-0.001 VCEEH(r,1,1,1.4)→IC=-0.03 1 2 5 10 -0.02 -0.01 0.00 0.01 0.02 r VNEEH(r,1,1,0.5)→IN=0.006 VNEEH(r,1,1,0.61)→IN=-0.00… view at source ↗
Figure 4
Figure 4. Figure 4: Three Scalar potentials ViEEH(r, M = 1, q = 1, αi ) and their integration Ii for l = 0-scalar mode in the narrow band of q(= 1) ∈ [0.95, 1.0065]. (Left) r ∈ [rC = 1.092, 10]. (Middle) r ∈ [rN = 0.8475, 10]. (Right) r ∈ [rH = 0.4438, 10]. where the s(l = 0)-mode potential is given by VEEH(r, M, q, α) = f(r) h 2M r 3 − 2q 2 r 4 + 12µq4 5r 8 + m2 effi . (20) Now we may introduce the scalar potentials around i… view at source ↗
Figure 5
Figure 5. Figure 5: (Left) Sufficient conditions for instability [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Left) Negative region of the metric function [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Radial profile of scalar field ϕi(r), metric functions Ni(r) and δi(r) with ϕ0,i = 0.5 and seven coupling constants α i for each i = L, C, H. From top to bottom, they corre￾spond to the scalarized L-horizon (qL = 0.5, qs,L = 0.6, 0.65, 0.68, 0.7, 0.74, 0.87, 0.97), C￾horizon (qC = 1, qs,C = 0.19, 0.22, 0.28, 0.34, 0.38, 0.47, 0.54), and H-horizon (qH = 2, qs,H = 0.07, 0.15, 0.27, 0.34, 0.43, 0.49, 0.56) so… view at source ↗
Figure 8
Figure 8. Figure 8: (left) Black hole mass Mi , (middle) Hawking temperature Ti , and (right) horizon area ah,i as functions of the scalar charge qs,i for L-horizon (qL = 0.5, αL = 50), C-horizon (qC = 1, αC = 5), and H-horizon (qH = 2, αH = 0.7) solutions, respectively. 5 Thermodynamic analysis In this section, we proceed to analyze the thermodynamic behavior of the hairy black hole solutions in different qi regions. The Haw… view at source ↗
Figure 9
Figure 9. Figure 9: Horizon radius ri (left) and horizon scalar ϕ0,i (right) as functions of the scalar charge qs,i for L-horizon (q = 0.5, α = 50), C-horizon (q = 1, α = 5), and H-horizon (q = 2, α = 0.7) solutions, respectively. The black dotted horizontal line in the right figure represents ϕ0,i = 0.5 at qs = qs,L = 0.97, qs,C = 0.54, and qs,H = 0.15 as we discuss before. Ti increases monotonically with increasing scalar c… view at source ↗
Figure 10
Figure 10. Figure 10: Profile of the effective potential Vs,H and corresponding time evolution of the scalar perturbation for scalarized H-horizon solutions for α = 0.7 (top) and α = 2 (bottom) with seven small horizon scalars ϕ0,H. a transition from unstable to stable configurations and a critical value q crit s,H can be inferred from the constant late-time behavior of the scalar perturbation. In the large-qs,H regime shown i… view at source ↗
Figure 11
Figure 11. Figure 11: Profile of the effective potential Vs,H and corresponding time evolution of the scalar perturbation for scalarized H-horizon solutions for α = 0.7 (top) and α = 2 (bottom) with seven large horizon scalars ϕ0,H. critical value of the horizon scalar ϕ0,C separating the unstable and stable regimes is shifted to larger values. In the large-qs,C regime, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p019… view at source ↗
Figure 12
Figure 12. Figure 12: Profile of the effective potential Vs,C and corresponding time evolution of the scalar perturbation for scalarized C-horizon solutions for α = 1.5 (top) and α = 5 (bottom) with seven small horizon scalars ϕ0,C. in the small-qs,i regime. For the H-horizon case, when α is relatively small, the black hole solutions remain stable at small qs,i, and increasing α triggers unstable modes in the small-qs,i regime… view at source ↗
Figure 13
Figure 13. Figure 13: Profile of the effective potential Vs,C and corresponding time evolution of the scalar perturbation for scalarized C-horizon solution for α = 1.5 (top) and α = 5 (bottom) with seven large horizon scalars ϕ0,C. known analyses. We note that for the EMS theory with quartic coupling, the cold branch with smaller scalar charge of nonlinear scalarized black holes is unstable, while the hot branch with larger sc… view at source ↗
Figure 14
Figure 14. Figure 14: Profile of the effective potential Vs,L and corresponding time evolution of the scalar perturbation for scalarized L-horizon solution for α = 20 (top) and α = 50 (bottom) with seven small horizon scalars ϕ0,L. these black hole solutions remain stable under scalar perturbations. We stress that these two analyses are completely independent and probe different notions of viability. In this sense, the two ana… view at source ↗
Figure 15
Figure 15. Figure 15: Profile of the effective potential Vs,L and corresponding time evolution of the scalar perturbation for scalarized L-horizon solution for α = 20 (top) and α = 50 (bottom) with seven large horizon scalars ϕ0,L. quasinormal mode (QNM) frequencies are purely imaginary. As qs,i increases further, one observes that oscillatory behavior gradually emerges, leading to complex QNM frequencies. Therefore, the time-… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Ruffini and J

    R. Ruffini and J. A. Wheeler, Phys. Today24, no. 1, 30 (1971). doi:10.1063/1.3022513

    doi:10.1063/1.3022513 1971

  2. [2]

    J. D. Bekenstein, Phys. Rev. Lett.28, 452-455 (1972) doi:10.1103/PhysRevLett.28.452

    work page doi:10.1103/physrevlett.28.452 1972

  3. [3]

    J. D. Bekenstein, Phys. Rev. D5, 1239-1246 (1972) doi:10.1103/PhysRevD.5.1239

    doi:10.1103/physrevd.5.1239 1972

  4. [4]

    C. A. R. Herdeiro and E. Radu, Int. J. Mod. Phys. D24, no. 09, 1542014 (2015) doi:10.1142/S0218271815420146 [arXiv:1504.08209 [gr-qc]]

    Pith/arXiv arXiv doi:10.1142/s0218271815420146 2015

  5. [5]

    J. D. Bekenstein, Phys. Rev. D51, no.12, R6608 (1995) doi:10.1103/PhysRevD.51.R6608

    work page doi:10.1103/physrevd.51.r6608 1995

  6. [6]

    D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett.120, no.13, 131103 (2018) doi:10.1103/PhysRevLett.120.131103 [arXiv:1711.01187 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevlett.120.131103 2018

  7. [7]

    H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Phys. Rev. Lett.120, no.13, 131104 (2018) doi:10.1103/PhysRevLett.120.131104 [arXiv:1711.02080 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevlett.120.131104 2018

  8. [8]

    Antoniou, A

    G. Antoniou, A. Bakopoulos and P. Kanti, Phys. Rev. Lett.120, no.13, 131102 (2018) doi:10.1103/PhysRevLett.120.131102 [arXiv:1711.03390 [hep-th]]

    Pith/arXiv arXiv doi:10.1103/physrevlett.120.131102 2018

  9. [9]

    C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual and J. A. Font, Phys. Rev. Lett.121, no.10, 101102 (2018) doi:10.1103/PhysRevLett.121.101102 [arXiv:1806.05190 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevlett.121.101102 2018

  10. [10]

    Y. S. Myung and D. C. Zou, Eur. Phys. J. C79, no.3, 273 (2019) doi:10.1140/epjc/s10052-019-6792-6 [arXiv:1808.02609 [gr-qc]]

    Pith/arXiv arXiv doi:10.1140/epjc/s10052-019-6792-6 2019

  11. [11]

    Heisenberg and H

    W. Heisenberg and H. Euler, Z. Phys.98, no.11-12, 714-732 (1936) doi:10.1007/BF01343663 [arXiv:physics/0605038 [physics]]

    Pith/arXiv arXiv doi:10.1007/bf01343663 1936

  12. [12]

    Y. N. Obukhov and G. F. Rubilar, Phys. Rev. D66, 024042 (2002) doi:10.1103/PhysRevD.66.024042 [arXiv:gr-qc/0204028 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.66.024042 2002

  13. [13]

    D. P. Sorokin, Fortsch. Phys.70(2022) no.7-8, 2200092 doi:10.1002/prop.202200092 [arXiv:2112.12118 [hep-th]]

    Pith/arXiv arXiv doi:10.1002/prop.202200092 2022

  14. [14]

    Yajima and T

    H. Yajima and T. Tamaki, Phys. Rev. D63, 064007 (2001) doi:10.1103/PhysRevD.63.064007 [arXiv:gr-qc/0005016 [gr-qc]]. 26

    Pith/arXiv arXiv doi:10.1103/physrevd.63.064007 2001

  15. [15]

    Y. S. Myung, [arXiv:2503.18239 [gr-qc]]

    Pith/arXiv arXiv

  16. [16]

    Karakasis, G

    T. Karakasis, G. Koutsoumbas, A. Machattou and E. Papantonopoulos, Phys. Rev. D106, no.10, 104006 (2022) doi:10.1103/PhysRevD.106.104006 [arXiv:2207.13146 [gr- qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.106.104006 2022

  17. [17]

    H. Guo, M. Park and Y. S. Myung, Chin. Phys. C50(2026) no.6, 065102 doi:10.1088/1674-1137/ae457c [arXiv:2508.16083 [gr-qc]]

    Pith/arXiv arXiv doi:10.1088/1674-1137/ae457c 2026

  18. [18]

    Zhang, D

    L. Zhang, D. C. Zou and Y. S. Myung, Eur. Phys. J. C85, no.12, 1463 (2025) doi:10.1140/epjc/s10052-025-15232-4 [arXiv:2510.07954 [gr-qc]]

    doi:10.1140/epjc/s10052-025-15232-4 2025

  19. [19]

    Zhang, Y

    L. Zhang, Y. S. Myung, D. C. Zou and C. M. Zhang, [arXiv:2605.15566 [gr-qc]]

    Pith/arXiv arXiv

  20. [20]

    Amaro and A

    D. Amaro and A. Mac´ ıas, Phys. Rev. D102, no.10, 104054 (2020) doi:10.1103/PhysRevD.102.104054

    doi:10.1103/physrevd.102.104054 2020

  21. [21]

    Bret´ on and L

    N. Bret´ on and L. A. L´ opez, Phys. Rev. D104, no.2, 024064 (2021) doi:10.1103/PhysRevD.104.024064 [arXiv:2105.12283 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.104.024064 2021

  22. [22]

    Allahyari, M

    A. Allahyari, M. Khodadi, S. Vagnozzi and D. F. Mota, JCAP02, 003 (2020) doi:10.1088/1475-7516/2020/02/003 [arXiv:1912.08231 [gr-qc]]

    Pith/arXiv arXiv doi:10.1088/1475-7516/2020/02/003 2020

  23. [23]

    Gao, Phys

    C. Gao, Phys. Rev. D104, no.6, 064038 (2021) doi:10.1103/PhysRevD.104.064038 [arXiv:2106.13486 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.104.064038 2021

  24. [24]

    J. L. Bl´ azquez-Salcedo, C. A. R. Herdeiro, J. Kunz, A. M. Pombo and E. Radu, Phys. Lett. B806, 135493 (2020) doi:10.1016/j.physletb.2020.135493 [arXiv:2002.00963 [gr- qc]]

    doi:10.1016/j.physletb.2020.135493 2020

  25. [25]

    Quasinormal modes of hot, cold and bald Einstein-Maxwell-scalar black holes

    J. Luis Bl´ azquez-Salcedo, C. A. R. Herdeiro, S. Kahlen, J. Kunz, A. M. Pombo and E. Radu, Eur. Phys. J. C81, no.2, 155 (2021) doi:10.1140/epjc/s10052-021-08952-w [arXiv:2008.11744 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-021-08952-w 2021

  26. [26]

    Luo and J

    Z. Luo and J. Li, Chin. Phys. C46, no.8, 085107 (2022) doi:10.1088/1674-1137/ac6574

    doi:10.1088/1674-1137/ac6574 2022

  27. [27]

    Dotti and R

    G. Dotti and R. J. Gleiser, Class. Quant. Grav.22, L1 (2005) doi:10.1088/0264- 9381/22/1/L01 [arXiv:gr-qc/0409005 [gr-qc]]. 27

    Pith/arXiv arXiv doi:10.1088/0264- 2005

  28. [28]

    Hod, Phys

    S. Hod, Phys. Lett. B798(2019), 135025 doi:10.1016/j.physletb.2019.135025 [arXiv:2002.01948 [gr-qc]]

    doi:10.1016/j.physletb.2019.135025 2019

  29. [29]

    G. Guo, P. Wang, H. Wu and H. Yang, Eur. Phys. J. C81(2021) no.10, 864 doi:10.1140/epjc/s10052-021-09614-7 [arXiv:2102.04015 [gr-qc]]

    Pith/arXiv arXiv doi:10.1140/epjc/s10052-021-09614-7 2021

  30. [30]

    J. L. Bl´ azquez-Salcedo, D. D. Doneva, J. Kunz and S. S. Yazadjiev, Phys. Rev. D98 (2018) no.8, 084011 doi:10.1103/PhysRevD.98.084011 [arXiv:1805.05755 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.98.084011 2018

  31. [31]

    H. O. Silva, C. F. B. Macedo, T. P. Sotiriou, L. Gualtieri, J. Sakstein and E. Berti, Phys. Rev. D99(2019) no.6, 064011 doi:10.1103/PhysRevD.99.064011 [arXiv:1812.05590 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.99.064011 2019

  32. [32]

    C. F. B. Macedo, J. Sakstein, E. Berti, L. Gualtieri, H. O. Silva and T. P. Sotiriou, Phys. Rev. D99(2019) no.10, 104041 doi:10.1103/PhysRevD.99.104041 [arXiv:1903.06784 [gr-qc]]

    Pith/arXiv arXiv doi:10.1103/physrevd.99.104041 2019

  33. [33]

    Y. S. Myung and D. C. Zou, Eur. Phys. J. C79(2019) no.8, 641 doi:10.1140/epjc/s10052-019-7176-7 [arXiv:1904.09864 [gr-qc]]. 28

    Pith/arXiv arXiv doi:10.1140/epjc/s10052-019-7176-7 2019