Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields

A. V. Tugay; V. I. Zhdanov; Yu. V. Taistra

REVIEW 3 major objections 6 minor 1 cited by

Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper claims that static, spherically symmetric, asymptotically flat scalar-field configurations with naked singularities are linearly unstable against radial perturbations when the scalar charge Q is sufficiently small, supporting…

desk verdict A plausible but not yet bulletproof numerical instability claim for a specific class of scalar-field naked singularities; the boundary condition at the singularity is the load-bearing node. read the letter →

arxiv 2507.09677 v1 pith:OPNSAHHM submitted 2025-07-13 gr-qc

A. V. Tugay , V. I. Zhdanov , Yu. V. Taistra This is my paper

classification gr-qc MSC 83C0583C5783C75 PACS 04.20.-q04.20.Dw

keywords cosmiccensorshipnakedsingularitiesscalarfieldslinearstabilitymonopoleperturbationseffectivepotentialshootingmethodasymptoticallyflatspacetimes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that static, spherically symmetric, asymptotically flat solutions of Einstein gravity with a minimally coupled scalar field—solutions that have a naked singularity at the center—are linearly unstable to radial perturbations when the scalar charge is small. The configurations are parameterized uniquely by total mass and a scalar charge $Q$, and the scalar potential is a power law $V(\phi)=V_0\phi^{2n}$ with $n>2$, giving the Coulomb asymptotic $\phi\approx Q/r$ at infinity. By reducing the linearized field equations to one master equation and searching numerically for exponentially growing modes, the paper finds such modes for $Q$ below a threshold $Q_{\rm max}$. The existence of divergent modes would mean these naked-singularity solutions are not stable end-states, which is what the Penrose cosmic censorship conjecture expects. For $Q$ above the threshold no divergent radial modes are found, although the paper does not claim proven stability there.

What carries the argument

The central object is the master equation $\partial^2\Phi/\partial t^2 - \partial^2\Phi/\partial r_*^2 + W_{\rm eff}(r)\Phi = 0$, obtained from the linearized field equations for the single perturbation function $\Phi = r\,\delta\phi$, with $r_*$ the tortoise coordinate defined by $dr_*/dr = e^{(\lambda_0-\nu_0)/2}$. The effective potential $W_{\rm eff}(r)$ is singular near the center, and after the substitution $\Phi\sim e^{-i\omega t}$ the instability problem becomes a Sturm–Liouville eigenvalue problem: one seeks $\Omega = -i\omega > 0$ such that $d^2\Phi/dr_*^2 = (\Omega^2 + W_{\rm eff})\Phi$ has a solution with the regular center behavior $\Phi\approx C_1 r$ and exponential decay at infinity. The numerical detection uses a shooting method that scans $\Omega$ and locates zeros of $A(\Omega) = \lim e^{-\Omega r_*}\Phi(r_*)$, which marks the transition from bounded to growing perturbations.

What would settle it

Re-solve the eigenvalue problem (23) with the general small-$r$ behavior $\Phi = C_1 r + C_2 r\ln r$ and a fixed condition at a small radius $\epsilon$ (for example $\Phi(\epsilon)=0$ or $\Phi'(\epsilon)=0$) for a representative case such as $n=3$, $V_0=1$, $Q=1$: if no $\Omega>0$ eigenvalue survives across a family of such boundary conditions, the claimed instability would not hold as stated. A complementary check is a full nonlinear evolution of a small radial perturbation for the same parameters, which should show exponential growth if the claim is right.

Watch

Extended reading notes

Core claim

The central discovery is that the linearized Einstein–Klein–Gordon system for monopole perturbations of these naked-singularity backgrounds has solutions that grow exponentially in time whenever the scalar charge is sufficiently small. For $V_0=1$ the divergent modes exist for $Q<1.53$ ($n=3$), $Q<2.15$ ($n=4$), and $Q<2.52$ ($n=5$), with analogous thresholds at $V_0=0.1$; the threshold $Q_{\rm max}$ increases with $n$ and depends on $V_0$ in a way that admits a common intersection point. The paper interprets the existence of these divergent modes as linear instability of the configurations, confirming the Penrose cosmic censorship conjecture in this particular case. For large $Q$ no divergent radial modes are found, which the authors present cautiously as motivation for further analysis rather than proof of stability.

Load-bearing premise

The instability conclusion rests on the boundary condition chosen at the naked singularity: the paper keeps only the regular branch $\Phi\approx C_1 r$ and discards the logarithmic branch $\Phi\approx C_2 r\ln r$, without proving that no other physically admissible boundary condition there would remove the growing modes.

Editorial extensions

If this is right

If the claim is correct, small-$Q$ naked-singularity configurations are not viable stationary solutions: any small radial perturbation grows exponentially, so the configuration would either collapse to a black hole, disperse, or settle into another state.
The threshold $Q_{\rm max}(V_0,n)$ provides a concrete boundary in parameter space: below it radial perturbations diverge, and above it this particular instability is absent.
The result complements earlier axial-perturbation analysis of the same systems: axial modes are stable, but monopole (radial) modes can be unstable, so stability verdicts depend on the perturbation channel.
Configurations with $Q > Q_{\rm max}$ remain possibly unstable to polar perturbations or nonlinear effects; the paper explicitly leaves these channels open.
The reduction to a single master equation and the shooting search for $\Omega>0$ modes carry over directly to other scalar potentials and related modified-gravity backgrounds with the same qualitative asymptotics.

Reading between the lines

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

An implication left implicit in the paper is that the exponential growth of monopole perturbations likely drives small-$Q$ configurations toward either black-hole formation or dispersal; a full nonlinear evolution would decide which, connecting the linear instability to dynamical cosmic censorship.
Because the mode spectrum depends on the boundary condition at the naked singularity, a different self-adjoint extension could shift or erase the instability; testing the logarithmic branch of $\Phi$ near $r=0$ is a natural check.
The common intersection of the $Q_{\rm max}(V_0)$ curves for different $n$ around $V_0\approx 0.015$ suggests a universal threshold that might have an analytic explanation from the shape of $W_{\rm eff}$.
The master-equation plus shooting construction could serve as a general numerical stability test for any static spherically symmetric solution with a singular effective potential, giving a practical criterion for naked-singularity instability.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This paper does something useful: it takes the SSS scalar-field configurations with power-law potentials from the authors' prior work and analyzes the one perturbation channel that had been skipped, the radial (monopole) channel. Previous work by the same group had shown axial stability for a broad class; here they derive a master equation for Φ = r δϕ, solve the eigenvalue problem by shooting, and find an exponentially growing mode for small scalar charge Q, with instability dying out above a Qmax that they tabulate. The result is genuinely new and the derivation is coherent. I also want to give credit where due: they do not overclaim. They explicitly caution that the large-Q branch is only 'no detected divergent modes,' not a proof of stability, and they flag the need for polar perturbation analysis.

The soft spots are real, though. The main one is exactly what the stress-test note says. Near r=0, W_eff ≈ -1/(4r_*²), so the radial operator is in the limit-circle case at the singular endpoint. The paper imposes Φ = C1 r, discarding the Φ ~ C2 r ln r branch, and calls it the only possible regular choice, citing earlier works. But it does not prove that. Both branches are square-integrable, and different self-adjoint boundary conditions can shift the spectrum. If a different admissible extension has no negative eigenvalue, the instability claim evaporates. The authors need to address the Ishibashi–Wald prescriptions they cite and show which boundary condition the dynamics picks, or at least show the instability persists for all physically admissible extensions. This is not a minor footnote; it is the hinge of the paper.

The second issue is numerical. The instability is established by shooting, but there are no reported step sizes, convergence tests, or error estimates, and no code. The Qmax values are quoted to two decimals. For a numerical instability claim, that is thin. I would not demand a full reproducibility package for a letter, but I would want the authors to show that the zero of A(Ω) is stable under refinement and is not contaminated by the singular endpoint.

On the positive side, the eigenvalues are computed outputs, not fitted, so there is no circularity in the small-Q instability. The self-citations are relevant and mostly to prior independent publications.

Bottom line: if the boundary-condition question is resolved in the authors' favor, this is a solid contribution to the cosmic censorship literature. Right now it is a well-posed but under-supported numerical result. Send it to a referee with expertise in self-adjoint extensions and singular endpoints; it deserves serious refereeing, not a desk rejection. I would use it as a cautionary example in a reading group about boundary conditions for perturbations of naked singularities.

Referee Report

3 major / 6 minor

Summary. The paper studies linear radial (monopole) stability of static, spherically symmetric, asymptotically flat solutions of Einstein gravity with a minimally coupled scalar field and power-law potential V = V0 phi^{2n} for n = 3, 4, 5. The authors reduce the linearized field equations to a one-dimensional master equation for Phi = r delta_phi, impose the boundary condition Phi = C1 r at the central naked singularity (Eq. (24)) and exponential decay at infinity (Eq. (25)), and solve the resulting eigenvalue problem numerically by shooting. They report unstable modes with Omega > 0 for sufficiently small values of the scalar charge Q, determine Qmax for several n and V0, and interpret this as supporting the Penrose cosmic censorship conjecture. For sufficiently large Q they report finding no divergent modes.

Significance. If the instability result is correct, the paper provides a concrete, nontrivial example in which naked-singularity backgrounds are linearly unstable in the monopole sector, extending the earlier axial-stability analysis of Ref. [18]. The derivation of the master equation is explicit and appears internally consistent, and the numerical scan over n, V0, and Q is a useful first map of the stability regions. The main strengths are the clean reduction of the perturbation problem and the identification of a threshold Qmax. However, two load-bearing issues currently make the central claim conditional rather than established: the boundary condition at the singular endpoint is assumed without proof, and the large-Q stability statement rests on numerical shooting with no convergence data.

major comments (3)

[Section 3, Eq. (24)] The statement that the boundary condition Phi(r) = C1 r as r -> 0 is 'the only possible' choice to guarantee regularity and smallness is not proved. From Eq. (14), r_* ~ r^2/(2 sqrt(Y_f)) near r = 0, and the leading singular term of Eq. (22) gives W_eff ~ -1/(4 r_*^2). For the one-dimensional operator -d^2/dr_*^2 + W_eff, the endpoint r_* = 0 is therefore limit-circle: both Frobenius branches Phi ~ C1 r and Phi ~ C2 r ln r are square-integrable with respect to dr_*, and self-adjointness requires an additional boundary condition. A different admissible self-adjoint extension, including choices within the Ishibashi-Wald prescriptions cited as Refs. [3-6], can shift the spectrum and may remove the negative eigenvalue found for Q < Qmax. The authors need either to prove that regularity and finite-energy criteria single out Eq. (24) or to show that the instability is robust across all admissible boundary conditions. Without this, the central instability claim is conditional on an unproven choice.
[Section 3, Figs. 6-8] The statement that configurations with Q > Qmax are linearly stable with respect to monopole perturbations is stronger than the numerical evidence supports. The manuscript reports no step sizes, no error estimates for the shooting function A(Omega), no convergence tests as the inner boundary r1 -> 0 and the outer boundary -> infinity, and no code or data release. Since the values Qmax ~ 1.53, 2.15, 2.52 and the Qmax(V0) curves in Fig. 8 are obtained from this shooting calculation, the absence of detected divergent modes for large Q is only an absence of detected modes, not a proof. A convergence study and representative A(Omega) curves are needed to support the threshold claim.
[Conclusions, Section 4] The paper's own conclusions contradict the stability statement in Section 3. Section 3 says 'configurations with Q > Qmax are linearly stable with respect to the monopole perturbations,' while the Conclusions say the large-Q result 'should not be interpreted as a definitive evidence of stability yet.' The manuscript should either prove or clearly label the large-Q claim as a numerical absence of unstable modes, and the abstract's phrase 'we have not found divergent modes' should be matched by a consistent wording throughout the text.

minor comments (6)

[Figure 1 caption] The caption labels seem inconsistent with the metric in Eq. (3): it writes 'gtt = e^{lambda0(r)}' and 'grr = e^{nu0(r)}', but Eq. (3) has gtt = e^{nu} and grr = -e^{lambda}. Please correct the caption.
[Section 2, first paragraph] The term 'Schwarzshild-like' is a typo for 'Schwarzschild-like'.
[Section 2, Eq. (15)] The notation nu0(r) = -mu0(r) = -rg/r introduces mu0 without using it later; please either define and use it or drop it.
[Section 3, shooting method] The shooting method is described only schematically; please specify how A(Omega) is evaluated numerically, the integration scheme, and the location of the matching point, so the computation can be reproduced.
[Figure 8] The claimed common intersection point Qmax = 3.9 at V0 = 0.015 is interesting but unexplained; please comment on whether it reflects an analytical property or a numerical coincidence.
[Abstract and Conclusions] The wording 'confirming the well-known Penrose conjecture' is stronger than what a linearized analysis of a particular family of static solutions can establish; 'consistent with' or 'supporting' would be more precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the instability eigenvalues are computed outputs, not fitted inputs, and the background citations are independent ingredients rather than assumptions tailored to produce the result.

full rationale

The derivation chain is: (i) construct the static background from equations (10)-(12) with power-law potential (13), integrating backwards from the Coulomb asymptotic (15) as described in Refs. [20,21]; (ii) reduce radial perturbations to the one-dimensional master equation (20)-(23) with effective potential W_eff from (22); (iii) impose boundary conditions (24)-(25); (iv) use a shooting method to find zeros of A(Omega) in (27). The instability criterion Omega>0 is an eigenvalue of this Sturm-Liouville problem, obtained by solving A(Omega)=0. Nothing in this procedure is fitted to the predicted instability: Q_max is computed, not imposed. The background properties imported from previous work, such as r^2 e^{nu0-lambda0} -> Yf > 0 in (14) and the logarithmic behavior near r=0, are inputs needed to build W_eff; they do not already encode the sign of Omega or the existence of an unstable mode. The boundary condition (24), while physically debatable because the discarded Phi ~ C2 r ln r branch could correspond to a different self-adjoint extension, is an assumption about admissible perturbations, not a circular reduction: equation (24) is not equivalent to the eigenvalue equation (23) nor to the Penrose conjecture. Self-citations to Refs. [18,20,21,22] are used for background construction and methodological precedent, but the central instability claim is verified by the new numerical eigenvalue computation reported in this paper. Therefore no specific circular step can be exhibited, and the paper is not circular in the sense defined here.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numbers are fitted to data; Q, V0, n and rg are physical and scale parameters of the solution family, not fitted constants. The main unproven inputs are the borrowed background properties from [20,21] and the regular boundary condition at the naked singularity. No new particles, fields, or forces are introduced.

assumptions (4)

domain assumption SSS background solutions with V = V0 phi^(2n), n > 2, are uniquely parameterized by total mass M and scalar charge Q and have a naked singularity at r=0 with r^2 e^(nu0-lambda0) tending to a positive constant Y_f.
Borrowed from the authors' earlier work [20,21]. The entire perturbation analysis is built on these background properties, including the singular behavior near the center.
ad hoc to paper Admissible perturbations at the naked singularity satisfy Phi(r) = C1 r as r approaches 0, excluding the Phi ~ C2 r ln r branch.
Stated in Eq. (24) and called regular, following [6,17,18]. No uniqueness theorem is given, and the instability result depends on this boundary condition.
standard math Perturbations are square-integrable at infinity, giving the decaying asymptotic Phi ~ exp(-Omega r*) with Omega > 0.
Standard quantum-mechanical boundary condition for bound modes, stated around Eq. (25). This selects the discrete spectrum that defines instability.
domain assumption The scalar field is minimally coupled with a power-law potential V = V0 phi^(2n) for n=3,4,5.
The model itself. The central claim is restricted to this class, so its validity does not automatically extend to other potentials or non-minimal couplings.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields." pith.science (2026). https://pith.science/paper/OPNSAHHM

@misc{pith2026250709677,
  author       = {Pith},
  title        = {Pith review of: Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPNSAHHM}},
  note         = {Machine review of arXiv:2507.09677}
}

abstract

General relativistic static spherically symmetric (SSS) asymptotically flat configurations with scalar fields typically contain naked singularities at the center. We consider minimally coupled scalar fields with power-law potentials leading to the Coulomb asymptotic of the field $\phi(r)\approx Q/r$ for large values of the radial variable r. The configurations are uniquely defined by total mass and a Q-parameter characterizing the strength of the scalar field at spatial infinity. The focus is on the linear stability against radial (monopole) perturbations of the SSS configurations satisfying conditions of asymptotic flatness. Our numerical investigations show the existence of divergent modes of small perturbations against the static background, at least for sufficiently small values of Q. This means instability of the configurations, confirming the well-known Penrose conjecture about the nonexistence of naked singularities - in this particular case. On the other hand, we have not found divergent modes of linear radial perturbations for sufficiently large Q.

Figures

Figures reproduced from arXiv: 2507.09677 by the authors.

**Figure 2.** Typical dependencies of the background scalar field [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The effective potential Weff (r) for the perturbations in case of n = 3 for the background configuration parameters: Q = 1.5 (blue short dashes), Q = 1 (red long dashes), Q = 0.5 (black solid line).V0 = 1 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The effective potential Weff for the same values of Q as in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 7.** Figure 7: Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: Dependence Qmax(V0) for different power-law exponents in (13) n = 5 (blue short dashes), n = 4 (red long dashes), n = 3 (black solid line). For each n, the unstable modes exist in the Q−V0 plane below the corresponding curve and disappear above it. It is interesting t…

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

On the stability of exceptional Brans-Dicke wormholes
gr-qc 2025-07 conditional novelty 6.0 of 10

Charged exceptional Brans-Dicke wormholes with omega = 0 are stable against radial perturbations; neutral ones are unstable.

Reference graph

Works this paper leans on

29 extracted references · 25 canonical work pages · cited by 1 Pith paper

[18]

However, the stability against polar perturbations in the specific problem considered still needs a detailed analysis

that in a fairly general case axial perturbations lead to the wave equation with a positive effective potential showing linear stability of a background configuration. However, the stability against polar perturbations in the specific problem considered still needs a detailed analysis. Acknowledgements. A.V.T and V.I.Z. acknowledge partial support from sc...

work page 2023
[1]

INTRODUCTION The evolution and structure of space-time singularities present an unsolved problem of classical General Rela- tivity (GR). There is no general answer to the question, except a few exact and/or approximate special solutions, how the singularities move after they are formed, can they evolve into black holes, can they disappear etc. The existin...

arXiv 2025
[2]

e−ν ∂ϕ ∂t 2 + e−λ ∂ϕ ∂r 2 + 2V (ϕ) # , (4) re−λ ∂ν ∂r + e−λ − 1 = = κ 2 r2

SPHERICALL Y SYMMETRIC CONFIGURA TIONS The action of the Einsteinian General Relativity in the presence of a minimally-coupled real SF ϕ is1 S = Z d4 x√−g − R 2κ + 1 2 ∂µϕ∂µϕ − V (ϕ) , (1) where ϕ = ϕ(t, r), κ = 8πG (c = 1). The energy-momentum of scalar field ϕ corresponding to (1) is Tµν = ∂µϕ ∂νϕ − gµν 1 2 ∂αϕ ∂αϕ − V (ϕ) . (2) We work with the spheric...

work page
[3]

tortoise

RADIAL PER TURBA TIONS Let us consider linearized stability against small radial perturbations of the static background by putting ν = ν0(r) +δν (t, r), λ = λ0(r) +δλ(t, r), ϕ = ϕ0(r) +δϕ(t, r). Given SSS solutions ν0(r), λ0(r), ϕ0(r), our aim is to check the existence of divergent modes, i.e. solutions for δν (t, r), δλ(t, r), δϕ(t, r), which satisfy app...

work page
[4]

3 Figure 1

Numerical simulations show that the smaller Q, the higher maximum of eλ0(r) near r ≃ rg and the closer to rg is the region, where the metric is approximately equal to the Schwarzschild one. 3 Figure 1. Typical dependencies of the background metric components upon the radial variable: gtt = eλ0(r) (lower curve, practically the same for all n = 3, 4, 5) and...

work page
[5]

Ishibashi and R

A. Ishibashi and R. M. Wald, Dynamics in non-globally- hyperbolic static spacetimes: II. General analysis of pre- scriptions for dynamics, Classical and Quantum Gravity 20, 3815 (2003), arXiv:gr-qc/0305012 [gr-qc]

arXiv 2003
[6]

The choice n = 3, 4, 5 in (13) ensures the Coulomb asymptotic behavior of the field at spatial infinity

CONCLUSIONS We studied asymptotically flat static spherically sym- metric configurations with power law SF potentials (13) within General Relativity. The choice n = 3, 4, 5 in (13) ensures the Coulomb asymptotic behavior of the field at spatial infinity. According to general results [20] such configuration has a naked singularity at the center. Leav- ing ...

work page
[7]

Christodoulou, Violation of cosmic censorship in the gravitational collapse of a dust cloud, Communications in Mathematical Physics 93, 171 (1984)

D. Christodoulou, Violation of cosmic censorship in the gravitational collapse of a dust cloud, Communications in Mathematical Physics 93, 171 (1984)

work page 1984

Show all 29 references

[8]

Penrose, Gravitational Collapse and Space-Time Sin- gularities, Physical Review Letters 14, 57 (1965)

R. Penrose, Gravitational Collapse and Space-Time Sin- gularities, Physical Review Letters 14, 57 (1965)

1965
[9]

Golden Oldie

R. Penrose, “Golden Oldie”: Gravitational Collapse: The Role of General Relativity, General Relativity and Grav- itation 34, 1141 (2002)

2002
[10]

R. M. Wald, Dynamics in nonglobally hyperbolic, static space-times, Journal of Mathematical Physics 21, 2802 (1980)

1980
[11]

Ishibashi and A

A. Ishibashi and A. Hosoya, Who’s afraid of naked sin- gularities? Probing timelike singularities with finite en- ergy waves, Phys. Rev. D 60, 104028 (1999), arXiv:gr- qc/9907009 [gr-qc]

1999
[12]

M. D. Roberts, Scalar field counterexamples to the cos- mic censorship hypothesis, General Relativity and Grav- itation 21, 907 (1989)

1989
[13]

G. W. Gibbons, S. A. Hartnoll, and A. Ishibashi, On the Stability of Naked Singularities with Negative Mass, Progress of Theoretical Physics 113, 963 (2005), arXiv:hep-th/0409307 [hep-th]

2005 arXiv
[14]

K. A. Bronnikov, S. V. Bolokhov, M. V. Skvortsova, R. Ibadov, and F. Y. Shaymanova, On the stabil- ity of electrovacuum space-times in scalar–tensor grav- ity, European Physical Journal C 84, 1027 (2024), arXiv:2407.12144 [gr-qc]

2024 arXiv
[15]

Ori and T

A. Ori and T. Piran, Naked singularities in self-similar spherical gravitational collapse, Physical Review Letters 59, 2137 (1987)

1987
[16]

P. S. Joshi and I. H. Dwivedi, Naked singularities in spherically symmetric inhomogeneous Tolman-Bondi dust cloud collapse, Physical Review D 47, 5357 (1993)

1993
[17]

P. S. Joshi, D. Malafarina, and R. Narayan, Distinguish- ing black holes from naked singularities through their ac- cretion disc properties, Classical and Quantum Gravity 31, 015002 (2013)

2013
[19]

Y. C. Ong, Space–time singularities and cosmic censor- ship conjecture: A Review with some thoughts, Interna- tional Journal of Modern Physics A 35, 2030007 (2020)

2020
[20]

scalar charge

for a wide family of the scalar field potentials, which includes the power-law case V (ϕ) = V0ϕ2n . (13) We consider (13) with n >2, which corresponds to the Coulomb asymptotic of the field for large values of radial variable (see, e.g., [20], Appendix 1). When considering the...
[21]

K. A. Bronnikov, J. C. Fabris, and A. Zhidenko, On the stability of scalar-vacuum space-times, European Physi- cal Journal C 71, 1791 (2011), arXiv:1109.6576 [gr-qc]

2011 arXiv
[22]

K. A. Bronnikov and A. V. Khodunov, Scalar field and gravitational instability, General Relativity and Gravita- tion 11, 13 (1979)

1979
[23]

Jetzer and D

P. Jetzer and D. Scialom, Dynamical instability of the static real scalar field solutions to the Einstein-Klein- Gordon equations, Physics Letters A 169, 12 (1992)

1992
[24]

M. A. Clayton, L. Demopoulos, and J. L´ egar´ e, The dy- namical stability of the static real scalar field solutions to the Einstein-Klein-Gordon equations revisited, Physics Letters A 248, 131 (1998), arXiv:gr-qc/9809014 [gr-qc]

1998 arXiv
[25]

O. S. Stashko, O. V. Savchuk, and V. I. Zhdanov, Quasi- normal modes of naked singularities in presence of non- linear scalar fields, Phys. Rev. D 109, 024012 (2024), arXiv:2307.04295 [gr-qc]

2024 arXiv
[26]

L. D. Landau and E. M. Lifshitz, COURSE OF THE- ORETICAL PHYSICS. V. 2: Field Theory.(Pergamon Press, 1971)

1971
[27]

V. I. Zhdanov and O. S. Stashko, Static spherically symmetric configurations with n nonlinear scalar fields: Global and asymptotic properties, Phys. Rev. D 101, 064064 (2020)

2020
[28]

O. S. Stashko, V. I. Zhdanov, and A. N. Alexandrov, Thin accretion discs around spherically symmetric con- figurations with nonlinear scalar fields, Phys. Rev. D104, 104055 (2021), arXiv:2107.05111 [gr-qc]

2021 arXiv
[29]

V. I. Zhdanov, O. S. Stashko, and Y. V. Shtanov, Spherically symmetric configurations in the quadratic f (R ) gravity, Phys. Rev. D 110, 024056 (2024), arXiv:2403.16741 [gr-qc]

2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [18]

However, the stability against polar perturbations in the specific problem considered still needs a detailed analysis

that in a fairly general case axial perturbations lead to the wave equation with a positive effective potential showing linear stability of a background configuration. However, the stability against polar perturbations in the specific problem considered still needs a detailed analysis. Acknowledgements. A.V.T and V.I.Z. acknowledge partial support from sc...

work page 2023

[2] [1]

INTRODUCTION The evolution and structure of space-time singularities present an unsolved problem of classical General Rela- tivity (GR). There is no general answer to the question, except a few exact and/or approximate special solutions, how the singularities move after they are formed, can they evolve into black holes, can they disappear etc. The existin...

arXiv 2025

[3] [2]

e−ν ∂ϕ ∂t 2 + e−λ ∂ϕ ∂r 2 + 2V (ϕ) # , (4) re−λ ∂ν ∂r + e−λ − 1 = = κ 2 r2

SPHERICALL Y SYMMETRIC CONFIGURA TIONS The action of the Einsteinian General Relativity in the presence of a minimally-coupled real SF ϕ is1 S = Z d4 x√−g − R 2κ + 1 2 ∂µϕ∂µϕ − V (ϕ) , (1) where ϕ = ϕ(t, r), κ = 8πG (c = 1). The energy-momentum of scalar field ϕ corresponding to (1) is Tµν = ∂µϕ ∂νϕ − gµν 1 2 ∂αϕ ∂αϕ − V (ϕ) . (2) We work with the spheric...

work page

[4] [3]

tortoise

RADIAL PER TURBA TIONS Let us consider linearized stability against small radial perturbations of the static background by putting ν = ν0(r) +δν (t, r), λ = λ0(r) +δλ(t, r), ϕ = ϕ0(r) +δϕ(t, r). Given SSS solutions ν0(r), λ0(r), ϕ0(r), our aim is to check the existence of divergent modes, i.e. solutions for δν (t, r), δλ(t, r), δϕ(t, r), which satisfy app...

work page

[5] [4]

3 Figure 1

Numerical simulations show that the smaller Q, the higher maximum of eλ0(r) near r ≃ rg and the closer to rg is the region, where the metric is approximately equal to the Schwarzschild one. 3 Figure 1. Typical dependencies of the background metric components upon the radial variable: gtt = eλ0(r) (lower curve, practically the same for all n = 3, 4, 5) and...

work page

[6] [5]

Ishibashi and R

A. Ishibashi and R. M. Wald, Dynamics in non-globally- hyperbolic static spacetimes: II. General analysis of pre- scriptions for dynamics, Classical and Quantum Gravity 20, 3815 (2003), arXiv:gr-qc/0305012 [gr-qc]

arXiv 2003

[7] [6]

The choice n = 3, 4, 5 in (13) ensures the Coulomb asymptotic behavior of the field at spatial infinity

CONCLUSIONS We studied asymptotically flat static spherically sym- metric configurations with power law SF potentials (13) within General Relativity. The choice n = 3, 4, 5 in (13) ensures the Coulomb asymptotic behavior of the field at spatial infinity. According to general results [20] such configuration has a naked singularity at the center. Leav- ing ...

work page

[8] [7]

Christodoulou, Violation of cosmic censorship in the gravitational collapse of a dust cloud, Communications in Mathematical Physics 93, 171 (1984)

D. Christodoulou, Violation of cosmic censorship in the gravitational collapse of a dust cloud, Communications in Mathematical Physics 93, 171 (1984)

work page 1984