Non-equatorial scalar rings supported by rapidly spinning Gauss-Bonnet black holes

Shahar Hod

REVIEW 3 major objections 3 minor 44 references

Non-equatorial scalar rings supported by rapidly spinning Gauss-Bonnet black holes

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Rapidly rotating Kerr black holes in Einstein-Gauss-Bonnet theory can support thin non-equatorial rings of a massive scalar field when the dimensionless spin exceeds about 0.78.

desk verdict A sharp algebraic observation about where the Kerr Gauss-Bonnet invariant is minimal, wrapped in an overclaimed proof of non-equatorial scalar rings. read the letter →

arxiv 2412.15332 v1 pith:UVEZU2I2 submitted 2024-12-19 gr-qc astro-ph.HEhep-th

Shahar Hod This is my paper · ORCID

classification gr-qcastro-ph.HEhep-th PACS 04.70.-s04.50.Kd

keywords blackholesscalarizationGauss-BonnetcouplingKerrspacetimescalarringsnon-equatorialmattereffectivemassno-hairtheorem

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 claims that rapidly rotating Kerr black holes in Einstein-Gauss-Bonnet theory can support pairs of thin non-equatorial rings made of a massive scalar field that is negatively coupled to the Gauss-Bonnet invariant. The proof works by analyzing the effective mass squared, $\mu_{\rm eff}^2 = \mu^2 - \eta G_{\rm Kerr}$, and showing that for dimensionless spin $\bar a > \bar a_{\rm crit} \simeq 0.78$ the Gauss-Bonnet invariant has a negative minimum at an off-equatorial polar angle near the horizon. Tuning the scalar mass $\mu$ and the negative coupling $\eta$ so that $-\eta/\mu^2$ approaches a spin-dependent critical value makes $\mu_{\rm eff}^2$ negative inside a narrow ring, which is interpreted as the onset of spontaneous scalarization. If correct, this establishes a new class of black-hole matter configurations and gives an analytic existence line marking where bald Kerr black holes develop scalar clouds.

What carries the argument

The load-bearing object is the effective scalar mass squared, $\mu_{\rm eff}^2(r,\theta)=\mu^2-\eta G_{\rm Kerr}(r,\theta)$, with $G_{\rm Kerr}$ given by Eq. (12). The paper performs a two-dimensional extremum analysis of $G_{\rm Kerr}$: for $\bar a>\bar a_{\rm crit}\simeq0.78$ the invariant attains a negative global minimum at the off-equatorial angle $(\cos^2\theta)_{\min}$ from Eq. (20), with the minimum located at $r\to r_+$. This converts the scalarization question into the geometry of a near-horizon potential well controlled by the small parameter $\epsilon$, and produces the closed-form critical relation (22), the existence line (35), and the ring-width formulas (29) and (30).

What would settle it

Numerically solve the linearized Klein-Gordon equation (13) on the Kerr background with $\mu_{\rm eff}^2$ from (14), imposing regularity at the horizon and decay at infinity, for parameters satisfying the critical ratio (35); if no normalizable mode exists, the sufficiency of the onset criterion (17) is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is an existence proof for linearized scalar bound states: for a Kerr black hole with dimensionless spin $\bar a > \bar a_{\rm crit} = \sqrt{\{7+\sqrt{7}\cos[{1\over3}\arctan(3\sqrt{3})] - \sqrt{21}\sin[{1\over3}\arctan(3\sqrt{3})]\}/12} \simeq 0.78$, and for a massive scalar field in the theory with coupling function $f(\varphi)=\eta\varphi^2/2$ and $\eta<0$, the composed system admits pairs of non-equatorial rings. The rings sit at polar angles determined by $(\cos^2\theta)_{\min}$ from Eq. (20), with radial location approaching the horizon radius $r_+$ in the spin limit. Their existence requires the large-mass, large-coupling limit $-\eta\to\infty$, $\mu\to\infty$ with the ratio $-\eta/\mu^2$ fixed at the critical value in Eq. (35), equivalently Eq. (22). In the near-critical regime the classically allowed region has angular width proportional to $\sqrt{\epsilon}$ and radial width proportional to $\epsilon$, so the rings become arbitrarily thin as the critical ratio is approached from above.

Load-bearing premise

The paper assumes that the effective mass squared touching zero from below is enough to guarantee a normalizable scalar bound state, but it never constructs the actual solution.

Editorial extensions

If this is right

For every $\bar a$ in the super-critical range, the critical ratio (35) gives the specific negative coupling and scalar mass that place non-equatorial scalar rings around the Kerr black hole.
The ring angle moves monotonically from the equator at $\bar a=\bar a_{\rm crit}$ to about $61.2^{\circ}$ away from the equator in the extremal limit $\bar a\to1$.
The classically allowed widths shrink to zero as the critical ratio is approached from above ($\epsilon\to0$): angular width scales like $\sqrt{\epsilon}$ and radial width like $\epsilon$.
Equation (35) marks the sharp boundary between bald Kerr black holes and spontaneously scalarized hairy configurations in the large-mass, large-coupling regime.
The presence of a finite scalar-field mass is what makes the ring widths arbitrarily thin, a feature that massless or minimally coupled configurations would not share.

Reading between the lines

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

The existence claim is established at the linearized level; whether the fully nonlinear field equations sustain these rings at the same critical ratio is not shown in the paper.
The extremum mechanism is somewhat generic: any curvature invariant that develops a negative off-equatorial minimum near a rapidly rotating horizon could support similar non-equatorial rings, so the phenomenon may extend beyond Gauss-Bonnet gravity.
Because the rings sit close to the horizon and require large couplings and masses, their direct astrophysical signature is likely weak; their significance may be mainly as an analytic marker of the scalarization threshold.
A numerical construction of the actual bound state would test whether the onset criterion (17) is sufficient, since the paper itself does not produce a normalizable solution.

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

The genuinely new thing here is the observation that for dimensionless spin a_bar above about 0.78, the global minimum of M^4 G_Kerr sits at a non-equatorial polar angle, with explicit formulas for the minimum value and location. That is a real qualitative step beyond the existing spinning scalarization literature, and the near-critical analysis in Section IV—ring widths scaling as sqrt(epsilon) in angle and epsilon in radius, relation (26), coefficients (27)-(28)—is clean algebra. The paper does its best work there.

The soft spot is exactly where the reader's stress-test lands. The abstract says "it is proved" that rapidly spinning Kerr black holes can support non-equatorial scalar rings. That is not what is shown. The bridge from "mu_eff^2 is negative somewhere" to "a bound state exists" is Eq. (17), the claim that in the large-mass regime onset is marked by min{mu_eff^2} -> 0^-, cited from earlier work. That is at most a necessary condition. No normalizable solution of the Klein-Gordon equation (13) is constructed, no WKB quantization is done, and the radial problem on the half-line has known threshold behavior: a shallow, narrow well need not bind. The paper's own near-critical scalings actually make this concrete—as epsilon -> 0 the well depth and width shrink in a combination that does not obviously support a zero-energy mode. So the central existence claim is unproven.

There is also a circular feel to the headline relation (22): it is the condition min{mu_eff^2} = 0 restated after substituting the asserted minimum of G_Kerr. That does not make it wrong, but it means the derivation inherits the unverified sufficiency of Eq. (17). I would also note that the minimization leading to (19) is presented without derivation; it looks plausible, but it is asserted rather than shown.

To be fair, none of this kills the geometric observation. The paper has proved a potential well exists in a non-equatorial ring, not that a scalar bound state lives in it. Reframing the claim from "support rings" to "the effective potential admits a negative non-equatorial region, with the following near-critical shape, and we conjecture the corresponding bound state exists" would be honest and still worth publishing.

Who is this for? People working on Gauss-Bonnet scalarization of Kerr, and anyone interested in how onset criteria are used in this literature. I would send it to peer review—the observation deserves referee time—but the referee should push hard on the eigenvalue existence question, and the paper should come back substantially revised.

Referee Report

3 major / 3 minor

Summary. The manuscript studies a massive scalar field non-minimally coupled to the Gauss-Bonnet invariant in a fixed Kerr black-hole background. In the regime η̄<0 and μ̄≫1, it computes the global minimum of the Kerr Gauss-Bonnet invariant (Eq. 19) and, invoking the criterion min{μ_eff²}→0^- (Eq. 17), derives a critical relation between the coupling and the mass (Eq. 22). It then analyzes the shape of the negative region of μ_eff² near this critical line (Eq. 26) and obtains the angular and radial widths of the putative rings (Eqs. 29 and 30). The paper concludes that rapidly spinning Kerr black holes can support a pair of non-equatorial massive scalar rings.

Significance. If the main existence claim were established, the result would add a genuinely new qualitative feature to the Einstein-Gauss-Bonnet scalarization literature: off-equatorial, arbitrarily thin scalar rings in a fixed Kerr background. The explicit formulas for the global minimum of G_Kerr and the monotonic behavior displayed in Table I are useful analytical results. However, the central claim is an existence statement about solutions of the linearized Klein-Gordon equation, and the manuscript never constructs or proves the existence of a normalizable scalar cloud. The significance therefore rests entirely on an unverified sufficiency assumption, so the result as stated is not established.

major comments (3)

[Section III, Eq. (17)] The condition min{μ_eff²}→0^- is treated as sufficient for the existence of a bound-state solution of Eq. (13), but it is at most necessary. A normalizable mode is an eigenfunction of an elliptic operator with regular horizon behavior and decay at infinity; negativity of the effective potential at isolated points does not imply a zero-energy bound state. No solution of the Klein-Gordon equation is constructed, and no spectral argument is given. Consequently, the abstract's statement 'it is proved that ... Kerr black holes ... can support a pair of non-equatorial massive scalar rings' is not supported by the body of the paper.
[Section IV, Eqs. (23)-(26)] The near-critical scalings show that as ε→0 the negative region of μ_eff² has angular width Δ(cos²θ)~√ε and radial width Δr~ε r+, while the depth of the well scales as μ̄²ε. For fixed μ̄, the dimensionless well-strength combination depth × (width)² tends to zero as ε→0, so an arbitrarily shallow and narrow well need not bind a mode. If instead ε is kept large enough to make the well deep enough to bind, then min{μ_eff²} is not close to 0^-, contradicting the assumed onset criterion. The paper therefore needs a quantitative eigenvalue condition (for example a variational bound or an explicit mode solution) to establish that the critical line (22) actually corresponds to an existing bound state.
[Equation (22)] The critical relation (22) is obtained by substituting the global minimum (19) and the angular location (20) into the onset criterion (17). It is thus a restatement of the assumed criterion after algebraic substitution, not an independent derivation of an existence line. Moreover, the cited references [31,39,40] for Eq. (17) concern related but different settings; no argument is provided that the same criterion is valid for a massive scalar field in the non-separable Kerr background, where the radial and angular problems do not decouple.

minor comments (3)

[Abstract and Eq. (18)] The abstract and Eq. (18) use arctan(3√3) in the definition of a_crit, while Eqs. (19)-(22) and (20) use arctan(1/(3√3)). Please check which argument is correct and use a unified notation throughout.
[Throughout] There are several typographical errors, including 'attarctive' in Section IV, 'spa cetime' in the abstract, and 'regim e' in the introduction. These should be corrected in a revision.
[Section IV, Eq. (23)] The arrow in Eq. (22) is ambiguous: the text says the ratio tends to 1^+, while Eq. (23) parameterizes the ratio as (η̄/μ̄²)_crit (1+ε). Please state explicitly whether the approach is from above or below and how ε relates to the '+' in Eq. (22).

Circularity Check

2 steps flagged · score 8.0 of 10

Central 'proof' of supported scalar rings reduces to the imported min μ_eff^2 onset criterion and to identifying the negative-potential region with the ring.

self definitional [Sec. III, Eqs. (17), (19), (22); Sec. IV, Eq. (26); footnote [43]]
"the onset of the spontaneous scalarization phenomenon in the non-trivial field theory (6) is marked by the critical functional behavior min{μ2eff(r, θ; M, a)} → 0− [31,39,40] ... From Eqs. (18) and (19) one deduces that ... with the critical relation [42] ... the effective mass term (14) ... becomes negative ... in a pair of narrow non-equatorial rings ... the angular width is defined as the angular interval in which the effective mass term (14) ... is negative and therefore represents an attractive binding potential well."

The 'proved' critical relation (22)/(35) is exactly the condition min μ_eff^2 = 0 evaluated at the minimum of G_Kerr, i.e., precisely the assumed onset criterion (17) after substituting the analytically computed G_min (19). Section IV then computes the region where μ_eff^2 < 0 and, in footnote [43], defines the ring's width as the interval where μ_eff^2 is negative. No normalizable solution of the Klein-Gordon equation (13) is constructed. Thus the existence claim 'the black hole can support a scalar ring' is equivalent, by construction, to 'the effective potential is negative somewhere,' and the advertised critical ratio is the input criterion restated.
self citation load bearing [Sec. III, Eq. (17); References [31] and [40]]
"In particular, in the dimensionless large-mass regime ... the onset of the spontaneous scalarization phenomenon in the non-trivial field theory (6) is marked by the critical functional behavior [31,39,40] min{μ2eff(r, θ; M, a)} → 0− ... This physically intriguing property ... implies that ... rotating Kerr black holes in the dimensionless large-spin regime (18) can support pairs of non-equatorial thin matter rings."

The premise that min μ_eff^2 → 0^- marks the onset of scalarization is the sole bridge from 'the potential well is negative' to 'a bound-state scalar ring exists.' That premise is justified in this paper only by citing Refs. [31,39,40], of which [31] (S. Hod, Phys. Rev. D 102, 084060) and [40] (S. Hod, JHEP 02, 039) are the present author's own works. The manuscript supplies no independent construction of an eigenfunction of Eq. (13), so the central claim is forced through a same-author citation chain rather than by a self-contained derivation.

full rationale

The paper contains genuine new algebra: the closed-form minimization of the Kerr Gauss-Bonnet invariant, the critical spin (18), the off-equatorial location (20), and the near-critical widths (29)-(30) are non-trivial computations about the effective potential. The circularity is in the step from those potential-well computations to the headline claim that the black hole 'can support' non-equatorial scalar rings. The main proved relation, Eq. (22) (and its summary form Eq. (35)), is obtained by imposing min μ_eff^2 = 0, which is precisely the onset criterion (17) imported from Refs. [31,39,40]; substituting the computed G_min makes the 'prediction' a restatement of the input. The paper never constructs a normalizable bound-state solution of the generalized Klein-Gordon equation (13), and footnote [38] plus footnote [43] explicitly identify the supported ring with the region where μ_eff^2 < 0, so 'the ring exists' reduces by definition to 'the effective potential is negative somewhere.' Because the load-bearing onset criterion is supported by two same-author citations and is not independently established in the manuscript, the central existence claim is forced through a self-citation chain and a definitional identification. The analytic minimization is real, but it does not rescue the existence statement: if Eq. (17) is only a necessary condition, the paper proves a potential well, not a bound state. Hence a score of 8: the result is forced by definition and by the imported criterion, while the peripheral algebra retains independent content.

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

The paper introduces no new entities. Its free-parameter count is zero: eta, mu, M, a are theory parameters, and the constants in Eqs. (19)-(22) arise from algebra. The load-bearing axioms are the onset criterion (17) and the unproved minimization (19)-(20).

assumptions (4)

ad hoc to paper The onset of spontaneous scalarization in the large-mass regime is marked by min{mu_eff^2} -> 0^- (Eq. 17).
This criterion is imported from refs. [31,39,40], two of which are authored by the present author, and is used as the basis for the critical line (22). It is a necessary condition whose sufficiency for a normalizable ring mode is not established in the text.
ad hoc to paper The claimed global minimum of M^4 * G_Kerr(r,theta) is given by Eq. (19) at r -> r+ and (cos^2 theta)_min of Eq. (20).
The minimization result, which carries the paper's main new content, is asserted in Section III without a derivation. The reader must accept it on faith or rederive it.
domain assumption The massive scalar field obeys the linearized Klein-Gordon equation (13) with the effective mass (14).
Standard linearized scalar field theory in the chosen Einstein-Gauss-Bonnet action (6); acceptable background.
standard math The expression (12) for the Gauss-Bonnet invariant of Kerr is correct.
Taken from prior literature (ref. [22]); standard result in general relativity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Non-equatorial scalar rings supported by rapidly spinning Gauss-Bonnet black holes." pith.science (2026). https://pith.science/paper/UVEZU2I2

@misc{pith2026241215332,
  author       = {Pith},
  title        = {Pith review of: Non-equatorial scalar rings supported by rapidly spinning Gauss-Bonnet black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UVEZU2I2}},
  note         = {Machine review of arXiv:2412.15332}
}

abstract

Black-hole spacetimes that possess stationary equatorial matter rings are known to exist in general relativity. We here reveal the existence of black-hole spacetimes that support {\it non}-equatorial matter rings. In particular, it is proved that rapidly-rotating Kerr black holes in the dimensionless large-spin regime ${\bar a}>{\bar a}_{\text{crit}}= \sqrt{\big\{{{7+\sqrt{7}\cos\big[{1\over3}\arctan\big(3\sqrt{3}\big)\big]- \sqrt{21}\sin\big[{1\over3}\arctan\big(3\sqrt{3}\big)\big]}\big\}/12}}\simeq0.78$ can support a pair of non-equatorial massive scalar rings which are negatively coupled to the Gauss-Bonnet curvature invariant of the spinning spacetime (here ${\bar a}\equiv J/M^2$ is the dimensionless angular momentum of the central supporting black hole). We explicitly prove that these non-equatorial scalar rings are characterized by the dimensionless functional relation $-{{57+28\sqrt{21}\cos\big[{1\over3}\arctan\big({{1}\over{3\sqrt{3}}}\big)\big]} \over{8(1+\sqrt{1-{\bar a}^2})^6}} \cdot{{\bar\eta}\over{{\bar\mu}^2}}\to 1^{+}$ in the large-mass ${\bar\mu}\equiv M\mu\gg1$ regime (here $\{{\bar\eta}<0,\mu\}$ are respectively the non-trivial coupling parameter of the composed Einstein-Gauss-Bonnet-massive-scalar field theory and the proper mass of the supported non-minimally coupled scalar field).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 23 canonical work pages

[1]

Ruﬃni and J

R. Ruﬃni and J. A. Wheeler, Physics Today 24, 30 (1971). 7

work page 1971
[2]

J. D. Bekenstein, Contribution to the 2nd International Sakharov Conference on Physics, 216-219 [e-Print: gr-qc/9 605059]

work page
[3]

J. D. Bekenstein, Phys. Rev. D 5, 1239 (1972)

1972
[4]

T. P. Sotiriou, Class. Quant. Grav. 32, 214002 (2015)

work page 2015
[5]

C. A. R. Herdeiro and E. Radu, Int. J. Mod. Phys. D 24, 1542014 (2015)

2015
[6]

T. P. Sotiriou and V. Faraoni, Phys. Rev. Lett. 108, 081103 (2012)

work page 2012
[7]

A. E. Mayo and J. D. Bekenstein and, Phys. Rev. D 54, 5059 (1996)

1996
[8]

Hod, Phys

S. Hod, Phys. Lett. B 771, 521 (2017) [arXiv:1911.08371]; S. Hod, Phys. Rev. D 96, 124037 (2017) [arXiv:2002.05903]

arXiv 2017

Show all 44 references

[9]

T. P. Sotiriou and S.-Y. Zhou, Phys. Rev. Lett. 112, 251102 (2014); T. P. Sotiriou and S.-Y. Zhou, Phys. Rev. D 90, 124063 (2014)

2014
[10]

T. P. Sotiriou, Lect. Notes Phys. 892, 3 (2015) [arXiv:1404.2955]

2015 arXiv
[11]

D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. 120, 131103 (2018)

2018
[12]

H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, a nd E. Berti, Phys. Rev. Lett. 120, 131104 (2018)

2018
[13]

See [14–20] for the physically related case of charged b lack hole in composed Einstein-Maxwell-scalar ﬁeld theori es that can support spatially regular bound-state conﬁgurations o f static scalar ﬁelds which are directly (non-minimally) co upled to the spatially-dependent Max...
[14]

C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Fon t, Phys. Rev. Lett. 121, 101102 (2018)

2018
[15]

P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu , and N. Sanchis-Gual, Class. Quant. Grav. 36, 134002 (2019) [arXiv:1902.05079]

2019 arXiv
[16]

Hod, Phys

S. Hod, Phys. Lett. B 798, 135025 (2019) [arXiv:2002.01948]

2019 arXiv
[17]

Hod, Phys

S. Hod, Phys. Rev. D 101, 104025 (2020) [arXiv:2005.10268]

2020 arXiv
[18]

Khodadi, A

M. Khodadi, A. Allahyari, S. Vagnozzi, and D. F. Mota, JC AP 09, 026 (2020) [arXiv:2005.05992]

2020 arXiv
[19]

Hod, The Euro

S. Hod, The Euro. Phys. Jour. C 80, 1150 (2020)

2020
[20]

Hod, Jour

S. Hod, Jour. of High Energy Phys. 08, 272 (2022) [arXiv:2206.12074]

2022 arXiv
[21]

P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. L ett. 123, 011101 (2019)

2019
[22]

A. Dima, E. Barausse, N. Franchini, and T. P. Sotiriou, P hys. Rev. Lett. 125, 231101 (2020) [arXiv:2006.03095]

2020 arXiv
[23]

Hod, Phys

S. Hod, Phys. Rev. D 100, 064039 (2019) [arXiv:1912.07630]

2019 arXiv
[24]

Hod, The Euro

S. Hod, The Euro. Phys. Jour. C 79, 966 (2019)

2019
[25]

D. D. Doneva, L. G. Collodel, C. J. Kr¨ uger, and S. S. Yazadjiev, Phys. Rev. D 102, 104027 (2020)

2020
[26]

C. A. R. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou, an d N. Yunes, Phys. Rev. Lett. 126, 011103 (2021) [arXiv:2009.03904]

2021 arXiv
[27]

Berti, L

E. Berti, L. G. Collodel, B. Kleihaus, and J. Kunz, Phys. Rev. Lett. 126, 011104 (2021)

2021
[28]

Hod, Phys

S. Hod, Phys. Rev. D 105, 024074 (2022)

2022
[29]

Hod, Phys

S. Hod, Phys. Rev. D 86, 104026 (2012) [arXiv:1211.3202]; S. Hod, The Euro. Phys. J ournal C 73, 2378 (2013) [arXiv:1311.5298]; S. Hod, Phys. Rev. D 90, 024051 (2014) [arXiv:1406.1179]

2012 arXiv
[30]

C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. 112, 221101 (2014)

2014
[31]

Hod, Phys

S. Hod, Phys. Rev. D 102, 084060 (2020) [arXiv:2006.09399]

2020 arXiv
[32]

It is worth noting that, as explicitly proved in [31], th e critical black-hole spin (1) corresponds to the regime l ≫ 1. In particular, one ﬁnds [31] that for smaller l values the critical rotation parameter that allows for the e xistence of negatively coupled scalar clouds i...
[33]

Note that in these units the mass µ of the non-minimally coupled scalar ﬁeld has the dimensions of length −1

We use natural units in which G = c = ¯h = 1. Note that in these units the mass µ of the non-minimally coupled scalar ﬁeld has the dimensions of length −1. The black-hole-scalar-ﬁeld composed mass parameter ¯ µ ≡ M µ is dimensionless
[34]

We shall assume, without loss of generality, the relati on ¯a ≡ J/M 2 > 0 for the angular momentum parameter of the supporting Kerr black hole
[35]

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, (W. H. Freeman, San Francisco, 1973)

1973
[36]

Chandrasekhar, The Mathematical Theory of Black Holes , (Oxford University Press, New York, 1983)

S. Chandrasekhar, The Mathematical Theory of Black Holes , (Oxford University Press, New York, 1983)

1983
[37]

D. D. Doneva, L. G. Collodel, C. J. Kr¨ uger, S. S. Yazadji ev, The Eur. Phys. Jour. C 80, 1205 (2020) [arXiv:2009.03774]

2020 arXiv
[38]

In particular, we shall explicitly prove below t hat, by adjusting properly the mass term of the non-minimally cou pled scalar ﬁeld [see Eq

It is worth emphasizing that the fact that the non-minim ally coupled scalar ﬁeld has a ﬁnite mass allows one to obtain non-equatorial scalar rings whose widths can be made arbitr arily thin. In particular, we shall explicitly prove below t hat, by adjusting properly the mass ...
[39]

C. A. R. Herdeiro, A. M. Pombo, and E. Radu, Universe 7, 483 (2021) [arXiv:2111.06442]

2021 arXiv
[40]

Hod, Jour

S. Hod, Jour. of High Energy Phys. 02, 039 (2022) [arXiv:2201.03503]

2022 arXiv
[41]

Note that the inequality in (20) is valid in the super-cr itical regime (18)
[42]

The composed black-hole-ﬁeld system also has excited bound states which are characterized by the near-critical relati on (23) with ǫ > 0

Note that the critical dimensionless ratio (22) corres ponds to the fundamental composed black-hole-ﬁeld conﬁgur ation. The composed black-hole-ﬁeld system also has excited bound states which are characterized by the near-critical relati on (23) with ǫ > 0
[43]

Note that the angular width is deﬁned as the angular inte rval in which the eﬀective mass term (14) in the scalar equati on (13) is negative and therefore represents an attractive bin ding potential well of the composed black-hole-ﬁeld system
[44]

Note that the radial width is deﬁned as the radial interv al in which the eﬀective mass term (14) is negative and theref ore represents an attractive black-hole-ﬁeld binding potenti al well

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Ruﬃni and J

R. Ruﬃni and J. A. Wheeler, Physics Today 24, 30 (1971). 7

work page 1971

[2] [2]

J. D. Bekenstein, Contribution to the 2nd International Sakharov Conference on Physics, 216-219 [e-Print: gr-qc/9 605059]

work page

[3] [3]

J. D. Bekenstein, Phys. Rev. D 5, 1239 (1972)

1972

[4] [4]

T. P. Sotiriou, Class. Quant. Grav. 32, 214002 (2015)

work page 2015

[5] [5]

C. A. R. Herdeiro and E. Radu, Int. J. Mod. Phys. D 24, 1542014 (2015)

2015

[6] [6]

T. P. Sotiriou and V. Faraoni, Phys. Rev. Lett. 108, 081103 (2012)

work page 2012

[7] [7]

A. E. Mayo and J. D. Bekenstein and, Phys. Rev. D 54, 5059 (1996)

1996

[8] [8]

Hod, Phys

S. Hod, Phys. Lett. B 771, 521 (2017) [arXiv:1911.08371]; S. Hod, Phys. Rev. D 96, 124037 (2017) [arXiv:2002.05903]

arXiv 2017