Abstract
We study the existence of scalar fields outside neutral reflecting shells. We consider static massive scalar fields non-minimally coupled to the Gauss–Bonnet invariant. We analytically investigated properties of scalar fields through the scalar field equation. In the small scalar field mass regime, we derive a compact resonance formula for the allowed masses of scalar fields in the composed scalar field and shell configurations.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The famous black hole no hair theorem has attracted a lot of attention from physicists and mathematicians for decades. If true, it states that asymptotically flat black holes cannot support static scalar field hairs outside horizons, for progress see references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and reviews [20, 21]. This no hair property was usually attributed to the existence of absorbing boundary conditions at black hole horizons. So it is of great interest to examine whether such no hair behavior can appear in horizonless spacetimes.
In the horizonless spacetime, Hod recently proved a new type no hair theorem that static scalar field hairs cannot exist outside asymptotically flat neutral reflecting stars [22, 23]. Furthermore, such no hair theorem also holds for systems constructed with static scalar fields and neutral horizonless reflecting shells [24, 25]. In the asymptotically dS spacetimes, the static scalar field hair also cannot exist outside neutral horizonless reflecting stars [26]. And further studies showed that such no scalar hair theorem is a general property for horizonless objects with reflecting boundary conditions [27,28,29,30,31,32].
However, asymptotically flat black holes can support static scalar fields non-minimally coupled to electromagnetic Maxwell fields, which violates the spirit of no hair theorems [33, 34]. Similarly, Hod proved that scalar fields can exist outside charged horizonless reflecting shells when considering non-minimally couplings between static scalar fields and electromagnetic Maxwell fields [35]. In particular, Hod derived a remarkably compact resonance formula for the allowed masses of the spatially regular scalar fields supported by charged shell configurations. In the background of neutral horizonless reflecting shells, exterior scalar fields also can exist when including non-minimally couplings between scalar fields and the Gauss–Bonnet invariant [36]. Inspired by analysis in [35], we plan to carry out an analytical investigation on configurations composed of scalar fields and neutral reflecting shells in the scalar-Gauss–Bonnet gravity.
This work is organized as follows. We firstly introduce the system with scalar fields non-minimally coupled to the Gauss–Bonnet invariant outside neutral horizonless reflecting shells. Then we shall derive a remarkably compact resonance formula for the allowed masses of scalar fields supported by neutral reflecting shells. We will summarize our main results in the last section.
2 Scalar field equations in the reflecting shell background
We consider scalar fields non-minimally coupled to the Gauss–Bonnet invariant. The Lagrangian density of the scalar-Gauss–Bonnet gravity is given by [37,38,39,40,41,42,43]
Here R is the Ricci scalar curvature, \(\Psi \) is the scalar field with mass \(\mu \) and \(f(\Psi )\mathcal {R}_{GB}^{2}\) describes the coupling, where \(\mathcal {R}_{GB}^{2}\) is the Gauss–Bonnet invariant [39, 40]
and \(f(\Psi )\) is a general function of \(\Psi \). When neglecting matter fields’ backreaction on the metric, the Gauss–Bonnet invariant term is \(\mathcal {R}_{GB}^{2}=\frac{48 M^{2}}{r^{6}}\). We put the function \(f(\Psi )\) in a simple form \(f(\Psi )=\eta \Psi ^2\) in the linear limit, where \(\eta \) is the coupling strength parameter. We emphasize that this choice of the coupling function is physically motivated since it guarantees the existence of hairless solutions.
The spherically symmetric spacetime is characterized by the curved line element [40]
The radial coordinate depending function g(r) is the metric solution. The angular coordinates are labeled as \(\theta \) and \(\phi \) respectively. And the radius of the shell is defined as \(r_{s}\).
With variation methods, we obtain the scalar field equation [37,38,39,40,41,42,43]
We choose to study a stationary massive scalar field in the form
with l, m representing the integer harmonic parameters. \(S_{lm}(\theta )\) is the angular scalar eigenfunction with the eigenvalue \(l(l+1)\), where l is the spherical harmonic index [35]. For simplicity, we label the characteristic radial function \(\psi _{lm}(r)\) as \(\psi (r)\).
The Eq. (4) and field decomposition (5) yield the ordinary differential equation
Here we take the metric outside shells in the flat spacetime limit with \(g=1\) and \(g'=1\). In particular, the present model includes a coupling of the scalar field to the Gauss–Bonnet invariant but the metric functions used in Eq. (6) are valid for a flat spacetime. Thus, the present model is only a toy-model.
At the shell radius, the scalar field satisfies reflecting surface conditions
At the infinity, the scalar field is spatially regular, leading to the vanishing condition
3 Analytical formula for the allowed masses of scalar fields
We investigate on properties of scalar fields outside reflecting shells through the ordinary differential equation (6). The equation can be analytically studied in two regions \(r\ll 1/\mu \) and \(r\gg (\sqrt{\eta }M)^{1/2}\). In the small scalar field mass regime \(\mu r\ll 1\), we can analyze the system in the overlapping region \((\sqrt{\eta }M)^{1/2}\ll r\ll 1/\mu \) [35]. In the overlapping region, we apply matching methods to obtain a formula of the masses of the non-minimally coupled scalar fields outside reflecting shells.
In the limit \(r\ll 1/\mu \), the ordinary differential equation (6) can be expressed as
The general mathematical solution of the Eq. (9) is
where \(J_{\nu }(z)\) and \(Y_{\nu }(z)\) are Bessel functions of the first and second kinds respectively. And the coefficients \(A_{1}\) and \(A_{2}\) are normalization constants.
In the limit \(z\rightarrow 0\), the Bessel functions asymptotically behave as s (see Eqs. 9.1.7 and 9.1.9 of [44])
In the radial overlap region \((\sqrt{\eta }M)^{1/2}\ll r\ll 1/\mu \), there is the relation \(\frac{\sqrt{\eta }M}{r^{2}}\ll 1\) and solution (10) of the scalar field equation (9) can be expressed as
In the limit \(r\gg (\sqrt{\eta }M)^{1/2}\), the ordinary differential equation (6) can be approximated by
The general mathematical solution of (14) can be expressed by the Bessel functions in the form
where the coefficients \(B_{1}\) and \(B_{2}\) are normalization constants.
According to behaviors (11) and (12), Eq. (15) can be mathematically expressed as
for the scalar field in the overlap region \((\sqrt{\eta }M)^{1/2}\ll r\ll 1/\mu \). Therefore, two analytically derived mathematical expressions (13) and (16) for the scalar field \(\psi (r)\) are both valid in the intermediate radial region
This fact allows us to match the solutions (13) and (16) in the region (17) and obtain relations between coefficients \(A_{i}\) and \(B_{i}\) as
In the large-argument (\(z\rightarrow \infty \)), the Bessel functions behave as (see Eqs. 9.2.1 and 9.2.2 of [44])
and the solution (15) has the asymptotic functional behavior
The asymptotic behavior (22) and the boundary condition (8) yield the relation
With relations (18), (19) and (23), we derive the relation between \(A_{i}\) in the form
We investigate on properties of scalar fields in the regime \(\mu r\ll 1\). So there is the relation \(\mu r_{s}\ll 1\). The expression (10) holds at the shell surface \(r_{s}\) satisfying \(\mu r_{s}\ll 1\). The boundary condition (7) and the solution (10) yield expression of \(A_{2}/A_{1}\) as
Substituting the ratio \(A_{2}/A_{1}\) of (25) into relation (24), we arrive at the resonance equation
The relation (17) yields the relation
According to Eq. (26), the relation (27) implies
where \(\{j_{\nu .n}\}_{n=1}^{n=\infty }\) are positive zeros of the Bessel function \(J_{\nu }(x)\).
The Bessel functions can be expanded around the positive zeros in the form (see Eq. 9.1.27d of [44])
and
where \(\Delta _{n}\) is the small quantity defined as
Substituting the expansions (29) and (30) into (26), one obtains the formula
which shows that the allowed values of scalar field masses are discrete.
4 Conclusions
We studied the existence of scalar fields outside neutral reflecting shells in the flat spacetime limit. We considered static massive scalar fields non-minimally coupled to the Gauss–Bonnet invariant. We investigated on properties of scalar fields outside reflecting shells through the ordinary differential equation (6). The system is amenable to an analytical analysis in two regions \(r\ll 1/\mu \) and \(r\gg (\sqrt{\eta }M)^{1/2}\), where \(\eta \) is the coupling parameter, M is the shell mass and \(\mu \) is the scalar field mass. In the small scalar field mass regime \(\mu r\ll 1\), we can analyze the system in the overlapping region \((\sqrt{\eta }M)^{1/2}\ll r\ll 1/\mu \). In the overlapping region, we applied matching methods to derive a remarkably compact resonance formula (32) for the allowed masses of the supported spatially regular scalar fields outside neutral horizonless reflecting shells.
Data Availability
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This work is based on analytical studies. So there is no numerical data and all analysis has been presented in the paper.]
References
J.D. Bekenstein, Transcendence of the law of baryon-number conservation in black hole physics. Phys. Rev. Lett. 28, 452 (1972)
J.E. Chase, Event horizons in static scalar-vacuum space-times. Commun. Math. Phys. 19, 276 (1970)
C. Teitelboim, Nonmeasurability of the baryon number of a black-hole. Lett. Nuovo Cimento 3, 326 (1972)
R. Ruffini, J.A. Wheeler, Introducing the black hole. Phys. Today 24, 30 (1971)
S. Hod, Stationary resonances of rapidly-rotating Kerr black holes. Eur. Phys. J. C 73, 2378 (2013)
S. Hod, The superradiant instability regime of the spinning Kerr black hole. Phys. Lett. B 758, 181 (2016)
C. Herdeiro, V. Paturyan, E. Radu, D.H. Tchrakian, Reissner–Nordström black holes with non-Abelian hair. Phys. Lett. B 772, 63–69 (2017)
M. Richartz, C.A.R. Herdeiro, E. Berti, Synchronous frequencies of extremal Kerr black holes: resonances, scattering and stability. Phys. Rev. D 96, 044034 (2017)
Y. Brihaye, C. Herdeiro, E. Radu, D.H. Tchrakian, Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension. JHEP 1711, 037 (2017)
C.L. Benone, L.C.B. Crispino, C. Herdeiro, E. Radu, Kerr–Newman scalar clouds. Phys. Rev. D 90, 104024 (2014)
C. Herdeiro, E. Radu, H. Rúnarsson, Non-linear QQ-clouds around Kerr black holes. Phys. Lett. B 739, 302 (2014)
E. Winstanley, Classical Yang–Mills black hole hair in anti-de Sitter space. Lect. Notes Phys. 769, 49–87 (2009)
Y. Peng, Hair mass bound in the black hole with non-zero cosmological constants. Phys. Rev. D 98, 104041 (2018)
Y. Peng, Hair distributions in noncommutative Einstein–Born–Infeld black holes. Nucl. Phys. B 941, 1–10 (2019)
Y. Brihaye, T. Delplace, C. Herdeiro, E. Radu, An analytic effective model for hairy black holes. Phys. Lett. B 782, 124–130 (2018)
M. Rogatko, Uniqueness of higher-dimensional phantom field wormholes. Phys. Rev. D 97(2), 024001 (2018)
J.C. Degollado, C.A.R. Herdeiro, Stationary scalar configurations around extremal charged black holes. Gen. Relativ. Gravit. 45, 2483 (2013)
P.V.P. Cunha, C.A.R. Herdeiro, E. Radu, H.F. Rúnarsson, Shadows of Kerr black holes with scalar hair. Phys. Rev. Lett. 115, 211102 (2015)
Y. Brihaye, C. Herdeiro, E. Radu, Inside black holes with synchronized hair. Phys. Lett. B 760, 279 (2016)
J.D. Bekenstein, Black hole hair: 25-years after. arXiv:gr-qc/9605059
C.A.R. Herdeiro, E. Radu, Asymptotically flat black holes with scalar hair: a review. Int. J. Mod. Phys. D 24(09), 1542014 (2015)
S. Hod, No-scalar-hair theorem for spherically symmetric reflecting stars. Phys. Rev. D 94, 104073 (2016)
S. Hod, No nonminimally coupled massless scalar hair for spherically symmetric neutral reflecting stars. Phys. Rev. D 96, 024019 (2017)
S. Hod, Charged massive scalar field configurations supported by a spherically symmetric charged reflecting shell. Phys. Lett. B 763, 275 (2016)
S. Hod, Marginally bound resonances of charged massive scalar fields in the background of a charged reflecting shell. Phys. Lett. B 768, 97–102 (2017)
S. Bhattacharjee, S. Sarkar, No-hair theorems for a static and stationary reflecting star. Phys. Rev. D 95, 084027 (2017)
Y. Peng, B. Wang, Y. Liu, Scalar field condensation behaviors around reflecting shells in Anti-de Sitter spacetimes. Eur. Phys. J. C 78(8), 680 (2018)
Y. Peng, Scalar field configurations supported by charged compact reflecting stars in a curved spacetime. Phys. Lett. B 780, 144–148 (2018)
S. Hod, Charged reflecting stars supporting charged massive scalar field configurations. Eur. Phys. J. C 78, 173 (2017)
Y. Peng, Static scalar field condensation in regular asymptotically AdS reflecting star backgrounds. Phys. Lett. B 782, 717–722 (2018)
Y. Peng, On instabilities of scalar hairy regular compact reflecting stars. JHEP 10, 185 (2018)
Y. Peng, Hair formation in the background of noncommutative reflecting stars. Nucl. Phys. B 938, 143–153 (2019)
C.A.R. Herdeiro, E. Radu, N. Sanchis-Gual, J.A. Font, Phys. Rev. Lett. 121, 101102 (2018)
P.G.S. Fernandes, C.A.R. Herdeiro, A.M. Pombo, E. Radu, N. Sanchis-Gual, Class. Quantum Gravity 36, 134002 (2019)
S. Hod, Non-minimally coupled massive scalar field configurations supported by charged reflecting shells: analytic treatment in the weak coupling regime. Phys. Lett. B 826, 136926 (2022)
Y. Peng, Analytical investigations on formations of hairy neutral reflecting shells in the scalar-Gauss–Bonnet gravity. Eur. Phys. J. C 80(3), 202 (2020)
T.P. Sotiriou, S.-Y. Zhou, Black hole hair in generalized scalar-tensor gravity. Phys. Rev. Lett. 112, 251102 (2014)
D.D. Doneva, S.S. Yazadjiev, New Gauss–Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories. Phys. Rev. Lett. 120, 131103 (2018)
H.O. Silva, J. Sakstein, L. Gualtieri, T.P. Sotiriou, E. Berti, Spontaneous scalarization of black holes and compact stars from a Gauss–Bonnet coupling. Phys. Rev. Lett. 120, 131104 (2018)
S. Hod, Spontaneous scalarization of Gauss–Bonnet black holes: analytic treatment in the linearized regime. Phys. Rev. D 100, 064039 (2019)
G. Antoniou, A. Bakopoulos, P. Kanti, Evasion of no-hair theorems and novel black-hole solutions in Gauss–Bonnet theories. Phys. Rev. Lett. 120, 131102 (2018)
C.A.R. Herdeiro, E. Radu, N. Sanchis-Gual, J.A. Font, Spontaneous scalarisation of charged black holes. Phys. Rev. Lett. 121, 101102 (2018)
P.V.P. Cunha, C.A.R. Herdeiro, E. Radu, Spontaneously scalarised Kerr black holes. Phys. Rev. Lett. 123, 011101 (2019)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)
Acknowledgements
We would like to thank the anonymous referee for the constructive suggestions to improve the manuscript. This work was supported by the Shandong Provincial Natural Science Foundation of China under Grant No. ZR2022MA074. This work was also supported by a grant from Qufu Normal University of China under Grant No. xkjjc201906, the Youth Innovations and Talents Project of Shandong Provincial Colleges and Universities (Grant No. 201909118), Taishan Scholar Project of Shandong Province (Grant No. tsqn202103062) and the Higher Educational Youth Innovation Science and Technology Program Shandong Province (Grant No. 2020KJJ004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3. SCOAP3 supports the goals of the International Year of Basic Sciences for Sustainable Development.
About this article
Cite this article
Peng, Y. Analytical investigations on non-minimally coupled scalar fields outside neutral reflecting shells. Eur. Phys. J. C 83, 140 (2023). https://doi.org/10.1140/epjc/s10052-023-11310-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-023-11310-7