Large regular reflecting stars have no scalar field hair
Abstract
We investigate the gravity system constructed with static scalar fields coupled to asymptotically flat regular reflecting stars. We consider the matter field’s backreaction on the reflecting star. We analytically show that there is an upper bound on the radius of the reflecting star. When the star radius is above the bound, the reflecting star cannot support the existence of scalar field hairs. That means large reflecting stars cannot have scalar field hairs.
1 Introduction
According to the no hair theorem [1, 2, 3, 4], the asymptotically flat black hole can be determined by the three conserved charges (the mass, angular momentum and charge of the black hole), see Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and reviews [20, 21]. The belief on the no hair theorem was partly based on the physical argument that exterior matter and radiation fields would either go away to the infinity or be swallowed by the black hole horizon. So no field can exist outside a black hole horizon, except when it is associated with the three conserved charges of black hole spacetimes.
Whether there is also no hair theorem in the horizonless spacetime is an interesting question to be answered. Recently, it was found that the no scalar hair behavior appears in the background of regular neutral reflecting stars. It was firstly proved in [22] that the static scalar field cannot exist outside asymptotically flat neutral compact reflecting stars without a horizon. When considering the nonminimal coupling between the massless scalar field and the gravity, it was shown that the no scalar hair theorem holds in certain range of the coupling model parameter [23]. And in the background of asymptotically dS gravity, it was found that the massive scalar, vector and tensor field cannot condense outside the regular neutral reflecting stars [24].
Can the no hair theorem obtained in the neural horizonless gravity still holds in the charged horizonless spacetime? In the charged horizonless reflecting shell spacetime, it was analytically found that the static scalar field cannot exist outside the shell when the shell radius is large enough [25, 26, 27]. Moreover, it was shown that charged horizonless reflecting stars cannot support the existence of the scalar field when the star radius is above an upper bound [28, 29, 30, 31, 32]. And the no scalar hair behavior also appears in the large size regular star with other surface boundary conditions [33, 34]. However, all the front discussion has been carried out in the probe limit. In this work, we try to generalize the discussion by considering the matter fields’ backreaction on the metric and also examine whether there are upper bounds on hairy star radii.
This paper is structured as follows. In Sect. 2, we construct the gravity system composed of a static scalar field and a asymptotically flat regular reflecting star away from the probe limit. In Sect. 3, we analytically show that there is an upper bound for the reflecting star radius. Above the bound, the scalar field cannot condense outside the star surface. Our summary is in the last section.
2 The gravity model composed of scalar fields and reflecting stars
However, all front discussions have been carried out without scalar fields’ backreaction. When considering scalar fields’ backreaction on the metric, Eq. (4) is coupled with Eqs. (3), (5) and (6). In this case of nonlinear coupled equations, approaches in [25, 26, 27, 28] fail and more precise numerical methods are needed. In this work, we analytically show that the nontrivial scalar field solution of nonlinear equations cannot exist when the star radius is above an upper bound. In other words, we prove no scalar field hair theorem for large reflecting stars in the nonlinear regime.
3 Upper bounds on radii of scalar hairy reflecting stars
 case 1:

\(\mu r_{s} \leqslant max \left\{ \mu R_{1},\mu R_{2},\mu R_{3} \right\} \);
 case 2:

\(\mu r_{s} \geqslant max \left\{ \mu R_{1},\mu R_{2},\mu R_{3} \right\} \) with \(\mu r_{s}\leqslant 2\sqrt{3}qQ\).
4 Conclusions
We studied static massive scalar field condensations outside static asymptotically flat spherically symmetric regular reflecting stars. We constructed a complete gravity model by considering the matter field’s backreaction on the background. We provided upper bounds on the star radius in the form \(\mu r_{s} \leqslant max \left\{ 2\sqrt{3}qQ,\mu R_{1},\mu R_{2},\mu R_{3} \right\} \), where \(\mu \) is the scalar field mass, q is the charge coupling parameter, Q is the total charge and \(R_{i}\) depends on the gravity theories. When the star radius is above the bound, the static massive scalar field cannot condense outside the reflecting star. That means the large regular reflecting star cannot have massive scalar field hairs in the asymptotically flat gravity.
Notes
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. ZR2018QA008.
References
 1.J.D. Bekenstein, Transcendence of the law of baryonnumber conservation in black hole physics. Phys. Rev. Lett. 28, 452 (1972)ADSCrossRefGoogle Scholar
 2.J.E. Chase, Event horizons in static scalarvacuum spacetimes. Commun. Math. Phys. 19, 276 (1970)ADSMathSciNetCrossRefGoogle Scholar
 3.C. Teitelboim, Nonmeasurability of the baryon number of a blackhole. Lett. Nuovo Cimento 3, 326 (1972)CrossRefGoogle Scholar
 4.R. Ruffini, J.A. Wheeler, Introducing the black hole. Phys. Today 24, 30 (1971)ADSCrossRefGoogle Scholar
 5.S. Hod, Stationary scalar clouds around rotating black holes. Phys. Rev. D 86, 104026 (2012)ADSCrossRefGoogle Scholar
 6.S. Hod, Stationary resonances of rapidlyrotating Kerr black holes. Euro. Phys. J. C 73, 2378 (2013)ADSCrossRefGoogle Scholar
 7.S. Hod, Kerr–Newman black holes with stationary charged scalar clouds. Phys. Rev. D 90, 024051 (2014)ADSCrossRefGoogle Scholar
 8.S. Hod, The largemass limit of cloudy black holes. Class. Quant. Grav. 32, 134002 (2015)ADSMathSciNetCrossRefGoogle Scholar
 9.S. Hod, The superradiant instability regime of the spinning Kerr black hole. Phys. Lett. B 758, 181 (2016)ADSCrossRefGoogle Scholar
 10.C.A.R. Herdeiro, E. Radu, Kerr black holes with scalar hair. Phys. Rev. Lett. 112, 221101 (2014)ADSCrossRefGoogle Scholar
 11.C.L. Benone, L.C.B. Crispino, C. Herdeiro, E. Radu, Kerr–Newman scalar clouds. Phys. Rev. D 90, 104024 (2014)ADSCrossRefGoogle Scholar
 12.C. Herdeiro, E. Radu, H. Runarsson, Nonlinear QQclouds around Kerr black holes. Phys. Lett. B 739, 302 (2014)ADSMathSciNetCrossRefGoogle Scholar
 13.C. Herdeiro, E. Radu, Construction and physical properties of Kerr black holes with scalar hair. Class. Quant. Grav. 32, 144001 (2015)ADSMathSciNetCrossRefGoogle Scholar
 14.Yan Peng, Hair mass bound in the black hole with nonzero cosmological constants. Phys. Rev. D 98, 104041 (2018)ADSCrossRefGoogle Scholar
 15.Yan Peng, Hair distributions in noncommutative EinsteinBornInfeld black holes. Nucl. Phys. B 941, 1–10 (2019)ADSCrossRefGoogle Scholar
 16.Yan Peng, The extreme orbital period in scalar hairy kerr black holes. Phys. Lett. B 792, 1–3 (2019)MathSciNetCrossRefGoogle Scholar
 17.J.C. Degollado, C.A.R. Herdeiro, Stationary scalar configurations around extremal charged black holes. Gen. Relat. Gravit. 45, 2483 (2013)ADSMathSciNetCrossRefGoogle Scholar
 18.P.V.P. Cunha, C.A.R. Herdeiro, E. Radu, H.F. Runarsson, Shadows of Kerr black holes with scalar hair. Phys. Rev. Lett. 115, 211102 (2015)ADSCrossRefGoogle Scholar
 19.Y. Brihaye, C. Herdeiro, E. Radu, Inside black holes with synchronized hair. Phys. Lett. B 760, 279 (2016)ADSCrossRefGoogle Scholar
 20.J.D. Bekenstein, Black hole hair: 25years after, arXiv:grqc/9605059
 21.Carlos A.R. Herdeiro, Eugen Radu, Asymptotically flat black holes with scalar hair: a review. Int. J. Mod. Phys. D 24(09), 1542014 (2015)ADSMathSciNetCrossRefGoogle Scholar
 22.S. Hod, Noscalarhair theorem for spherically symmetric reflecting stars. Phys. Rev. D 94, 104073 (2016)ADSMathSciNetCrossRefGoogle Scholar
 23.S. Hod, No nonminimally coupled massless scalar hair for spherically symmetric neutral reflecting stars. Phys. Rev. D 96, 024019 (2017)ADSMathSciNetCrossRefGoogle Scholar
 24.Srijit Bhattacharjee, Sudipta Sarkar, Nohair theorems for a static and stationary reflecting star. Phys. Rev. D 95, 084027 (2017)ADSMathSciNetCrossRefGoogle Scholar
 25.S. Hod, Charged massive scalar field configurations supported by a spherically symmetric charged reflecting shell. Phys. Lett. B 763, 275 (2016)ADSCrossRefGoogle Scholar
 26.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)ADSCrossRefGoogle Scholar
 27.Yan Peng, Bin Wang, Yunqi Liu, Scalar field condensation behaviors around reflecting shells in Antide Sitter spacetimes. Eur. Phys. J. C 78(8), 680 (2018)ADSCrossRefGoogle Scholar
 28.Yan Peng, Scalar field configurations supported by charged compact reflecting stars in a curved spacetime. Phys. Lett. B 780, 144–148 (2018)ADSMathSciNetCrossRefGoogle Scholar
 29.Shahar Hod, Charged reflecting stars supporting charged massive scalar field configurations. Eur. Phys. J. C 78, 173 (2017)ADSCrossRefGoogle Scholar
 30.Yan Peng, Static scalar field condensation in regular asymptotically AdS reflecting star backgrounds. Phys. Lett. B 782, 717–722 (2018)ADSCrossRefGoogle Scholar
 31.Yan Peng, On instabilities of scalar hairy regular compact reflecting stars. JHEP 10, 185 (2018)ADSMathSciNetCrossRefGoogle Scholar
 32.Yan Peng, Hair formation in the background of noncommutative reflecting stars. Nucl. Phys. B 938, 143–153 (2019)ADSCrossRefGoogle Scholar
 33.Yan Peng, Scalar condensation behaviors around regular Neumann reflecting stars. Nucl. Phys. B 934, 459–465 (2018)ADSCrossRefGoogle Scholar
 34.Y. Peng, No hair theorem for spherically symmetric regular compact stars with Dirichlet boundary conditions. arXiv:1901.11415 [grqc]
 35.D. Núñez, H. Quevedo, D. Sudarsky, Black holes have no short hair. Phys. Rev. Lett. 76, 571 (1996)ADSMathSciNetCrossRefGoogle Scholar
 36.S. Hod, Hairy black holes and null circular geodesics. Phys. Rev. D 84, 124030 (2011). arXiv:1112.3286 [grqc]ADSCrossRefGoogle Scholar
 37.Pallab Basu, Chethan Krishnan, P.N. Bala Subramanian, Hairy black holes in a box. JHEP 11, 041 (2016)MathSciNetCrossRefGoogle Scholar
 38.Yan Peng, Studies of a general flat space/boson star transition model in a box through a language similar to holographic superconductors. JHEP 1707, 042 (2017)ADSCrossRefGoogle Scholar
 39.Yan Peng, Bin Wang, Yunqi Liu, On the thermodynamics of the black hole and hairy black hole transitions in the asymptotically flat spacetime with a box. Eur. Phys. J. C 78(3), 176 (2018)ADSCrossRefGoogle Scholar
 40.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Holographic superconductors. JHEP 0812, 015 (2008)ADSMathSciNetCrossRefGoogle Scholar
 41.Hua Bi Zeng, Yu. Tian, Zhe Yong Fan, ChiangMei Chen, Nonlinear transport in a two dimensional holographic superconductor. Phys. Rev. D 93, 121901 (2016)ADSCrossRefGoogle Scholar
 42.X.H. Ge, B. Wang, S.F. Wu, G.H. Yang, Analytical study on holographic superconductors in external magnetic field. JHEP 08, 108 (2010)ADSCrossRefGoogle Scholar
 43.Yi Ling, Peng Liu, Wu JianPin, Note on the butterfly effect in holographic superconductor models. Phys. Lett. B 768, 288 (2017)ADSCrossRefGoogle Scholar
 44.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Holographic superconductors. JHEP 12, 015 (2008)ADSMathSciNetCrossRefGoogle Scholar
 45.Nicolas SanchisGual, Juan Carlos Degollado, Pedro J. Montero, Jos A. Font, Carlos Herdeiro, Explosion and final state of an unstable Reissner–Nordström black hole. Phys. Rev. Lett 116, 141101 (2016)ADSCrossRefGoogle Scholar
 46.Sam R. Dolan, Supakchai Ponglertsakul, Elizabeth Winstanley, Stability of black holes in Einsteincharged scalar field theory in a cavity. Phys. Rev. D 92, 124047 (2015)ADSMathSciNetCrossRefGoogle Scholar
 47.Pallab Basu, Chethan Krishnan, P.N. Bala Subramanian, Phases of global AdS black holes. JHEP 06, 139 (2016)ADSMathSciNetCrossRefGoogle Scholar
 48.Marek Rogatko, Karol I. Wysokinski, Viscosity of holographic fluid in the presence of dark matter sector. JHEP 1608, 124 (2016)ADSMathSciNetCrossRefGoogle Scholar
 49.Wu Chen, Xu Renli, Decay of massive scalar field in a black hole background immersed in magnetic field. Eur. Phys. J. C 75(8), 391 (2015)CrossRefGoogle Scholar
 50.Yan Peng, Qiyuan Pan, Yunqi Liu, A general holographic insulator/superconductor model with dark matter sector away from the probe limit. Nucl. Phys. B 915, 69–83 (2017)ADSCrossRefGoogle Scholar
 51.Darío Núñez, Hernando Quevedo, Daniel Sudarsky, Black holes have no short hair. Phys. Rev. Lett. 76, 571–574 (1996)ADSMathSciNetCrossRefGoogle Scholar
 52.Takashi Torii, KeiIchi Maeda, Takashi Tachizawa, NonAbelian black holes and catastrophe theory. 1. Neutral type. Phys. Rev. D 51, 1510–1524 (1995)ADSMathSciNetCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}.