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Scalarization of Horizonless Reflecting Stars

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Scalarization of Horizonless Reflecting Stars Physics Letters B 804 (2020) 135372 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Scalarization of horizonless reflecting stars: Neutral scalar fields non-minimally coupled to Maxwell fields Yan Peng School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China a r t i c l e i n f o a b s t r a c t Article history: We analyze condensation behaviors of neutral scalar fields outside horizonless reflecting stars in the Received 25 January 2020 Einstein-Maxwell-scalar gravity. It was known that minimally coupled neutral scalar fields cannot Received in revised form 26 February 2020 exist outside horizonless reflecting stars. In this work, we consider non-minimal couplings between Accepted 16 March 2020 scalar fields and Maxwell fields, which is included to aim to trigger formations of scalar hairs. We Available online 24 March 2020 analytically demonstrate that there is no hair theorem for small coupling parameters below a bound. For Editor: M. Trodden large coupling parameters above the bound, we numerically obtain regular scalar hairy configurations supported by horizonless reflecting stars. © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction the context of scalar-tensor theories [51]. At present, lots of spon- taneous scalarization models were constructed in the background The recent observation of gravitational waves may provide a of black holes [52–60]. way to test the nature of astrophysical black holes [1–3]. In clas- As mentioned above, some candidate quantum-gravity models sical general relativity, one famous property of black holes is no suggested that quantum effects may prevent the formation of hori- hair theorem, which states that a nontrivial static scalar field can- zons and a reflecting wall may lay above the would-be horizon po- not exist in the exterior region of asymptotically flat black holes, sition [14–21]. Interestingly, it was found that neutral scalar field see references [4–11] and reviews [12,13]. However, some can- hairs cannot exist outside such horizonless reflecting stars (even didate quantum-gravity models suggested that, due to quantum the star is charged) [26]. When considering scalar-Gauss-Bonnet effects [14–18], the classical absorbing horizon should be replaced couplings, we showed that the coupling can lead to the formation by a reflecting surface [19–25]. Interestingly, no hair behaviors also of neutral scalar field hairs in the background of horizonless re- appear for such horizonless reflecting stars [26–38]. In particular, flecting stars [61]. In fact, black hole spontaneous scalarization is even for static scalar fields non-minimally coupled to the Ricci cur- a very universal property, which also can be induced by another vature, no hair theorem still holds in backgrounds of black holes type of non-minimal couplings between scalar fields and Maxwell and horizonless reflecting stars [39–44]. fields [62–67]. As a further step, it is very interesting to exam- Intriguingly, for static scalar fields non-minimally coupled to ine whether scalar-Maxwell couplings can trigger condensations of the Gauss-Bonnet invariant, scalar field hairs can exist outside neutral scalar fields outside horizonless reflecting stars. asymptotically flat black holes [45–48]. Spinning hairy black holes This work is organized as follows. We start by introducing a were also numerically obtained in the scalar-Gauss-Bonnet the- model with a neutral scalar field coupled to the Maxwell field in ory [49]. In addition, analytical formula of the scalar-Gauss-Bonnet the charged horizonless reflecting star spacetime. For small cou- coupling parameter was explored in [50]. These scalarization mod- pling parameters, the neutral scalar field cannot exist. In contrast, 2 for large coupling parameters, we get numerical solutions of scalar els are constructed by introducing an additional term f (ψ)RGB, 2 hairy horizonless reflecting stars. Main conclusions are presented where f (ψ) is a function of the scalar field ψ and RGB is the Gauss-Bonnet invariant. In the scalar-Gauss-Bonnet gravity, under in the last section. scalar perturbations, the bald black hole is thermodynamically un- stable and it may evolve into a hairy black hole [46,47]. This 2. Investigations on the coupling parameter between scalar fields and Maxwell fields intriguing mechanism of hair formations is usually called sponta- neous scalarization, which was found long ago for neutron stars in We take the Lagrange density with scalar fields non-minimally coupled to Maxwell fields in the asymptotically flat background. It E-mail address: [email protected]. is defined by the following expression [62–64] https://doi.org/10.1016/j.physletb.2020.135372 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 Y. Peng / Physics Letters B 804 (2020) 135372 ν 2 2 2 L = R −∇ ∇ν − μ ψ + f ()I. (1) 2M Q N(rpeak) = 1 − + > 0. (10) r r2 Here R is the scalar curvature. is the static neutral scalar field peak peak I = ρσ with mass μ. f () is a function coupled to Fρσ F . In the With (9) and (10), we get the relation 2 linearized regime, there is I =−Q and the general coupling r4 α Q 2 l(l + 1) function can be expressed as f () = 1 − α2, where α is the − − μ2 0. (11) r4 r2 model parameter describing coupling strength [62–64]. In the limit peak peak of α → 0, it returns to the usual Einstein-Maxwell-scalar gravity. According to the inequality (11), we deduce a bound on the The scalar field differential equation is coupling parameter f I 2 4 2 ∇ν∇ − 2 + = μ r + l(l + 1)r 2r4 + l(l + 1)r2 ν μ 0. (2) peak peak μ s s 2 α 2 2 Q Q The charged static spherically symmetric background is μ2(M + M2 − Q 2)4 + l(l + 1)(M + M2 − Q 2)2 2 > . (12) dr Q 2 ds2 =−N(r)dt2 + + r2(dθ 2 + sin2θdφ2). (3) N(r) If compact reflecting stars are surrounded with static neutral In the weak-field limit, the metric function N(r) is scalar hairs, the parameter α should be above the bound (12). In other words, we obtain a no hair theorem for small coupling pa- 2 2M Q rameters N(r) = 1 − + (4) r r2 μ2(M + M2 − Q 2)4 + l(l + 1)(M + M2 − Q 2)2 with M and Q representing the star mass and star charge respec- α . (13) Q 2 tively. We point out that this background metric is valid on the condition αψ 2 1 It implies that neutral massive static exterior scalar fields usually We take the scalar field decomposition cannot exist in cases of large field mass, large spherical harmonic index, large star mass or small star charge. In particular, for α = 0, imφ (r,θ,φ)= e Slm(θ)Rlm(r). (5) the no hair condition (13) always holds, which means that charged lm stars cannot support minimally coupled neutral scalar hairs. In the following, we numerically show that scalar hairs can be induced For simplicity, we label R (r) as ψ(r). With relations (2), (3), lm by large non-minimal coupling parameters satisfying (12). Q 2 (5) and I =− , we derive the ordinary differential equation [33, r4 68,69] 3. Neutral scalar field hairs non-minimally coupled to Maxwell fields 2 2 2 N α Q l(l + 1) μ ψ + ( + )ψ + ( − − )ψ = 0. (6) r N r4 N r2 N N We numerically solve the equation (6)together with boundary + conditions (7). Besides parameters rs, M, Q , l and α, we also need Here l is the spherical harmonic index and l(l 1) is the charac- initial values of ψ(rs) and ψ (rs) to integrate the equation. The re- teristic eigenvalue of the angular scalar eigenfunction Slm(θ). flecting condition of (7)gives the value ψ(rs) = 0. According to We label rs as the radial coordinate of the star surface. Since the symmetry ψ → kψ of equation (6), we firstly set ψ (r ) = 1 we focus on the compact star without a horizon, the star sur- s without loss of generality. Since the equation (6)also satisfies the face is outside the gravitational radius, which can be expressed symmetry r → γ r, μ → μ/γ , M → γ M, Q → γ Q , we use di- 2 2 as rs > M + M − Q . At the star surface, we take scalar re- mensionless parameters μrs, μM, μQ , l and α to describe the flecting surface boundary conditions ψ(rs) = 0. In the far region, system. For given values of μrs, μM, μQ and l, using standard the physical massive static scalar fields asymptotically behave as shooting methods, we search for the proper α with the vanishing →∞ ∼ 1 −μr ψ(r ) r e . So the scalar field satisfies bound-state condi- condition ψ(∞) = 0. The equation of motion of the scalar field is tions linear with respect to ψ and the solution is scale free. After getting numerical solutions, we modify the boundary condition ψ (rs) = 1 = ∞ = 2 2 ψ(rs) 0,ψ() 0. (7) so that αψ 1 holds. In this work, we take very small αψ sat- 2 −7 isfying αψ < 10 . According to boundary conditions (7), one concludes that the In the case of μrs = 2.7, μM = 1.5, μQ = 1.0 and l = 0, we function ψ(r) must possess (at least) one extremum point r = r peak choose various α to try to get the physical solution with ψ(∞) = = between the star surface r rs and spatial infinity.
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