No-go theorem for spatially regular boson stars made of static nonminimally coupled massive scalar fields

We present a compact theorem which reveals the fact that static spatially regular massive scalar fields with nonminimal coupling to gravity cannot form spherically symmetric asymptotically flat horizonless matter configurations. In particular, the no-go theorem rules out the existence of boson stars made of static scalar fields with generic values of the physical parameter $\xi$ which quantifies the coupling between the spacetime curvature and the massive bosonic fields.


I. INTRODUCTION
The non-linearly coupled Einstein-massive-scalar field equations have a physically interesting and mathematically elegant structure that has attracted the attention of researches during the last five decades (see [1][2][3][4][5][6][7][8][9][10][11][12] and references therein). In particular, this physical system is known to possess asymptotically flat bound-state solutions in the form of rotating hairy black holes that support spatially regular stationary massive scalar field configurations [11,12].
Intriguingly, the phenomenologically rich Einstein-massive-scalar system also admits horizonless bound-state solutions known as boson stars. These physically interesting compact objects are made of spatially regular stationary massive scalar fields. The physical and mathematical properties of these self-gravitating horizonless matter configurations have been extensively explored by physicists and mathematicians (see [1,2] and references therein). In particular, the intriguing possibility has been considered in the physics literature that boson stars may contribute to the elusive dark matter distributions in galaxies [1,2]. In addition, compact bound-state scalar configurations (boson stars) have been considered in the literature as physically exotic horizonless mimickers of classical black-hole spacetimes [1,2].
The main goal of the present paper is to reveal the intriguing fact that, as opposed to stationary matter fields, static self-gravitating massive scalar fields with non-minimal coupling to gravity cannot form spatially regular horizonless bound-state matter configurations (boson stars). In this context, it is interesting to note that the asymptotic analysis of the non-linearly coupled Einstein-scalar field equations presented in [13] can be used to rule out the existence of static boson stars (spatially regular matter configurations) made of nonminimally coupled massive scalar fields in the dimensionless regime ξ < 0 of the nonminimal coupling parameter [14].
In the present paper we shall explore the physical and mathematical properties of the Einstein-scalar equations in the complementary physical regime ξ ≥ 0 of the dimensionless nonminimal coupling parameter. In particular, below we shall present a remarkably compact no-go theorem for static boson stars which reveals the fact that the non-linearly coupled Einstein-massive-scalar field equations do not admit static spatially regular bound-state solutions made of self-gravitating scalar fields with nonminimal coupling to gravity.

II. DESCRIPTION OF THE SYSTEM
We consider a spatially regular asymptotically flat spacetime whose curvature is sourced by a static self-gravitating massive scalar field ψ with nonminimal coupling to gravity. The curved spherically symmetric spacetime is described by the line element [8,15,16] ds 2 = −e ν dt 2 + e λ dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) , where ν = ν(r) and λ = λ(r). The metric functions describing the spatially regular matter configurations are characterized by the near-origin functional behaviors [17] e λ (r → 0) = 1 + O(r 2 ) ; 0 < e ν < ∞ with [18] λ ′ (r → 0) → 0 ; In addition, the metric functions of the asymptotically flat spacetime are characterized by the far-region (r → ∞) functional relations [8] ν where M is the asymptotically measured finite ADM mass of the spatially regular static matter configuration. The self-gravitating scalar field is characterized by the action [8,19] where µ [20] is the mass of the matter field and the dimensionless nonminimal coupling parameter ξ quantifies the coupling between the scalar curvature R(r) [see Eq. (22) below] of the spatially regular spacetime and the massive scalar field ψ. For later purposes we note that asymptotically flat spacetimes are characterized by the far-region functional behavior [8] R of the scalar curvature. In addition, spatially regular nonminimally coupled scalar matter configurations are characterized by asymptotically finite radial eigenfunctions. In particular, as discussed in [8], the bounded functional relation for the scalar eigenfunction guarantees that the corresponding physically acceptable curved spacetimes are characterized by finite and positive values of the asymptotic effective gravitational constant G eff = G(1 − 8πGξψ 2 ) [8].
The spatial and temporal functional behaviors of the nonminimally coupled massive scalar fields with the action (5) are governed by the compact partial differential equation [8] In particular, for spherically symmetric static matter configurations, one obtains from (1) and (8) the characteristic ordinary differential equation which determines the spatial behavior of the scalar eigenfunction in the curved spacetime (1). The components of the energy-momentum tensor T α β , which characterizes the nonminimally coupled static scalar fields with the action (5), are given by the functional expressions [8] and As explicitly proved in [8], physically acceptable spacetimes are characterized by finite components of the energymomentum tensor: III. THE NO-GO THEOREM FOR THE STATIC NONMINIMALLY COUPLED MASSIVE SCALAR FIELD CONFIGURATIONS (BOSON STARS) IN THE REGIME ξ ≥ 0 As emphasized above, the asymptotic (r ≫ M ) analysis of the Einstein-scalar field equations presented in [13,21] rules out the existence of static boson stars made of self-gravitating massive scalar fields in the dimensionless regime ξ < 0 of the nonminimal coupling parameter. In the present section we shall complete the no-go theorem for the static sector of horizonless boson stars. In particular, we shall explicitly prove that nonminimally coupled massive scalar fields with a dimensionless field-curvature coupling parameter in the physical regime ξ ≥ 0 cannot form spatially regular asymptotically flat static matter configurations.
We shall now prove that the eigenfunction ψ(r), which characterizes the spatial behavior of the spherically symmetric static scalar field configurations, has a non-monotonic dependence on the areal coordinate r. In particular, if the radial scalar eigenfunction is characterized by the near-origin relation ψ(r → 0) → 0, then its asymptotic behavior [see Eq.
would immediately imply that it must possess (at least) one extremum point r peak ∈ (0, ∞) in the curved spacetime (1). It therefore remains to be proved that nonminimally coupled massive scalar fields which are characterized by the near-origin behavior ψ(r → 0) = 0 (17) must also posses an extremum point. For later purposes, we note that one finds from the energy-momentum components (10) and (12) the simple functional relation In particular, taking cognizance of Eqs. (2), (3), (13), and (18), one deduces the characteristic near-origin functional behavior for spatially regular scalar matter configurations. Likewise, taking cognizance of Eqs. (2), (3), (13), (17), (19), and the relation [see Eqs. (10) and (11)] one finds the bounded near-origin functional behavior for spatially regular static field configurations. Substituting the Ricci scalar curvature [22] which characterizes the spherically symmetric self-gravitating massive scalar fields into Eq. (9), one obtains the ordinary differential equation which determines the radial behavior of the nonminimally coupled scalar matter configurations, where For later purposes we note that, taking cognizance of Eq. (16), one finds that F (r; ξ) is characterized by the simple asymptotic behavior The radially-dependent function F (r; ξ) is obviously positive definite for nonminimally coupled massive scalar fields in the dimensionless regimes ξ ≥ 1/6 and ξ ≤ 0 [see Eq. (24)]. We shall now prove that this property also characterizes the function F (r) in the complementary dimensionless regime 0 < ξ < 1/6 of the nonminimal coupling parameter. Let us assume that the function F (r) has a root at some point r 0 ∈ (0, ∞), in which case one finds from (23) the functional relation [23] 8πξ(6ξ − 1)(ψ ′ ) 2 = µ 2 e λ (1 + 8πξψ 2 ) at r = r 0 .
Taking cognizance of the fact that, in the dimensionless regime 0 < ξ < 1/6, the l.h.s of (26) is non-positive whereas the r.h.s of (26) is positive definite, one deduces that the equality sign in (23) cannot be respected at the assumed root r = r 0 of F (r). We therefore learn that the radial function F (r) has no roots. In particular, using the characteristic asymptotic relation (25), one finds that the radially-dependent function F (r; ξ) is characterized by the simple inequality in the entire curved spacetime. Taking cognizance of Eqs. (2), (3), (19), (21), and (23), one obtains the near-origin (r → 0) radial equation whose physically acceptable solution [24] is characterized by the small-r spatial behavior [25,26] ψ(r → 0) = B 1 + 1 6 where B is a constant. Interestingly, and most importantly for our analysis, one learns from Eqs. (24), (27), and (29) that spatially regular nonminimally coupled massive scalar fields with ξ ≥ 0 are characterized by the near-origin functional relations The asymptotic large-r and small-r functional behaviors (16) and (30), when combined together, reveal the important fact that the characteristic radial eigenfunction ψ(r) of the spatially regular nonminimally coupled massive scalar fields must possess, in accord with our previous assertion, (at least) one extremum point r peak ∈ (0, ∞) in the curved spacetime (1). In particular, at this extremum point the scalar eigenfunction is characterized by the functional relations Substituting (31) into the radial differential equation (23), one finds F · ψψ ′′ = µ 2 e λ (1 + 8πξψ 2 )ψ 2 at r = r peak .
Interestingly, taking cognizance of the analytically derived relations (27) and (31) one learns that, in the dimensionless physical regime ξ ≥ 0, the l.h.s of (32) is negative definite whereas the r.h.s of (32) is positive definite. Thus, for spatially regular static scalar configurations, the radial differential equation (23) cannot be satisfied at the characteristic extremum point (31). We therefore conclude that static spatially regular massive scalar fields whose nonminimal coupling parameter lies in the dimensionless physical regime ξ ≥ 0 cannot form spherically symmetric asymptotically flat horizonless matter configurations (boson stars).

IV. SUMMARY
Boson stars represent self-gravitating horizonless compact objects which are made of massive scalar fields. The physical and mathematical properties of these spatially regular bound-state matter configurations have attracted the attention of physicists and mathematicians during the last three decades (see [1,2] and references therein).
In the present paper we have presented a novel non-existence theorem for static boson stars. In particular, the static sector of the non-linearly coupled Einstein-matter field equations has been studied analytically for spatially regular self-gravitating massive scalar fields with nonminimal coupling to gravity. Interestingly, our compact no-go theorem explicitly rules out the existence of asymptotically flat horizonless spacetimes that contain spatially regular boson stars which are made of nonminimally coupled [27] massive scalar fields.