3.3 Charged boson stars

Charged boson stars result from the coupling of the boson field to the electromagnetic field [123]. The coupling between gravity and a complex scalar field with a U (1) charge arises by considering the action of Eq. 7View Equation with the following matter Lagrangian density
1[ ( ) ( )] 1 ℒ ℳ = − --gab ∇aϕ¯+ ieAa ¯ϕ (∇bϕ − ieAb ϕ) + V |ϕ|2 − -FabF ab, (60 ) 2 4
where e is the gauge coupling constant. The Maxwell tensor Fab can be decomposed in terms of the vector potential A a
Fab = ∇aAb − ∇bAa . (61 )
The system of equations obtained by performing the variations on the action forms the Einstein–Maxwell–Klein–Gordon system, which contains the evolution equations for the complex scalar field ϕ, the vector potential Aa and the spacetime metric gab [179].

Because a charged BS may be relevant for a variety of scenarios, we detail the resulting equations. For example, cosmic strings are also constructed from a charged, complex scalar field and obeys these same equations. It is only when we choose the harmonic time dependence of the scalar field that we distinguish from the harmonic azimuth of the cosmic string [218]. The evolution equations for the scalar field and for the Maxwell tensor are

dV gab∇a ∇bϕ − 2 ieAa ∇a ϕ − e2ϕAaAa − i eϕ ∇aAa = ----2ϕ (62 ) d |ϕ | ∇aF ab = − Jb = ie (¯ϕ∇b ϕ − ϕ∇b ¯ϕ) + 2e2 ϕϕ¯Ab . (63 )
Notice that the vector potential is not unique; we can still add any curl-free components without changing the Maxwell equations. The gauge freedom can be fixed by choosing, for instance, the Lorentz gauge ∇aAa = 0. Within this choice, which sets the first time derivative of the time component A0, the Maxwell equations reduce to a set of wave equations in a curved background with a non-linear current. This gauge choice resembles the harmonic gauge condition, which casts the Einstein equations as a system of non-linear, wave equations [219].

Either from Noether’s theorem or by taking an additional covariant derivative of Eq. (63View Equation), one obtains that the electric current J a follows a conservation law. The spatial integral of the time component of this current, which can be identified with the total charge Q, is conserved. This charge is proportional to the number of particles, Q = e N. The mass M and the total charge Q can be calculated by associating the asymptotic behavior of the metric with that of Reissner–Nordström metric,

( ) 2G M G Q2 −1 grr = 1 − ------+ ----2- for r → ∞ , (64 ) r 4π r
which is the unique solution at large distances for a scalar field with compact support.

We look for a time independent metric by first assuming a harmonically varying scalar field as in Eq. (33View Equation). We work in spherical coordinates and assume spherical symmetry. With a proper gauge choice, the vector potential takes a particularly simple form with only a single, non-trivial component A = (A (r),0,0,0 ) a 0. This choice implies an everywhere vanishing magnetic field so that the electromagnetic field is purely electric. The boundary conditions for the vector potential are obtained by requiring the electric field to vanish at the origin because of regularity, ∂rA0 (r = 0) = 0. Because the electromagnetic field depends only on derivatives of the potential, we can use this freedom to set A0 (∞ ) = 0 [123].

With these conditions, it is possible to find numerical solutions in equilibrium as described in Ref. [123]. Solutions are found for 2 2 2 2 &tidle;e ≡ e M Planck∕(8π m ) < 1∕2. For 2 &tidle;e > 1∕2 the repulsive Coulomb force is bigger than the gravitational attraction and no solutions are found. This bound on the BS charge in terms of its mass ensures that one cannot construct an overcharged BS, in analogy to the overcharged monopoles of Ref. [154]. An overcharged monopole is one with more charge than mass and is, therefore, susceptible to gravitational collapse by accreting sufficient (neutral) mass. However, because its charge is higher than its mass, such collapse might lead to an extremal Reissner–Nordström BH, but BSs do not appear to allow for this possibility. Interestingly, Ref. [190] finds that if one removes gravity, the obtained Q-balls may have no limit on their charge.

The mass and the number of particles are plotted as a function of ϕc for different values of &tidle;e in Figure 5View Image. Trivially, for &tidle;e = 0 the mini-boson stars of Section (2.4) are recovered. Excited solutions with nodes are qualitatively similar [123]. The stability of these objects has been studied in [119], showing that the equilibrium configurations with a mass larger than the critical mass are dynamically unstable, similar to uncharged BSs.

View Image

Figure 5: The mass (solid) and the number of particles (dashed) versus central scalar value for charged boson stars with four values of &tidle;e as defined in Section 3.3. The mostly-vertical lines crossing the four plots indicate the solution for each case with the maximum mass (solid) and maximum particle number (dashed). Reprinted with permission from [123]; copyright by Elsevier.

Recent work with charged BSs includes the publication of Maple [157] routines to study boson nebulae charge [63, 169, 168]

Other work generalizes the Q-balls and Q-shells found with a certain potential, which leads to the signum-Gordon equation for the scalar field [131, 132]. In particular, shell solutions can be found with a black hole in its interior, which has implications for black hole scalar hair (for a review of black hole uniqueness see [113]).

One can also consider Q-balls coupled to an electromagnetic field, a regime appropriate for particle physics. Within such a context, Ref. [76] studies the chiral magnetic effect arising from a Q-ball. Other work in Ref. [36] studies charged, spinning Q-balls.

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