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
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 Either from Noether’s theorem or by taking an additional covariant derivative of Eq. (63), one obtains
that the electric current
follows a conservation law. The spatial integral of the time component of this
current, which can be identified with the total charge
, is conserved. This charge is proportional to the
number of particles,
. The mass
and the total charge
can be calculated by
associating the asymptotic behavior of the metric with that of Reissner–Nordström metric,
We look for a time independent metric by first assuming a harmonically varying scalar field as in
Eq. (33). 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
. 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,
. Because the
electromagnetic field depends only on derivatives of the potential, we can use this freedom to set
[123].
With these conditions, it is possible to find numerical solutions in equilibrium as described in Ref. [123].
Solutions are found for . For
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 for different values of
in
Figure 5
. Trivially, for
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.
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.
http://www.livingreviews.org/lrr-2012-6 |
Living Rev. Relativity 15, (2012), 6
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