We consider spherically symmetric, equilibrium configurations corresponding to minimal energy solutions while requiring the spacetime to be static. In Schwarzschild-like coordinates, the general, spherically symmetric, static metric can be written as
in terms of two real metric functions, The equilibrium equations are obtained by substituting the metric of Eq. (34) and the harmonic ansatz
of Eq. (33
) into the spherically symmetric EKG system of Eqs. (27
– 32
) with
, resulting in
three first order partial differential equations (PDEs)
Notice that the form of the metric in Eq. (34) resembles Schwarzschild allowing the association
, where
is the ADM mass of the spacetime. This allow us to define a more
general mass aspect function
In isotropic coordinates, the spherically symmetric metric can be written as
where As above, boson stars are spherically symmetric solutions of the Eqs. (38 – 40
) with asymptotic behavior
given by Eqs. (41
– 45
). For a given value of the central amplitude of the scalar field
, there
exist configurations with some effective radius and a given mass satisfying the previous conditions for a
different set of
discrete eigenvalues
. As
increases, one obtains solutions with an increasing
number of nodes in
. The configuration without nodes is the ground state, while all those with any
nodes are excited states. As the number of nodes increases, the distribution of the mass as a function of the
radius becomes more homogeneous.
As the amplitude increases, the stable configuration has a larger mass while its effective radius
decreases. This trend indicates that the compactness of the boson star increases. However, at some point
the mass instead decreases with increasing central amplitude. Similar to models of neutron stars (see
Section 4 of [59]), this turnaround implies a maximum allowed mass for a boson star in the ground state,
which numerically was found to be
. The existence of a maximum mass for boson
stars is a relativistic effect, which is not present in the Newtonian limit, while the maximum of baryonic
stars is an intrinsic property.
Solutions for a few representative boson stars in the ground state are shown in Figure 2 in isotropic
coordinates. The boson stars becomes more compact for higher values of
, implying narrower profiles
for the scalar field, larger conformal factors, and smaller lapse functions, as the total mass
increases.
http://www.livingreviews.org/lrr-2012-6 |
Living Rev. Relativity 15, (2012), 6
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