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, and . The coordinate is an areal radius such that spheres of constant have surface area . For this reason, these coordinates are often called polar-areal coordinates.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 these equations hold for any stress-energy contributions and for a generic type of self-potentials . In order to close the system of Eqs. (35 – 37), we still have to prescribe this potential. The simplest case admitting localized solutions is the free field case of Eq. (12) for which the potential describes a field with mass and for which the equations can be written as In order to obtain a physical solution of this system, we have to impose the following boundary conditions, which guarantee regularity at the origin and asymptotic flatness. For a given central value of the field , we need only to adjust the eigenvalue to find a solution which matches the asymptotic behavior of Eqs. (44 – 45). This system can be solved as a shooting problem by integrating from towards the outer boundary . Eq. (39) is linear and homogeneous in and one is, therefore, able to rescale the field consistent with Eq. (45). We can get rid of the constants in the equations by re-scaling the variables in the following mannerNotice 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

which measures the total mass contained in a coordinate sphere of radius at time .In isotropic coordinates, the spherically symmetric metric can be written as

where is the conformal factor. A change of the radial coordinate can transform the solution obtained in Schwarzschild coordinates into isotropic ones, in particular where the first condition is the initial value to integrate the second equation backwards, obtained by imposing that far away from the boson star the spacetime resembles Schwarzschild solution.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.

Living Rev. Relativity 15, (2012), 6
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