enforce any assumed symmetries. Although the boson star is found by a harmonic ansatz for the time
dependence, here we choose to retain the full time-dependence. However, a considerably simplification
is provided by assuming that the spacetime is spherically symmetric. Following [141],
the most general metric in this case can be written in terms of spherical coordinates as
where
is the lapse function,
is the radial component of the shift vector and
represent components of the spatial metric, with
the metric of a unit
two-sphere. With this metric, the extrinsic curvature only has two independent components
. The constraint equations, Eqs. (18) and (19), can now be written as
where we have defined the auxiliary scalar-field variables
The evolution equations for the metric and extrinsic curvature components reduce to
Similarly, the reduction of the Klein–Gordon equation to first order in time and space leads to the
following set of evolution equations
This set of equations, Eqs. (23) – (32), describes general, time-dependent, spherically symmetric
solutions of a gravitationally-coupled complex scalar field. In the next section, we proceed to solve for
the specific case of a boson star.