### 2.3 The 3+1 decomposition of the spacetime

Although the spacetime description of general relativity is very elegant, the covariant form of Einstein equations is not suitable to describe how an initial configuration evolves towards the future. It is, therefore, more intuitive to instead consider a succession of spacetime geometries, where the evolution of a given slice is given by the Einstein equations (for more detailed treatments see [4, 24, 34, 96]). In order to convert the four-dimensional, covariant Einstein equations to a more intuitive “space+time” or 3+1 decomposition, the following steps are taken:
• specify the choice of coordinates. The spacetime is foliated by a family of spacelike hypersurfaces, which are crossed by a congruence of time lines that will determine our observers (i.e., coordinates). This congruence is described by the vector field , where is the lapse function which measures the proper time of the observers, is the shift vector that measures the displacement of the observers between consecutive hypersurfaces and is the timelike unit vector normal to the spacelike hypersurfaces.
• decompose every 4D object into its 3+1 components. The choice of coordinates allows for the definition of a projection tensor . Any four-dimensional tensor can be decomposed into 3+1 pieces using the spatial projector to obtain the spatial components, or contracting with for the time components. For instance, the line element can be written in a general form as
The stress-energy tensor can then be decomposed into its various components as
• write down the field equations in terms of the 3+1 components. Within the framework outlined here, the induced (or equivalently, the spatial 3D) metric and the scalar field are as yet still unknown (remember that the lapse and the shift just describe our choice of coordinates). In the original 3+1 decomposition (ADM formulation [9]) an additional geometrical tensor is introduced to describe the change of the induced metric along the congruence of observers. Loosely speaking, one can view the determination of and as akin to the specification of a position and velocity for projectile motion. In terms of the extrinsic curvature and its trace, , the Einstein equations can be written as
In a similar fashion, one can introduce a quantity for the Klein–Gordon equation which reduces it to an equation first order in time, second order in space
• 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.