3.1 Field equations in spherical symmetry
The Einstein equations are
and the matter equation is
Note that the matter equation of motion is contained within the contracted Bianchi identities. Choptuik
chose Schwarzschild-like coordinates,
where is the metric on the unit 2-sphere. This choice of coordinates is defined by
the radius r giving the surface area of 2-spheres as , and by t being orthogonal to r (polar-radial
coordinates). One more condition is required to fix the coordinate completely. Choptuik chose = 1 at
r = 0, so that t is the proper time of the central observer.
In the auxiliary variables
the wave equation becomes a first-order system,
In spherical symmetry there are four algebraically independent components of the Einstein equations.
Of these, one is a linear combination of derivatives of the other and can be disregarded. The
other three contain only first derivatives of the metric, namely , , and , and are
Because of spherical symmetry, the only dynamics is in the scalar field equations (22, 23). The metric can
be found by integrating the ODEs (24) and (25) for a and at any fixed t, given and .
Equation (26) can be ignored in this “fully constrained” evolution scheme.