Field equations in spherical symmetry

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 $2 2 2 2 dΩ = d 𝜃 + sin 𝜃d φ$ 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 $4 πr2$ , and by t being orthogonal to r (polar-radial coordinates). One more condition is required to fix the coordinate completely. Choptuik chose $α$ = 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,

( ) Φ = α-Π , (22 ) ,t a ,r 1 ( α ) Π,t = -2 r2--Φ . (23 ) r a ,r

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 $a,t$ , $a,r$ , and $α,r$ , and are