The Affine-Null Formulation Of The Gravitational Equations: Spherical Critical Collapse

A new evolution algorithm for the characteristic initial value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation is applied in the spherically symmetric case to the gravitational collapse of a massless scalar field. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to critical collapse. Unexpected quadratures lead to an evolution algorithm based upon two first order equations which can be integrated along the null rays. It is implemented as a global numerical evolution code based upon the Galerkin method. New results regarding the global properties of critical collapse are presented.