Null cone evolution of axisymmetric vacuum space–times

Abstract

The details of an algorithm for the global evolution of asymptotically flat, axisymmetric space–times, based upon a characteristic initial value formulation using null cones as evolution hypersurfaces is presented. A new static solution of the vacuum field equations which provides an important test bed for characteristic evolution codes is identified. It is also shown how linearized solutions of the Bondi equations can be generated by solutions of the scalar wave equation, thus providing a complete set of test beds in the weak field regime. These tools are used to establish that the algorithm is second order accurate and stable, subject to a Courant–Friedrichs–Lewy condition. In addition, the numerical versions of the Bondi mass and news function, calculated at scri on a compactified grid, are shown to satisfy the Bondi mass loss equation to second order accuracy. This verifies that numerical evolution preserves the Bianchi identities. Results of numerical evolution confirm the theorem of Christodoulou and Klainerman that in vacuum, weak initial data evolve to a flat space–time. For the class of asymptotically flat, axisymmetric vacuum space–times, for which no nonsingular analytic solutions are known, the algorithm provides highly accurate solutions throughout the regime in which neither caustics nor horizons form.