Null cone evolution of axisymmetric vacuum spacetimes

TL;DR

This paper introduces a second-order accurate, stable characteristic evolution algorithm for axisymmetric vacuum spacetimes, validated through tests with new solutions and linearized models, confirming evolution to flat spacetime.

Contribution

The paper presents a novel characteristic evolution algorithm for axisymmetric vacuum spacetimes, including new static solutions and validation methods using linearized solutions and mass loss calculations.

Findings

Algorithm is second order accurate and stable.

Numerical solutions satisfy the Bondi mass loss equation.

Results confirm evolution to flat spacetime for weak data.

Abstract

We present the details of an algorithm for the global evolution of asymptotically flat, axisymmetric spacetimes, based upon a characteristic initial value formulation using null cones as evolution hypersurfaces. We identify a new static solution of the vacuum field equations which provides an important test bed for characteristic evolution codes. We also show 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 spacetime. For the class of asymptotically flat, axisymmetric vacuum spacetimes, for which no nonsingular analytic solutions are known, the algorithm provides highly accurate solutions throughout the regime in which neither caustics nor horizons form.