Well-posed constrained evolution of 3+1 formulations of General Relativity

Details Well-posed constrained evolution of 3+1 formulations of General Relativity Well-posed constrained evolution of 3+1 formulations of General Relativity

2005-11-14

arxiv.org/abs/gr-qc/0511075v3

Phys.Rev. D75 (2007) 024026

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Accepted to PRD, typos corrected, added references, extension of the analysis

Scientific paper

We present an analysis of well-posedness of constrained evolution of 3+1 formulations of GR. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of ADM, BSSN and other 3+1 formulations derived from ADM by adding combinations of constraints to the right-hand-side of ADM and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical Arnowitt--Deser--Misner (ADM) and many other 3+1 formulations of general relativity.

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