Ill-posedness in the Einstein equations

Details Ill-posedness in the Einstein equations Ill-posedness in the Einstein equations

2000-06-22

arxiv.org/abs/gr-qc/0006082v1

J.Math.Phys. 41 (2000) 5535-5549

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, 3 figures, accepted for publication in Journal of Mathematical Physics (to appear August 2000)

Scientific paper

10.1063/1.533423

It is shown that the formulation of the Einstein equations widely in use in numerical relativity, namely, the standard ADM form, as well as some of its variations (including the most recent conformally-decomposed version), suffers from a certain but standard type of ill-posedness. Specifically, the norm of the solution is not bounded by the norm of the initial data irrespective of the data. A long-running numerical experiment is performed as well, showing that the type of ill-posedness observed may not be serious in specific practical applications, as is known from many numerical simulations.

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