Potential for ill-posedness in several second-order formulations of the Einstein equations

Simonetta Frittelli
Phys. Rev. D 70, 044029 – Published 23 August 2004
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Abstract

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false.

  • Received 2 April 2004

DOI:https://doi.org/10.1103/PhysRevD.70.044029

©2004 American Physical Society

Authors & Affiliations

Simonetta Frittelli*

  • Department of Physics, Duquesne University, Pittsburgh, Pennsylvania 15282, USA
  • Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

  • *Electronic address: simo@mayu.physics.duq.edu

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Issue

Vol. 70, Iss. 4 — 15 August 2004

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