Ill-posedness in the Einstein equations

Abstract

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.

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

References in corpus (1)

• Well-posed forms of the 3+1 conformally-decomposed Einstein equations

Cited by in corpus (17)

• Numerical Relativity and Compact Binaries
• Numerical Relativity: A review
• Modeling gravitational radiation from coalescing binary black holes
• Extraction of Gravitational Waves in Numerical Relativity
• An explicit harmonic code for black-hole evolution using excision
• Numerical Relativity of Compact Binaries in the 21st Century
• Cauchy boundaries in linearized gravitational theory
• Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system
• Dynamical shift conditions for the Z4 and BSSN hyperbolic formalisms
• Numerical stability for finite difference approximations of Einstein's equations
• Gravitational waveforms with controlled accuracy
• Absorbing boundary conditions for simulation of gravitational waves with spectral methods in spherical coordinates
• Constraint propagation in the family of ADM systems
• Gravitational Waves from a Fissioning White Hole
• Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations
• Numerical modeling of black holes as sources of gravitational waves in a nutshell
• Constraint and gauge shocks in one-dimensional numerical relativity