Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations

Abstract

We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the ADM formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as non-trivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent.

v1: 9 pages. v2: sign error corrected in eq.(8) and relevant discussion amended accordingly. v3: short section added, to appear in Phys Rev D. v4: typos corrected, published version

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Cited by in corpus (13)

• Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations
• On the well posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations
• Boundary Conditions for the Einstein Evolution System
• Symmetric hyperbolicity and consistent boundary conditions for second-order Einstein equations
• Optimal Constraint Projection for Hyperbolic Evolution Systems
• Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision
• Constraint-preserving boundary conditions in the Z4 Numerical Relativity formalism
• Absorbing boundary conditions for Einstein's field equations
• Algebraic stability analysis of constraint propagation
• Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity
• Boundary Conditions for the Gravitational Field
• Einstein boundary conditions for the Einstein equations in the conformal-traceless decomposition
• Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations