Problems which are well-posed in a generalized sense with applications to the Einstein equations

In the harmonic description of general relativity, the principle part of Einstein equations reduces to a constrained system of 10 curved space wave equations for the components of the space-time metric. We use the pseudo-differential theory of systems which are well-posed in the generalized sense to establish the well-posedness of constraint preserving boundary conditions for this system when treated in second order differential form. The boundary conditions are of a generalized Sommerfeld type that is benevolent for numerical calculation.Comment: Final version to appear in Classical and Qunatum Gravit

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Stability of quasi-linear hyperbolic dissipative systems

In this work we want to explore the relationship between certain eigenvalue condition for the symbols of first order partial differential operators describing evolution processes and the linear and nonlinear stability of their stationary solutions.Comment: 16 pages, Te

Hyperbolicity and Constrained Evolution in Linearized Gravity

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). Since computational solution of differential equations introduces almost inevitable errors, it is clearly "more correct" to introduce a scheme which actively maintains the constraints by solution ({\it constrained} evolution). This has shown promise in computational settings, but the analysis of the resulting mixed elliptic hyperbolic method has not been completely carried out. We present such an analysis for one method of constrained evolution, applied to a simple vacuum system, linearized gravitational waves. We begin with a study of the hyperbolicity of the unconstrained Einstein equations. (Because the study of hyperbolicity deals only with the highest derivative order in the equations, linearization loses no essential details.) We then give explicit analytical construction of the effect of initial data setting and constrained evolution for linearized gravitational waves. While this is clearly a toy model with regard to constrained evolution, certain interesting features are found which have relevance to the full nonlinear Einstein equations.Comment: 18 page

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Symposium: Copyright Owners\u27 Rights and Users\u27 Privileges on the Interne

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Symposium: Copyright\u27s Balance in an Internet Worl