1,411 research outputs found
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
Symposium Introduction: Copyright Protection and Reverse Engineering of Software
Symposium introductio
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
Foreword
Symposium: Copyright Owners\u27 Rights and Users\u27 Privileges on the Interne
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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