The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

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On the structure of conformal singularities in classical general relativity. II Evolution equations and a conjecture of K. P. Tod

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1993.0159 ◽

1993 ◽

Vol 443 (1919) ◽

pp. 493-515 ◽

Cited By ~ 17

Keyword(s):

Large Scale ◽

Evolution Equations ◽

Physical Space ◽

Space Time ◽

Symmetric Hyperbolic System ◽

First Order ◽

Initial Hypersurface ◽

The Cauchy Problem ◽

Data Subject ◽

The Universe

Consideration is given to the Cauchy problem for perfect fluid space-times which evolve from an initial singularity of conformal type. The evolution equations for the conformally transformed, unphysical geometry are shown to be expressible as a first order symmetric hyperbolic system, albeit with a singular forcing term. It is concluded that the 3-metric on the initial hypersurface of the unphysical space-time constitutes the freely specifiable initial data. Subject to Penroses’s Weyl Curvature Hypothesis, according to which the Weyl tensor was initially zero, it follows that the physical space-time is Robertson–Walker. This may provide a basis for a new explanation for the large-scale isotropy of the universe.

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The asymptotic characteristic initial value problem for Einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1981.0159 ◽

1981 ◽

Vol 378 (1774) ◽

pp. 401-421 ◽

Cited By ~ 99

Keyword(s):

Hyperbolic System ◽

Initial Value Problem ◽

Vacuum Field ◽

Symmetric Hyperbolic System ◽

Field Equations ◽

Initial Value ◽

Null Infinity ◽

First Order ◽

Asymptotic Characteristic ◽

Characteristic Initial Value Problem

The asymptotic characteristic initial value problem for Einstein’s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first-order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown. It is delineated how the same method can be applied to the standard Cauchy problems for Einstein’s vacuum and conformal vacuum equations.

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A Difference Method for Plane Problems in Magnetoelastodynamics

Journal of Applied Mechanics ◽

10.1115/1.3422774 ◽

1972 ◽

Vol 39 (3) ◽

pp. 689-695 ◽

Cited By ~ 4

Author(s):

W. W. Recker

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Numerical Method ◽

Difference Method ◽

Necessary Condition ◽

Two Dimensional ◽

Symmetric Hyperbolic System ◽

Von Neumann ◽

First Order ◽

Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

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CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

International Journal of Modern Physics D ◽

10.1142/s0218271800000037 ◽

2000 ◽

Vol 09 (01) ◽

pp. 13-34 ◽

Cited By ~ 6

Author(s):

GEN YONEDA ◽

HISA-AKI SHINKAI

Keyword(s):

General Relativity ◽

Hyperbolic System ◽

Hyperbolic Systems ◽

Equations Of Motion ◽

Well Posedness ◽

Symmetric Hyperbolic System ◽

The Cauchy Problem ◽

Long Time ◽

Vacuum Spacetime ◽

Gauge Conditions

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

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