Numerical relativity. II. Numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space‒times with two Killing vectors

1983 ◽  
Vol 386 (1791) ◽  
pp. 373-391 ◽  
Keyword(s):  
General Relativity ◽  
Numerical Methods ◽  
Plane Wave ◽  
Numerical Solution ◽  
Initial Value Problem ◽  
Vacuum Field ◽  
Field Equations ◽  
Initial Value ◽  
Wave Space ◽  
Characteristic Initial Value Problem

This is the second of a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity. Although the equations to be integrated have regular coefficients, the nonlinearity leads to the occurrence of singularities after a finite evolution time. In this paper we first discuss some novel techniques for integrating the equations right up to the singularities. The second half of the paper presents as examples the numerical evolution of the Schwarzschild and certain colliding plane wave space‒times.

On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations

1981 ◽  
Vol 375 (1761) ◽  
pp. 169-184 ◽  
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Vacuum Field ◽  
System Of Differential Equations ◽  
Symmetric Hyperbolic System ◽  
Field Equations ◽  
Initial Value ◽  
Null Infinity ◽  
Asymptotic Characteristic ◽  
Characteristic Initial Value Problem

The regular characteristic initial value problem for Einstein’s vacuum field equations where data are given on two intersecting null hypersurfaces is reduced to a characteristic initial value problem for a symmetric hyperbolic system of differential equations. This is achieved by making use of the spin-frame formalism instead of the harmonic gauge condition. The method is applied to the asymptotic characteristic initial value problem for Einstein’s vacuum field equations, where data are given on part of past null infinity and on an incoming null-hypersurface. A uniqueness theorem for this problem is proved by showing that a solution of the problem must satisfy a regular symmetric hyperbolic system of differential equations in a neighbourhood of past null infinity.

On the existence of analytic null asymptotically flat solutions of Einstein’s vacuum field equations

1982 ◽  
Vol 381 (1781) ◽  
pp. 361-371 ◽  
Keyword(s):  
Initial Value Problem ◽  
Analytic Solutions ◽  
Null Hypersurface ◽  
Vacuum Field ◽  
Field Equations ◽  
Initial Value ◽  
Null Infinity ◽  
Asymptotically Flat ◽  
Asymptotic Characteristic ◽  
Characteristic Initial Value Problem

This paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for Einstein’s field equations for analytic data on past null infinity and on an incoming null hypersurface.

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

1981 ◽  
Vol 378 (1774) ◽  
pp. 401-421 ◽  
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.

scholarly journals Revisiting the characteristic initial value problem for the vacuum Einstein field equations

2020 ◽  
Vol 52 (10) ◽  
Author(s):  
David Hilditch ◽  
Juan A. Valiente Kroon ◽  
Peng Zhao
Keyword(s):  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Initial Value ◽  
Bootstrap Argument ◽  
Null Hypersurfaces ◽  
Characteristic Initial Value Problem ◽  
Local Existence Of Solutions

AbstractUsing the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

A characteristic initial value problem in general relativity in the case of a perfect fluid with axial symmetry

1973 ◽  
Vol 332 (1591) ◽  
pp. 549-560 ◽  
Keyword(s):  
General Relativity ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Axial Symmetry ◽  
Empty Space ◽  
Space Time ◽  
Initial Value ◽  
Asymptotically Flat ◽  
Physical Variables ◽  
Characteristic Initial Value Problem

An initial value problem for an axisymmetric perfect fluid in general relativity is considered by specifying data on an initial null hypersurface. The approach is an adaptation of the techniques introduced by Bondi, van der Burg & Metzner (1962) for dealing with asymptotically flat, empty space-time. A solution to the problem is shown to exist provided that the metric and physical variables are expanded in positive powers of a radial parameter and certain restrictive conditions at the origin are satisfied.

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