scholarly journals On the Local Existence for the Characteristic Initial Value Problem in General Relativity

2011 ◽  
Vol 2012 (20) ◽  
pp. 4625-4678 ◽  
Author(s):  
Jonathan Luk
Keyword(s):  
General Relativity ◽  
Initial Value Problem ◽  
Local Existence ◽  
Initial Value ◽  
Characteristic Initial Value Problem

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.

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.

scholarly journals Improved existence for the characteristic initial value problem with the conformal Einstein field equations

2020 ◽  
Vol 52 (9) ◽  
Author(s):  
David Hilditch ◽  
Juan A. Valiente Kroon ◽  
Peng Zhao
Keyword(s):  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Asymptotic Region ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Initial Value ◽  
Null Infinity ◽  
Characteristic Initial Value Problem ◽  
Local Existence Of Solutions

Abstract We adapt Luk’s analysis of the characteristic initial value problem in general relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall’s reduction strategy. In analysing the conformal Einstein equations we make use of the Newman–Penrose formalism and a gauge due to J. Stewart.

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