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.

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.

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this 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.

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.

scholarly journals A Singular Initial-Value Problem for Second-Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Second Order ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Local Existence And Uniqueness ◽  
Singular Initial Value Problem

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

scholarly journals The Coupled Kuramoto-Sivashinsky-KdV Equations for Surface Wave in Multilayered Liquid Films

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Maomao Cai ◽  
Dening Li ◽  
Chontita Rattanakul
Keyword(s):  
Surface Wave ◽  
Existence Of Solutions ◽  
A Priori ◽  
Local Existence ◽  
Liquid Films ◽  
Energy Estimates ◽  
Initial Value ◽  
Kdv Equations ◽  
Linearized Problem ◽  
Local Existence Of Solutions

We study the coupled Kuramoto-Sivashinsky-KdV equations describing the surface waves on multilayered liquid films. A priori energy estimates for linearized problem are derived and local existence of solutions for initial-value problem is established.

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.

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