scholarly journals Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. I. The conformal field equations

1998 ◽  
Vol 58 (6) ◽  
Author(s):  
Jörg Frauendiener
Keyword(s):  
Initial Value Problem ◽  
Einstein Equations ◽  
Numerical Treatment ◽  
Field Equations ◽  
Initial Value ◽  
Conformal Field ◽  
Vacuum Einstein Equations

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.

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

Numerical treatment of a quasilinear initial value problem with boundary layer

2015 ◽  
Vol 93 (11) ◽  
pp. 1845-1859
Author(s):  
Musa Cakir ◽  
Erkan Cimen ◽  
Ilhame Amirali ◽  
Gabil M. Amiraliyev
Keyword(s):  
Boundary Layer ◽  
Initial Value Problem ◽  
Numerical Treatment ◽  
Initial Value

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.

scholarly journals The Einstein–Infeld–Hoffmann legacy in mathematical relativity I: The classical motion of charged point particles

2019 ◽  
Vol 28 (11) ◽  
pp. 1930017
Author(s):  
Michael K.-H. Kiessling ◽  
A. Shadi Tahvildar-Zadeh
Keyword(s):  
Initial Value Problem ◽  
Rest Mass ◽  
Relativity Theory ◽  
Field Energy ◽  
Gravitational Coupling ◽  
Field Equations ◽  
Initial Value ◽  
Classical Motion ◽  
Gravitational Fields ◽  
Charged Point

Einstein, Infeld and Hoffmann (EIH) claimed that the field equations of general relativity theory alone imply the equations of motion of neutral matter particles, viewed as point singularities in space-like slices of spacetime; they also claimed that they had generalized their results to charged point singularities. While their analysis falls apart upon closer scrutiny, the key idea merits our attention. This report identifies necessary conditions for a well-defined general-relativistic joint initial value problem of [Formula: see text] classical point charges and their electromagnetic and gravitational fields. Among them, in particular, is the requirement that the electromagnetic vacuum law guarantees a finite field energy–momentum of a point charge. This disqualifies the Maxwell(–Lorentz) law used by EIH. On the positive side, if the electromagnetic vacuum law of Bopp, Landé–Thomas and Podolsky (BLTP) is used, and the singularities equipped with a nonzero bare rest mass, then a joint initial value problem can be formulated in the spirit of the EIH proposal, and shown to be locally well-posed — in the special-relativistic zero-[Formula: see text] limit. With gravitational coupling (i.e. [Formula: see text]), though, changing Maxwell’s into the BLTP law and assigning a bare rest mass to the singularities is by itself not sufficient to obtain even a merely well-defined joint initial value problem: the gravitational coupling also needs to be changed, conceivably in the manner of Jordan and Brans–Dicke.

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