scholarly journals Diffusing nonlocal inflation: Solving the field equations as an initial value problem

2008 ◽  
Vol 78 (6) ◽  
Author(s):  
D. J. Mulryne ◽  
N. J. Nunes
Keyword(s):  
Initial Value Problem ◽  
Field Equations ◽  
Initial Value

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 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.

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 SPIN FLUIDS IN BIANCHI-I f(R)-COSMOLOGY WITH TORSION

2012 ◽  
Vol 09 (07) ◽  
pp. 1250054 ◽  
Author(s):  
STEFANO VIGNOLO ◽  
LUCA FABBRI
Keyword(s):  
Conservation Laws ◽  
Initial Value Problem ◽  
Bianchi Type ◽  
Hamiltonian Constraint ◽  
Type I ◽  
Field Equations ◽  
Initial Value ◽  
Bianchi I ◽  
Jordan And Einstein Frames ◽  
General Conservation

We study Weyssenhoff spin fluids in Bianchi type-I cosmological models, within the framework of torsional f(R)-gravity; the resulting field equations are derived and discussed in both Jordan and Einstein frames, clarifying the role played by the spin and the non-linearity of the gravitational Lagrangian f(R) in generating the torsional dynamical contributions. The general conservation laws holding for f(R)-gravity with torsion are employed to provide the conditions needed to ensure the preservation of the Hamiltonian constraint and the consequent correct formulation of the associated initial value problem. Examples are eventually given.

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.

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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