scholarly journals ESTIMATES FOR THE MAXWELL FIELD NEAR THE SPATIAL AND NULL INFINITY OF THE SCHWARZSCHILD SPACETIME

2009 ◽  
Vol 06 (02) ◽  
pp. 229-268 ◽  
Author(s):  
JUAN ANTONIO VALIENTE KROON
Keyword(s):  
Wide Class ◽  
Initial Value Problem ◽  
Maxwell Equations ◽  
Maxwell Field ◽  
Schwarzschild Spacetime ◽  
Initial Value ◽  
Null Infinity ◽  
Spacelike Infinity ◽  
Schwarzschild Background

It is shown how the gauge of the "regular finite initial value problem at spacelike infinity" can be used to construct a certain type of estimates for the Maxwell field propagating on a Schwarzschild background. These estimates are constructed with the objective of obtaining information about the smoothness near spacelike and null infinity of a wide class of solutions to the Maxwell equations.

scholarly journals The Maxwell field on the Schwarzschild space–time: behaviour near spatial infinity

2007 ◽  
Vol 463 (2086) ◽  
pp. 2609-2630 ◽  
Author(s):  
Juan Antonio Valiente Kroon
Keyword(s):  
Initial Data ◽  
Maxwell Equations ◽  
Geodesic Distance ◽  
Transport Equations ◽  
Maxwell Field ◽  
Spatial Infinity ◽  
Schwarzschild Solution ◽  
Null Infinity ◽  
Completely Regular ◽  
Critical Sets

The behaviour of the Maxwell field near one of the spatial infinities of the Schwarzschild solution is analysed by means of the transport equations implied by the Maxwell equations on the cylinder at spatial infinity. Initial data for the Maxwell equations will be assumed to be expandable in terms of powers of a coordinate ρ measuring the geodesic distance to spatial infinity (in the conformal picture) and such that the highest possible spherical harmonics at order p are 2 p -polar ones. It is shown that if the 2 p -polar harmonics at order p in the initial data satisfy a certain regularity condition, then the solutions to the transport equations at orders p and p +1 are completely regular at the critical sets where null infinity touches spatial infinity. On the other hand, the solutions to the transport equations of order p +2 contain, in general, logarithmic singularities at the critical sets.

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.

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.

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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