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Cauchy problems for the conformal vacuum field equations in general relativity. (English) Zbl 0555.35116
Author’s summary: Cauchy problems for Einstein’s conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The ”hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of class $$H^ s$$, $$s\geq 4$$, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of class $$H^ s$$. It provides a solution of Einstein’s vacuum field equations which has a smooth structure at past null infinity.
Reviewer: U.F.Wodarzik

MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C15 Exact solutions to problems in general relativity and gravitational theory 35L60 First-order nonlinear hyperbolic equations 35L45 Initial value problems for first-order hyperbolic systems
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