On the hyperbolicity of Einstein’s and other gauge field equations. (English) Zbl 0588.35058

Friedrich, Helmut

Commun. Math. Phys. 100, 525-543 (1985).

It is shown that gauge field equations, including the Yang-Mills and Einstein’s field equations, if suitably understood imply symmetric hyperbolic systems of propagation equations for any choice of gauge. In the equations appear certain freely specifiable ”gauge source functions”, which govern the propagation of the gauge off a suitable initial surface. The method is discussed in detail for the ”regular conformal field equations” for the conformally rescaled metric field in general relativity. Besides the coordinate and the frame gauge source functions may be given here arbitrarily the Ricci-scalar which constitutes the gauge source function which determines the propagation of the conformal factor. The technique has been applied to analyse the existence of radiative space-times by the author [”On Purely Radiative Space-Times”, ibid. 103, 35-65 (1986)]. Furthermore it led to the first general results on the global existence and the asymptotic behaviour of solutions of Einstein’s field equations with positive cosmological constant [see the author, ”On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure”. Preprint, Hamburg (1986)].

MSC:

35L45	Initial value problems for first-order hyperbolic systems
83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C30	Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory
58J45	Hyperbolic equations on manifolds
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

gauge field equations; Einstein’s field equations; symmetric hyperbolic systems; propagation equations; regular conformal field equations; existence of radiative space-times

References:

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