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On the hyperbolicity of Einstein’s and other gauge field equations. (English) Zbl 0588.35058
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
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[1] Friedrich, H.: 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. Proc. R. Soc. (London) A378, 401-421 (1981) · Zbl 0481.58026 · doi:10.1098/rspa.1981.0159
[2] Friedrich, H.: On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations. Proc. R. Soc. (London) A375, 169-184 (1981) · Zbl 0454.58017 · doi:10.1098/rspa.1981.0045
[3] Friedrich, H.: Cauchy problems for the conformal vacuum field equations in general relativity. Commun. Math. Phys.91, 445-472 (1983) · Zbl 0555.35116 · doi:10.1007/BF01206015
[4] Friedrichs, K. O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math.8, 345-392 (1954) · Zbl 0059.08902 · doi:10.1002/cpa.3160070206
[5] Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. II. New York: Interscience 1962 · Zbl 0099.29504
[6] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal.58, 181-205 (1975) · Zbl 0343.35056 · doi:10.1007/BF00280740
[7] Friedrich, H.: On purely radiative space-times, preprint, Hamburg 1985.
[8] Friedrich, H.: On some (con-) formal properties of Einstein’s field equations and their consequences. In: Proceedings of the conference on ?Asymptotic behaviour of mass and space-time geometry.? Corvallis 1983, Flaherty, F. J., ed. Berlin Heidelberg, New York: Springer 1984
[9] Lanczos, C.: Ein vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen. Phs. Z.23, 537-539 (1922) · JFM 48.1023.01
[10] Choquet-Bruhat, Y.: Théorème d’existence pour certain systèmes d’èquations aux derivées partielles non linéaires. Acta Math.88, 141-225 (1952) · Zbl 0049.19201 · doi:10.1007/BF02392131
[11] Mizohata, S.: The theory of partial differential equations. Cambridge: Cambridge University Press 1973 · Zbl 0263.35001
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