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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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[1] Penrose, R.: Asymptotic properties of fields and space-times. Phys. Rev. Lett.10, 66 (1963); Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc.A284, 159-203 (1965) · Zbl 0129.41202
[2] Bondi, H., van der Burg, M. G. J., Metzner, A. W. K.: Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems. Proc. R. Soc.A269, 21-52 (1962) · Zbl 0106.41903
[3] Sachs, R. K.: Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time. Proc. R. Soc. A270, 103-126 (1962) · Zbl 0101.43605
[4] Newman, E., Penrose, R.: An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys.3, 566-578 (1962) · Zbl 0108.40905
[5] Friedrich, H.: On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations. Proc. R. Soc. A375, 169-184 (1981) · Zbl 0454.58017
[6] Ehlers, J.: Isolated systems in general relativity. ann. N.Y. Acad. Sci.336, 279-294 (1980)
[7] Schmidt, B. G., Stewart, J. M.: The scalar wave equation in a Schwarzschild space-time. Proc. R. Soc. A367, 503-525 (1979); Porrill, J., Stewart, J. M.: Electromagnetic and gravitational fields in a Schwarzschild space-time. Proc. R. Soc. A376, 451-463 (1981) · Zbl 0425.35064
[8] Walker, M., Will, C. M.: Relativistic Kepler problem. I. Behaviour in the distant past of orbits with gravitational radiation damping. Phys. Rev. D.19, 3483-3494 (1979); II. Asymptotic behaviour of the field in the infinite past. Phys. Rev. D.19, 3495-3508 (1979) · Zbl 1267.83025
[9] 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. A378, 401-421 (1981) · Zbl 0481.58026
[10] Friedrich, H.: On the existence of asymptotically flat and empty spaces. In: Proceedings of the summer school on ?Gravitational radiation?. Les Houches 1982, Deruelle, N., Piran, T. (eds.). Amsterdam: North-Holland 1983
[11] Friedrichs, K. O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math.8, 345-392 (1954) · Zbl 0059.08902
[12] Choquet-Bruhat, Y.: Théorème d’existence pour certains systèmes d’equations aux dérivées partielles non lineaires. Acta Math.88, 141-225 (1952) · Zbl 0049.19201
[13] Friedrich, H.: On the existence of analytic null asymptotically flat solutions of Einstein’s vacuum field equations. Proc. R. Soc. A381, 361-371 (1982) · Zbl 0489.58032
[14] Friedrich, H., Stewart, J.: Characteristic initial data and wavefront singularities in general relativity. Proc. R. Soc. A,385, 345-371 (1983) · Zbl 0513.58043
[15] Choquet-Bruhat, Y., York, J. W.: The Cauchy problem. In: General relativity and gravitation, Vol. 1, Held, A, (ed.), pp 99-172, New York: Plenum 1980
[16] Fischer, A. E., Marsden, J. E.: The initial value problem and the dynamical formulation of general relativity. In: General relativity. Hawking, S. W., Israel, W. (eds.). Cambridge: University Press 1979
[17] Christodoulou, D., O’Murchadha, N.: The boost problem in general relativity. Commun. Math. Phys.80, 271-300 (1981) · Zbl 0477.35081
[18] York, J. W.: Cravitational degrees of freedom and the initial-value problem. Phys. Rev. Lett.26, 1656-1658 (1971); York, J. W.: Role of conformal three-geometry in the dynamics of gravitation. Phys. Rev. Lett.28, 1082-1085 (1972); York, J. W.: Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity. J. Math. Phys.13, 125-130 (1973)
[19] O’Murchadha, N., York, J. W.: The initial-value problem of general relativity. Phys. Rev.D10, 428-436 (1974)
[20] Beig, R., Schmidt, B. G.: Einstein’s equations near spatial infinity. commun. Math. Phys.87, 65-80 (1982) · Zbl 0504.53025
[21] Ashtekar, A., Hansen, R. O.: A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity. J. Math. Phys.19, 1542-1566 (1978)
[22] Ashtekar, A.: Asymptotic structure of the gravitational field at spatial infinity. In: General relativity and gravitation, Vol. 2, Held, A. (ed.), pp 37-69, New York: Plenum 1980
[23] Schmidt, B. G.: A new definition of conformal and projective infinity of space-times. Commun. Math. Phys.36, 73 (1974) · Zbl 0282.53042
[24] Geroch, R.: Asymptotic structure of space-time. In: Asymptotic structure of space-time, Esposito, F. P. Witten, L. (eds.). New York: Plenum 1977
[25] Geroch, R., Horowitz, G. T.: Asymptotically simple does not imply asymptotially Minkowskian. Phys. Rev. Lett.40, 203-206 (1978)
[26] Hawking, S. W., Ellis, G. F. R.: The large scale structure of space-time. Cambridge: University Press 1973 · Zbl 0265.53054
[27] Penrose, R.: Relativistic symmetry groups. In: Group theory in non-linear problems. Barut, A. O. (ed.). New York: Reidel 1974 · Zbl 0281.22021
[28] Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 2. New York: Interscience 1962 · Zbl 0099.29504
[29] Fischer, A. E., Marsden, J. E.: The Einstein evolution equations as a first-order quasilinear symmetric hyperbolic system. commun. Math. Phys.28, 1-38 (1972) · Zbl 0247.35082
[30] Taylor, M. G.: Pseudodifferential operators. Princeton: University Press 1981 · Zbl 0453.47026
[31] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal.58, 181-205 (1975) · Zbl 0343.35056
[32] Adams, R.: Sobolev spaces. New York: Academic Press 1975 · Zbl 0314.46030
[33] Dieudonné, J. Foundations of modern analysis. New York: Academic Press 1969 · Zbl 0176.00502
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