Abstract
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 classH S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.
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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)
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)
Sachs, R. K.: Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time. Proc. R. Soc. A270, 103–126 (1962)
Newman, E., Penrose, R.: An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys.3, 566–578 (1962)
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)
Ehlers, J.: Isolated systems in general relativity. ann. N.Y. Acad. Sci.336, 279–294 (1980)
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)
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)
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)
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
Friedrichs, K. O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math.8, 345–392 (1954)
Choquet-Bruhat, Y.: Théorème d'existence pour certains systèmes d'equations aux dérivées partielles non lineaires. Acta Math.88, 141–225 (1952)
Friedrich, H.: On the existence of analytic null asymptotically flat solutions of Einstein's vacuum field equations. Proc. R. Soc. A381, 361–371 (1982)
Friedrich, H., Stewart, J.: Characteristic initial data and wavefront singularities in general relativity. Proc. R. Soc. A,385, 345–371 (1983)
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
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
Christodoulou, D., O'Murchadha, N.: The boost problem in general relativity. Commun. Math. Phys.80, 271–300 (1981)
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)
O'Murchadha, N., York, J. W.: The initial-value problem of general relativity. Phys. Rev.D10, 428–436 (1974)
Beig, R., Schmidt, B. G.: Einstein's equations near spatial infinity. commun. Math. Phys.87, 65–80 (1982)
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)
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
Schmidt, B. G.: A new definition of conformal and projective infinity of space-times. Commun. Math. Phys.36, 73 (1974)
Geroch, R.: Asymptotic structure of space-time. In: Asymptotic structure of space-time, Esposito, F. P. Witten, L. (eds.). New York: Plenum 1977
Geroch, R., Horowitz, G. T.: Asymptotically simple does not imply asymptotially Minkowskian. Phys. Rev. Lett.40, 203–206 (1978)
Hawking, S. W., Ellis, G. F. R.: The large scale structure of space-time. Cambridge: University Press 1973
Penrose, R.: Relativistic symmetry groups. In: Group theory in non-linear problems. Barut, A. O. (ed.). New York: Reidel 1974
Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 2. New York: Interscience 1962
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)
Taylor, M. G.: Pseudodifferential operators. Princeton: University Press 1981
Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal.58, 181–205 (1975)
Adams, R.: Sobolev spaces. New York: Academic Press 1975
Dieudonné, J. Foundations of modern analysis. New York: Academic Press 1969
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Communicated by S.-T. Yau
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Friedrich, H. Cauchy problems for the conformal vacuum field equations in general relativity. Commun.Math. Phys. 91, 445–472 (1983). https://doi.org/10.1007/BF01206015
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DOI: https://doi.org/10.1007/BF01206015