Abstract
We show that any asymptotically flat initial data for the Einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface. Furthermore, the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development. We also extend former results on global solutions of the constraint equations. By virtue of this extension, the constraint and evolution parts of the problem fit together exactly. Several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems. This paper contains in addition a systematic exposition of the functional spaces employed.
Similar content being viewed by others
References
Lichnerowicz, A.: Problèmes globaux en mécanique relativiste. Paris: Hermann 1939;
Lichnerowicz, A.: J. Math. Pur. Appl.23, 37–63 (1944)
Choquet-Bruhat, Y.: Acta Math.88, 144–255 (1952) See also: Choquet-Bruhat, Y.: “Cauchy Problem”. In: Gravitation; An introduction to current research. Witten, L. (ed.). New York: John Wiley 1962
Choquet-Bruhat, Y.: Ann. Inst. Henri Poincaré8, 327–338 (1968)
Choquet-Bruhat, Y.: C.R. Acad. Sci. Paris272, 386–388 (1971)
Choquet-Bruhat, Y.: G.R.G.1, 359–362 (1971)
Leray, J.: Hyperbolic differential equations. Institute for Advanced Study, Notes (1953)
Dionne, P.: J. Anal. Math. Jerusalem10, 1–90 (1962)
Choquet-Bruhat, Y., Geroch, R.: Commun. Math. Phys.14, 329–335 (1969)
Fischer, A., Marsden, J.: Commun. Math. Phys.28, 1–38 (1972)
Hughes, T., Kato, T., Marsden, J.: Arch. Ration. Mech. Anal.63, 273–294 (1977)
York, J.: Phys. Rev. Lett.26, 1656–1658 (1971)
York, J.: Phys. Rev. Lett.28, 1082–1085 (1972)
York, J.: J. Math. Phys.14, 456–464 (1973)
O'Murchadha, N., York, J.: Phys. Rev. D10, 428–436 (1974)
Choquet-Bruhat, Y.: G.R.G.5, 49–60 (1974)
Cantor, M.: Commun. Math. Phys.57, 83–96 (1977)
Cantor, M.: J. Math. Phys.20, 1741–1744 (1979)
Chaljub-Simon, A., Choquet-Bruhat, Y.: C.R. Acad. Sci. Paris286, 917–920 (1978)
Chaljub-Simon, A., Choquet-Bruhat, Y.: Global solutions of the Lichnerowicz equation in general relativity on an asymptotically euclidean complete manifold. Preprint (1979)
Hawking, S., Ellis, G.: The large scale structure of spacetime. Cambridge: Cambridge University Press 1973
Sommers, P.: J. Math. Phys.19, 542–554 (1978)
Ashtekhar, A., Hansen, R.: J. Math. Phys.19, 1542–1566 (1978)
Persides, S.: J. Math. Phys.21, 135–151 (1980)
O'Murchadha, N.: Preprint (1978)
Adams, R.: Sobolev spaces. New York: Academic Press 1975
Choquet-Bruhat, Y., Christodoulou, D.: Elliptic systems in Hilbert spaces on manifolds which are euclidean at infinity. Acta Math. (to appear)
Choquet-Bruhat, Y., Christodoulou, D.: C.R. Acad. Sci. Paris290, 781–785 (1980)
Christodoulou, D.: The boost problem for weakly coupled quasilinear hyperbolic systems of the second order. J. Math. Pur. Appl. (to appear)
Christodoulou, D.: C.R. Acad. Sci. Paris290, 641–644 (1980)
Cantor, M.: Compositio Math.38, Fasc. 1, 3–35 (1979)
McOwen, R.: Commun. Pure Appl. Math.32, 783–795 (1979)
Choquet, G., Choquet-Bruhat, Y.: C.R. Acad. Sci. Paris287, 1047–1049 (1978)
Gilbarg, D., Trudinger, N.S.: “Elliptic partial differential equations of second order”, Heidelberg: Springer 1977
Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis manifolds and physics. Amsterdam: North Holland 1977
Arnowitt, R., Deser, S., Misner, C.: In: Gravitation; An introduction to current research. Witten, L. (ed.). New York: John Wiley 1962
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Christodoulou, D., O'Murchadha, N. The boost problem in general relativity. Commun.Math. Phys. 80, 271–300 (1981). https://doi.org/10.1007/BF01213014
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01213014