Well posed reduced systems for the Einstein equations

Choquet-Bruhat, Yvonne ; York, James

Banach Center Publications, Tome 38 (1997), p. 119-131 / Harvested from The Polish Digital Mathematics Library

Résumé

We review some well posed formulations of the evolution part of the Cauchy problem of General Relativity that we have recently obtained. We include also a new first order symmetric hyperbolic system based directly on the Riemann tensor and the full Bianchi identities. It has only physical characteristics and matter sources can be included. It is completely equivalent to our other system with these properties.

Détail

Publié le : 1997-01-01

Zbl 0896.53052

EUDML-ID : urn:eudml:doc:252237

Bibliographie

[000] [1] A. Abrahams, A. Anderson, Y. Choquet-Bruhat and J. W. York, Einstein and Yang-Mills theories in hyperbolic form without gauge fixing, Phys. Rev. Letters 75 (1995), 3377-3381. | Zbl 1020.83503

[001] [2] A. Abrahams, A. Anderson, Y. Choquet-Bruhat and J. W. York, Geometrical hyperbolic systems for general relativity and gauge theories, submitted to Class. Quantum Grav., gr-qc/9605014. | Zbl 0866.58059

[002] [3] A. Abrahams, A. Anderson, Y. Choquet-Bruhat and J. W. York, A non-strictly hyperbolic system for the Einstein equations with arbitrary lapse and shift, submitted to C.R. Acad. Sci. Paris A. | Zbl 1020.83503

[003] [4] L. Bel, C.R. Acad. Sci. Paris 246 (1958), 3105.

[004] [5] C. Bona, J. Masso, E. Seidel and J. Stela, A new formalism for numerical relativity, Phys. Rev. Letters 75 (1995), 600-603.

[005] [6] Y. Choquet (Foures)-Bruhat, Sur L'Intégration des Équations de la Relativité Générale, J. Rat. Mechanics and Anal. 5 (1956), 951-966. | Zbl 0075.21602

[006] [7] Y. Choquet-Bruhat and D. Christodoulou, Elliptic systems in $H_s,δ$ spaces on manifolds which are Euclidean at infinity,Acta. Math. 146 (1981), 129-150. | Zbl 0484.58028

[007] [8] Y. Choquet-Bruhat and T. Ruggeri, Hyperbolicity of the 3+1 system of Einstein equations, Commun. Math. Phys. 89 (1983), 269-275. | Zbl 0521.53034

[008] [9] Y. Choquet-Bruhat and J. W. York, The Cauchy problem in: General Relativity and Gravitation, A. Held (ed.), Plenum, New York, 1980, 99-172.

[009] [10] Y. Choquet-Bruhat and J. W. York, Geometrical well posed systems for the Einstein equations, C.R. Acad. Sci. Paris 321 (1995), Série I, 1089-1095. | Zbl 0839.53063

[010] [11] Y. Choquet-Bruhat and J. W. York, Mixed Elliptic and Hyperbolic Systems for the Einstein Equations, in: Gravitation, Electromagnetism and Geometric Structures, G. Ferrarese (ed.) Pythagora Editrice, Bologna, Italy, 1996, 55-73.

[011] [12] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton University Press, Princeton, 1993. | Zbl 0827.53055

[012] [13] H. Friedrich, Hyperbolic reductions for Einstein's equations, Class. Quantum Grav. 13 (1996), 1451-1459.

[013] [14] S. Frittelli and O. Reula, On the Newtonian limit of general relativity, Commun. Math. Phys. 166 (1994), 221-235. | Zbl 0812.35147

[014] [15] C. Lanczos, A remarkable property of the Riemann-Christoffel tensor in four dimensions, Ann. of Math. 39 (1938), 842-850. | Zbl 0019.37904

[015] [16] J. Leray, Hyperbolic Differential Equations, Institute for Advanced Study, Princeton, 1952.

[016] [17] J. Leray and Y. Ohya, Équations et systèmes non-linéaires, hyperboliques non-stricts, Math. Ann. 170 (1967), 167-205. | Zbl 0146.33701

[017] [18] A. Lichnerowicz, Problèmes globaux en Mécanique Relativiste, Hermann, Paris, 1939. | Zbl 0061.47002

[018] [19] J. W. York, Kinematics and dynamics of general relativity, in: Sources of Gravitaional Radiation, L. Smarr (ed.), Cambridge University Press, Cambridge, 1979, 83-126. | Zbl 0418.58016

[019] [20] J. W. York, Bel-Robinson Gravitational Superenergy and Flatness, in: Gravitation and Geometry, W. Rindler and A. Trautman (eds.), Bibliopolis, Naples, Italy, 1987, 497-505.