Abstract
We consider Cauchy data (g, π) on IR3 that are asymptotically Euclidean and that satisfy the vacuum constraint equations of general relativity. Only those (g, π) are treated that can be joined by a curve of “sufficiently bounded” initial data to the trivial data (δ, 0). It is shown that in the Cauchy developments of such data, the maximal slicing condition tr π=0 can always be satisfied. The proof uses the recently introduced “weighted Sobolev spaces” of Nirenberg, Walker, and Cantor.
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References
Arnowitt, R., Deser, S., Misner, C. W.: The dynamics of general relativity, In: Gravitation: an introduction to current research, (ed. L. Witten). New York: Wiley 1962
Brill, D., Deser, S.: Variational methods and positive energy in general relativity. Ann. Phys.50, 548–570 (1968)
Cantor, M.: Spaces of functions with asymptotic conditions on IRn. Ind. U. J. Math. to appear (1975)
Cantor, M.: Perfect fluid flows over IRn with asymptotic conditions, J. Funct. Anal.18, 73–84 (1975)
Cantor, M.: Growth of Solutions of elliptic equations with nonconstant coefficients on IRn. Preprint
Choquet-Bruhat, Y.: Sous-Varietes maximales, ou a courbure constante, de varietes lorentziennes, C. R. Acad. Sc. Paris,280, Ser A, 169–171 (1975)
Fischer, A., Marsden, J.: The Einstein equations of evolution — A geometric approach. J. Math. Phys.13, 546–568 (1972)
Fischer, A., Marsden, J.: The Einstein evolution equations as a first order quasi-linear hyperbolic system. Commun. math. Phys.28, 1–38 (1972)
Fischer, A., Marsden, J.: Linearization stability of nonlinear partial differential equations, Proc. Symp. Pure Math. A. M. S.27, 219–263 (1975) (also Bull. A. M. S.79, 997–1003 (1973),80, 479–484, and General Relativity and Gravitation5, 73–77 (1974)
O'Murchadha, N., York, J. W.: Initial value problem of general relativity (I, II). Phys. Rev. D10, 428–436, 437–446 (1974)
O'Murchadha, N., York, J. W.: Gravitational energy. Phys. Rev. D10, 2345–2357 (1974)
York, J. W.: The role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Letters28, 1082 (1972)
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Communicated by J. Ehlers
Research partially supported by National Science Foundation Grants GP-39060 and GP-15735
Research partially supported by National Science Foundation Grant GP-43909 to the University of North Carolina
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Cantor, M., Fischer, A., Marsden, J. et al. The existence of maximal slicings in asymptotically flat spacetimes. Commun.Math. Phys. 49, 187–190 (1976). https://doi.org/10.1007/BF01608741
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DOI: https://doi.org/10.1007/BF01608741