Abstract
This paper demonstrates the existence of non-trivial solutions (g, k) to the constraint equations of the initial value formulation of the Einstein field equations over ℝ3 withg ij − δ ij ∼ |x|−1 as |x| → ∞. Using the conformal methods of Lichnerowicz and York, this problem is divided into two parts. First, using weighted Sobolev spaces it is shown the set of pairs (g, k) withg a conformal metric andk transverse-traceless with respect tog forms a smooth vector bundleP with infinite dimensional fiber. Second, it is shown that the elements of a large open set inP uniquely determine a solution to the scalar constraint equation with the appropriate growth at infinity, and thereby determine solution to the constraint equations.
Similar content being viewed by others
References
Bourguignon, J.P., Ebin, D., Marsden, J.: Sur le noyau des operateurs pseudo-differential a symbole surjectif et non injectif. C. R. Acad. Sci. Paris A282, 867–870
Cantor, M.: Perfect fluid flows over ℝn with asymptotic conditions. J. Func. Anal.18, 73–84 (1975)
Cantor, M.: Some problems of global analysis on asymptotically simple manifolds. Comp. Math. (to appear)
Cantor, M.: Spaces of functions with asymptotic conditions. Indiana U. Math. J.24, 397–902 (1975)
Cantor, M., Fischer, A., Marsden, J., O'Murchadha, N., York, J.: The existence of maximal slicings in asymptotically flat spacetimes. Commun. math. Phys.49, 187–190 (1976)
Choquet-Bruhat, Y.: Global solutions of the equations of constraints in general relativity on closed manifolds. Symp. Math.XIII, 317–325 (1973)
Choquet-Bruhat, Y.: Private correspondence
Choquet-Bruhat, Y.: Probleme des constraintes sur une variete compacte. C. R. Acad. Sci. Paris274, 682–684 (1972)
Choquet-Bruhat, Y.: Sous-varietes, ou a courbure constante, de varietes lorentziennes. C. R. Acad. Sci. Paris280, 169–171 (1975)
Choquet-Bruhat, Y., Marsden, J.: Solution of the local mass problem in general relativity. C. R. Acad. Sci. Paris282, 609–612 (1976)
Fischer, A., Marsden, J.: The manifold of conformally equivalent metrics. Can. J. Math.29, 193–209 (1977)
Kazden, J., Warner, F.: Scalar curvature and conformal deformation of Riemannian structure. J. Diff. Geom.10, 113–134 (1975)
Lang, S.: Introduction to differentiable manifolds. New York: Interscience Publishers 1962
Lichnerowicz, A.: L'integration des equations de la gravitation probleme desn corps. J. Math. Pures Appl.23, 37–63 (1944)
Marsden, J.: Applications of global analysis in mathematical physics. Boston: Publish or Perish 1974
Nirenberg, L., Walker, H.: The null space of elliptic partial differential operators in ℝn. J. Math. Anal. Appl.47, 271–301 (1973)
O'Murchadha, N., York, J.: Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds. J. Math. Phys.14, 1551–1557 (1973)
O'Murchadha, N., York, J.: Initial-value problem of general relativity. I. General formulation and physical interpretation. Phys. Rev. D10, 428–436 (1974)
Protter, M., Weinberger, H.: Maximal principles in differential equations. Englewood Cliffs, NJ: Prentice Hall 1967
York, J.W.: Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity. J. Math. Phys.14, 456–464 (1973)
York, J.W.: Covariant decomposition of symmetric tensors in the theory of gravitation. Ann. Inst. H. Poincare21, 319–332 (1974)
Author information
Authors and Affiliations
Additional information
Communicated by R. Geroch
Rights and permissions
About this article
Cite this article
Cantor, M. The existence of non-trivial asymptotically flat initial data for vacuum spacetimes. Commun.Math. Phys. 57, 83–96 (1977). https://doi.org/10.1007/BF01651695
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01651695