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.
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Communicated by R. Geroch
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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
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DOI: https://doi.org/10.1007/BF01651695