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Limiting behavior of asymptotically flat gravitational fields

Authors:	Stephanie Novak Joshua N Goldberg

Institution:	(1) Department of Physics, Syracuse University, 13210 Syracuse, New York;(2) Present address: Corporate Research-Science Laboratories, Exxon Research and Engineering Company, P. O. Box 45, 07036 Linden, New Jersey

Abstract:	Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that $\Psi _0 = O(r^{ - 2 - \varepsilon _0 } )$ , (d/dr) $\Psi _0 = O(r^{ - 3 - \varepsilon _0 } )$ and (/x ^A) $\Psi _1 = O(r^{ - 2 - \varepsilon _0 } )$ , wherex ^A (A = 2, 3) are angular coordinates, they show that $\Psi _1 = O(r^{ - 2 - \varepsilon _1 } )$ , where ₁ 2 and ₁<₀; $< \varepsilon _0 ;\Psi _2 = O(r^{ - 2 - \varepsilon _2 } )$ , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption $\Psi _0 = O(r^{ - 3 - \varepsilon _0 } )$ , (d/dr) $\Psi _0 = O(r^{ - 4 - \varepsilon _0 } )$ , and (/x ^A) $\Psi _0 = O(r^{ - 3 - \varepsilon _0 } )$ does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.

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