Relativistic Kepler problem. II. Asymptotic behavior of the field in the infinite past

Abstract

The standard weak-field, slow-motion approximation to Einstein's relativistic theory of gravitation is used to express the curvature tensor, up to order ${r}^{\ensuremath{-}5}$ on a flat background space-time, as a functional of the motion of the source of this curvature. The behavior, in the distant past, of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is then used to compute the asymptotic behavior of the corresponding curvature tensor along past-directed null straight lines in the flat background. It is found, on the one hand, that the falloff of the curvature is fast enough to guarantee satisfaction of a condition to exclude incoming gravitational radiation. On the other hand, the falloff is slower than would have been expected if the conformally rescaled curvature tensor had been regular on the hypersurface at past null infinity of the flat background.