This thesis contains various constructions that attempt to answer and clarify the long-standing question of how to model the asymptotic behaviour of gravitational radiation in astrophysical processes within the theory of general relativity.

It is argued that the correct mathematical setup to address this question is the scattering problem (as opposed to the Cauchy problem), where data are posed in the infinite past. The choice for these data is informed by making certain heuristic connections to the Newtonian theory (Post-Newtonian theory). It is then proved that there exists a unique solution that attains these data in the limit, and the asymptotic properties of this solution are analysed.

In this way, it is found that the constructed solutions, which have clear physical interpretations afforded by the connection to the Newtonian theory, neither admit a smooth past null infinity nor a smooth future null infinity (they violate peeling near both), and moreover decay slower towards spatial infinity than typically assumed in the literature. In other words, our constructions show that many of the commonly-used concepts to model isolated physical systems are, in fact, not suitable for this purpose. Furthermore, it is shown that the non-smoothness of future null infinity, is, in a certain sense, exactly conserved and even determines the asymptotic behaviour of gravitational radiation at late times.

Our constructions either concern the nonlinear Einstein-Scalar field equations under the assumption of spherical symmetry, or the system of linearised gravity around Schwarzschild with no symmetry assumptions, but the methods employed are capable of generalisation to the nonlinear Einstein vacuum equations.