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This paper initiates a series of works dedicated to the rigorous study of theprecise structure of gravitational radiation near infinity. We begin with abrief review of an argument due to Christodoulou [1] stating that Penrose'sproposal of smooth conformal compactification of spacetime (or smooth nullinfinity) fails to accurately capture the structure of gravitational radiationemitted by $N$ infalling masses coming from past timelike infinity $i^-$.Modelling gravitational radiation by scalar radiation, we then take a firststep towards a rigorous, fully general relativistic understanding of thenon-smoothness of null infinity by constructing solutions to the sphericallysymmetric Einstein-Scalar field equations. Our constructions are motivated byChristodoulou's argument: They arise dynamically from polynomially decayingboundary data, $r\phi\sim t^{-1}$ as $t\to-\infty$, on a timelike hypersurface(to be thought of as the surface of a star) and the no incoming radiationcondition, $r\partial_v\phi=0$, on past null infinity $\mathcal{I}^-$. We showthat if the initial Hawking mass at $i^-$ is non-zero, then, in accordance withthe non-smoothness of $\mathcal I^+$, $\partial_v(r\phi)$ satisfies thefollowing asymptotic expansion near $\mathcal{I}^+$ for some constant $C\neq0$: $\partial_v(r\phi)=Cr^{-3}\log r+\mathcal O(r^{-3})$. We also show that thesame logarithmic terms appear in the linear theory, i.e. when considering thespherically symmetric linear wave equation on a fixed Schwarzschild background.  As a corollary, we can apply our results to the scattering problem onSchwarzschild: Putting smooth, compactly supported scattering data for the waveequation on $\mathcal I^-$ and on $\mathcal H^-$, we find that the asymptoticexpansion of $\partial_v(r\phi)$ near $\mathcal I^+$ generically containslogarithmic terms at second order, i.e. at order $r^{-4}\log r$.

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples