jats:titleAbstract</jats:title>jats:pThis paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by jats:italicN</jats:italic> infalling masses coming from past timelike infinity jats:inline-formulajats:alternativesjats:tex-math$$i^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mii</mml:mi> mml:mo-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>. Modelling gravitational radiation by scalar radiation, we then take a first step towards a jats:italicdynamical understanding</jats:italic> of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, jats:inline-formulajats:alternativesjats:tex-math$$r\phi \sim t^{-1}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mir</mml:mi> mml:miϕ</mml:mi> mml:mo∼</mml:mo> mml:msup mml:mit</mml:mi> mml:mrow mml:mo-</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> as jats:inline-formulajats:alternativesjats:tex-math$$t\rightarrow -\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mit</mml:mi> mml:mo→</mml:mo> mml:mo-</mml:mo> mml:mi∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, jats:inline-formulajats:alternativesjats:tex-math$$r\partial _v\phi =0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mir</mml:mi> mml:msub mml:mi∂</mml:mi> mml:miv</mml:mi> </mml:msub> mml:miϕ</mml:mi> mml:mo=</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, on past null infinity. We show that if the initial Hawking mass at jats:inline-formulajats:alternativesjats:tex-math$$i^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mii</mml:mi> mml:mo-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> is nonzero, then, in accordance with the non-smoothness of jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {I}}^+$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miI</mml:mi> </mml:mrow> mml:mo+</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, the asymptotic expansion of jats:inline-formulajats:alternativesjats:tex-math$$\partial _v(r\phi )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mi∂</mml:mi> mml:miv</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:mir</mml:mi> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> near jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {I}}^+$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miI</mml:mi> </mml:mrow> mml:mo+</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> reads jats:inline-formulajats:alternativesjats:tex-math$$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mi∂</mml:mi> mml:miv</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:mir</mml:mi> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mo=</mml:mo> mml:miC</mml:mi> mml:msup mml:mir</mml:mi> mml:mrow mml:mo-</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msup> mml:molog</mml:mo> mml:mir</mml:mi> mml:mo+</mml:mo> mml:miO</mml:mi> mml:mrow mml:mo(</mml:mo> mml:msup mml:mir</mml:mi> mml:mrow mml:mo-</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msup> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for some non-vanishing constant jats:italicC</jats:italic>. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {I}}^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miI</mml:mi> </mml:mrow> mml:mo-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> and on jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {H}}^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miH</mml:mi> </mml:mrow> mml:mo-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, we find that the asymptotic expansion of jats:inline-formulajats:alternativesjats:tex-math$$\partial _v(r\phi )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mi∂</mml:mi> mml:miv</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:mir</mml:mi> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> near jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {I}}^+$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:mrow mml:miI</mml:mi> </mml:mrow> mml:mo+</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> generically contains logarithmic terms at second order, i.e. at order jats:inline-formulajats:alternativesjats:tex-math$$r^{-4}\log r$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msup mml:mir</mml:mi> mml:mrow mml:mo-</mml:mo> mml:mn4</mml:mn> </mml:mrow> </mml:msup> mml:molog</mml:mo> mml:mir</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>