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

This 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 [1] 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 $N$ infalling masses coming from past timelike infinity $i^-$. Modelling gravitational radiation by scalar radiation, we then take a first step towards a rigorous, fully general relativistic understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein-Scalar field equations. Our constructions are motivated by Christodoulou's argument: They arise dynamically from polynomially decaying boundary 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 radiation condition, $r\partial_v\phi=0$, on past null infinity $\mathcal{I}^-$. We show that if the initial Hawking mass at $i^-$ is non-zero, then, in accordance with the non-smoothness of $\mathcal I^+$, $\partial_v(r\phi)$ satisfies the following asymptotic expansion near $\mathcal{I}^+$ for some constant $C\neq 0$: $\partial_v(r\phi)=Cr^{-3}\log r+\mathcal O(r^{-3})$. We also show that the same logarithmic terms appear 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 smooth, compactly supported scattering data for the wave equation on $\mathcal I^-$ and on $\mathcal H^-$, we find that the asymptotic expansion of $\partial_v(r\phi)$ near $\mathcal I^+$ generically contains logarithmic terms at second order, i.e. at order $r^{-4}\log r$.