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

Understand

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 $\mathcal{I}^-$.