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

TL;DR

This work challenges the assumption of a smooth null infinity by constructing explicit, fully relativistic counterexamples within the spherically symmetric Einstein–Maxwell–Scalar system. Through characteristic and timelike boundary data with no incoming radiation, the authors demonstrate logarithmic terms in the asymptotics of the radiation field near future null infinity, signaling non-smooth conformal structure and violating the Sachs peeling chain. They also connect these nonlinear phenomena to scattering on Schwarzschild and related spacetimes, and extend the analysis to linear and extremal settings, with companion results translating i0 asymptotics to i+ via Kerrburger2. The results illuminate the physical relevance of non-smooth infinity, suggest robust logarithmic corrections in a broad class of spacetimes, and outline a program to extend the analysis beyond spherical symmetry and into tensorial perturbations via Teukolsky-type equations.

Abstract

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, $rφ\sim t^{-1}$ as $t\to-\infty$, on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, $r\partial_vφ=0$, 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 $\mathcal I^+$, $\partial_v(rφ)$ satisfies the following asymptotic expansion near $\mathcal{I}^+$ for some constant $C\neq 0$: $\partial_v(rφ)=Cr^{-3}\log r+\mathcal O(r^{-3})$. 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 $\partial_v(rφ)$ near $\mathcal I^+$ generically contains logarithmic terms at second order, i.e. at order $r^{-4}\log r$.

Figures (11)

• Figure 1: Schematic depiction of Christodoulou's argument. Integrating $\partial_u(r^4\beta)\sim|u|^{-1}$ from initial data gives rise to logarithmic terms.
• Figure 2: The Penrose diagram of the solution of Theorem \ref{['thm.intro:timelikecase']}. We impose polynomially decaying data on a timelike boundary $\Gamma$ and no incoming radiation from past null infinity $\mathcal{I}^-$. Note that, with our choice of coordinates ($u=v$ on $\Gamma$), $\Gamma$ becomes a straight line.
• Figure 3: The Penrose diagram of the solution of Theorem \ref{['thm.intro:nullcase']}. We impose polynomially decaying data on an ingoing null hypersurface $\mathcal{C}_{\mathrm{in}}$ and no incoming radiation from past null infinity $\mathcal{I}^-$.
• Figure 4: The Penrose diagram of Schwarzschild. By Theorem \ref{['thm.intro:scattering']}c), smooth compactly supported scattering data on $\mathcal{H}^-$ and $\mathcal{I}^-$ generically lead to the setup of Theorem \ref{['thm.intro:nullcase']} with $p=3$. The region $\mathcal{D}$ as depicted corresponds to Figure \ref{['fig:3']}. As a consequence, the solution fails to be conformally regular on $\mathcal{I}^+$.
• Figure 5: The Penrose diagram of $\mathcal{D}_{U}$. It contains no black or white holes and, correspondingly, no trapped or anti-trapped surfaces (cf. \ref{['eq:nu-']}, \ref{['eq:lambda+']}). See also Dafermos2005b for an explanation of these notions.

Theorems & Definitions (73)

• Remark 1.1
• Conjecture 1.1
• Conjecture 1.2
• Remark 2.1
• Theorem 2.1
• Remark 2.2
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.3