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 (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 N infalling masses coming from past timelike infinity \(i^-\). Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding 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, \(r\phi \sim t^{-1}\) as \(t\rightarrow -\infty \), 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. We show that if the initial Hawking mass at \(i^-\) is nonzero, then, in accordance with the non-smoothness of \({\mathcal {I}}^+\), the asymptotic expansion of \(\partial _v(r\phi )\) near \({\mathcal {I}}^+\) reads \(\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})\) for some non-vanishing constant C. 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 \({\mathcal {I}}^-\) 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 \(r^{-4}\log r\).

这篇论文开创了一系列致力于严格研究近无穷远引力辐射精确结构的工作。我们首先简要回顾一下由于 Christodoulou（在：第九届马塞尔格罗斯曼会议，世界科学出版公司，新加坡，2002 年）提出的论点，指出彭罗斯关于时空（或平滑零无穷大）的平滑共形紧致化的提议未能准确捕捉来自过去似时间的无限\(i^-\)的N个下落质量发出的引力辐射结构。通过标量辐射模拟引力辐射，然后我们迈出了动力学理解的第一步通过构造由多项式衰减边界数据产生的球对称爱因斯坦标量场方程的解，零无穷大的非光滑性，\(r\phi \sim t^{-1}\)为\(t\rightarrow - \infty \)，在类似时间的超曲面（被认为是恒星的表面）和无入射辐射条件\(r\partial _v\phi =0\) 上，在过去的零无穷大上。我们证明，如果\(i^-\)处的初始霍金质量不为零，那么根据\({\mathcal {I}}^+\)的非光滑性，\(\部分 _v(r\phi )\)靠近\({\mathcal {I}}^+\)读取\(\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})\)对于一些非零常数C。事实上，相同的对数项已经出现在线性理论中，即在考虑固定 Schwarzschild 背景上的球对称线性波动方程时。作为推论，我们可以将我们的结果应用于 Schwarzschild 上的散射问题：在\({\mathcal {I}}^-\)和\({\ mathcal {H}}^-\)，我们发现\(\partial _v(r\phi )\)靠近\({\mathcal {I}}^+\)的渐近展开一般包含二阶对数项，即在订单\(r^{-4}\log r\)。