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 (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, rphi sim t^{-1} as trightarrow -infty , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, rpartial _vphi =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(rphi ) near {mathcal {I}}^+ reads partial _v(rphi )=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(rphi ) near {mathcal {I}}^+ generically contains logarithmic terms at second order, i.e. at order r^{-4}log r.
Highlights
This work is concerned with the rigorous mathematical analysis of gravitational waves near infinity
In the first part (Part I), we give some historical background on the concept of smooth null infinity and review an important argument against smooth null infinity due to Christodoulou, which forms the main motivation for the present work
The ideas developed in these works were combined by Penrose’s notion of asymptotic simplicity [6], a concept that can be found in most advanced textbooks on general relativity
Summary
One can recover the good r−3-weight by commuting times with vector fields which, in Eddington–Finkelstein coordinates, to leading order all look like r2∂v.16 Using these commuted wave equations, one can adapt the methods of this paper to obtain similar results for higher -modes, with logarithms appearing in the expansions of ∂v(rφ ) at orders which depend in a more subtle way on the precise setup. We note that these commuted wave equations, which we will dub approximate conservation laws, are closely related to the higherorder Newman–Penrose quantities for the scalar wave equation (see the introduction of [40] or the recent [43]). See the recent [44] and [45], where a generalisation of the well-known Price’s law is obtained for Kerr backgrounds
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