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

Abstract

AbstractThis 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 $$i^-$$ 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, $$r\phi \sim t^{-1}$$ r ϕ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, $$r\partial _v\phi =0$$ r ∂ v ϕ = 0 , on past null infinity. We show that if the initial Hawking mass at $$i^-$$ i - is nonzero, then, in accordance with the non-smoothness of $${\mathcal {I}}^+$$ I + , the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + reads $$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$ ∂ v ( r ϕ ) = C r - 3 log r + 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 $${\mathcal {I}}^-$$ I - and on $${\mathcal {H}}^-$$ H - , we find that the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + generically contains logarithmic terms at second order, i.e. at order $$r^{-4}\log r$$ r - 4 log r .