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

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 $$i^-$$ 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 .

The Case Against Smooth Null Infinity II: A Logarithmically Modified Price's Law

“…The choice for these data was in turn motivated by an argument due to D. Christodoulou [2], which showed that the assumption of Sachs peeling and, thus, of (conformally) smooth null infinity, is incompatible with the no incoming radiation condition and the prediction of the quadrupole formula for N infalling masses from i − . Indeed, we proved that the solutions from [1] described above are not only in agreement with the quadrupole formula (which predicts that ∂ u (rφ) ∼ |u| −2 near i 0 ), but also lead to logarithmic terms in the asymptotic expansion of ∂ v (rφ) as I + is approached, thus contradicting the statement of Sachs peeling that such expansions can be expanded in powers of 1/r. Roughly speaking, we obtained for the spherically symmetric mode φ 0 that if the limit lim…”

“…

In this paper, we expand on results from our previous paper "The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples" [1] by showing that the failure of "peeling" (and, thus, of smooth null infinity) in a neighbourhood of i 0 derived therein translates into logarithmic corrections at leading order to the wellknown Price's law asymptotics near i + . This suggests that the non-smoothness of I + is physically measurable.More precisely, we consider the linear wave equation g φ = 0 on a fixed Schwarzschild background (M > 0), and we show the following: If one imposes conformally smooth initial data on an ingoing null hypersurface (extending to H + and terminating at I − ) and vanishing data on I − (this is the no incoming radiation condition), then the precise leading-order asymptotics of the solution φ are given by rφ| I + = Cu −2 log u + O(u −2 ) along future null infinity, φ| r=R>2M = 2Cτ −3 log τ + O(τ −3 ) along hypersurfaces of constant r, and φ| H + = 2Cv −3 log v+O(v −3 ) along the event horizon.

…”

“…In this paper, we expand on results from our previous paper "The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples" [1] by showing that the failure of "peeling" (and, thus, of smooth null infinity) in a neighbourhood of i 0 derived therein translates into logarithmic corrections at leading order to the wellknown Price's law asymptotics near i + . This suggests that the non-smoothness of I + is physically measurable.…”

“…We initiated the study of such data in [1], where we constructed two classes of solutions 1 satisfying the no incoming radiation condition (as a condition on data on I − ). The first class had polynomially decaying boundary data on a timelike boundary Γ terminating at i − , whereas the second class had polynomially decaying characteristic initial data on an ingoing null hypersurface C in terminating at I − .…”

A New Gauge for Asymptotically Flat Spacetime

“…Smoothness of null infinity is historically a complicated problem in general relativity, see eg. [33] for a recent discussion. More work is clearly needed to understand our result fully, here we will make one elementary comment about Schwarzschild black holes.…”