2021/05/17 by Kehrberger, Lionor M. A. · 3 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2105.08084
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 i0 derived therein translates into logarithmic corrections at leading order to the well-known 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 \Boxg 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+=C u-2log u+O(u-2) along future null infinity, ϕ|r=R>2M=2Cτ-3logτ+O(τ-3) along hypersurfaces of constant r, and ϕ|H+=2Cv-3log v+O(v-3) along the event horizon. Moreover, the constant C is given by C=4M I0(past)[ϕ], where I0(past)[ϕ]:=limu→ -∞ r2∂u(rϕℓ=0) is the past Newman--Penrose constant of ϕ on I-. Thus, the precise late-time asymptotics of ϕ are completely determined by the early-time behaviour of the spherically symmetric part of ϕ near I-. Similar results are obtained for polynomially decaying timelike boundary data. The paper uses methods developed by Angelopoulos--Aretakis--Gajic and is essentially self-contained.