The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $$\ell $$-Modes of Linear Waves on a Schwarzschild Background

Abstract

AbstractIn this paper, we derive the early-time asymptotics for fixed-frequency solutions $$\phi _\ell $$ ϕ ℓ to the wave equation $$\Box _g \phi _\ell =0$$ □ g ϕ ℓ = 0 on a fixed Schwarzschild background ($$M>0$$ M > 0 ) arising from the no incoming radiation condition on $${\mathscr {I}}^-$$ I - and polynomially decaying data, $$r\phi _\ell \sim t^{-1}$$ r ϕ ℓ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on either a timelike boundary of constant area radius $$r>2M$$ r > 2 M (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of $$\partial _v(r\phi _\ell )$$ ∂ v ( r ϕ ℓ ) along outgoing null hypersurfaces near spacelike infinity $$i^0$$ i 0 contains logarithmic terms at order $$r^{-3-\ell }\log r$$ r - 3 - ℓ log r . In contrast, in case (II), we obtain that the asymptotic expansion of $$\partial _v(r\phi _\ell )$$ ∂ v ( r ϕ ℓ ) near spacelike infinity $$i^0$$ i 0 contains logarithmic terms already at order $$r^{-3}\log r$$ r - 3 log r (unless $$\ell =1$$ ℓ = 1 ). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity $$i^+$$ i + that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate a logarithmically modified Price’s law for each $$\ell $$ ℓ -mode. On the other hand, the data of case (II) lead to much stronger deviations from Price’s law. In particular, we conjecture that compactly supported scattering data on $${\mathscr {H}}^-$$ H - and $${\mathscr {I}}^-$$ I - lead to solutions that exhibit the same late-time asymptotics on $${\mathscr {I}}^+$$ I + for each $$\ell $$ ℓ : $$r\phi _\ell |_{{\mathscr {I}}^+}\sim u^{-2}$$ r ϕ ℓ | I + ∼ u - 2 as $$u\rightarrow \infty $$ u → ∞ .