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

Abstract

In this paper, we derive the early-time asymptotics for fixed-frequency solutions \(\phi _\ell \) to the wave equation \(\Box _g \phi _\ell =0\) on a fixed Schwarzschild background (\(M>0\)) arising from the no incoming radiation condition on \({\mathscr {I}}^-\) and polynomially decaying data, \(r\phi _\ell \sim t^{-1}\) as \(t\rightarrow -\infty \), on either a timelike boundary of constant area radius \(r>2M\) (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of \(\partial _v(r\phi _\ell )\) along outgoing null hypersurfaces near spacelike infinity \(i^0\) contains logarithmic terms at order \(r^{-3-\ell }\log r\). In contrast, in case (II), we obtain that the asymptotic expansion of \(\partial _v(r\phi _\ell )\) near spacelike infinity \(i^0\) contains logarithmic terms already at order \(r^{-3}\log r\) (unless \(\ell =1\)). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity \(i^+\) 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 \(\ell \)-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 \({\mathscr {H}}^-\) and \({\mathscr {I}}^-\) lead to solutions that exhibit the same late-time asymptotics on \({\mathscr {I}}^+\) for each \(\ell \): \(r\phi _\ell |_{{\mathscr {I}}^+}\sim u^{-2}\) as \(u\rightarrow \infty \).