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

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