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, rphi _ell sim t^{-1} as trightarrow -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(rphi _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(rphi _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 : rphi _ell |_{{mathscr {I}}^+}sim u^{-2} as urightarrow infty .