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# General Relativity and Quantum Cosmology

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

(Submitted on 31 May 2021 (v1), last revised 30 Jul 2021 (this version, v2))

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 $\mathcal I^-$ and polynomially decaying data, $r\phi_\ell\sim t^{-1}$ as $t\to-\infty$, on either a timelike boundary of constant area radius (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 logarithmic modifications to 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 $\mathcal H^-$ and $\mathcal I^-$ lead to solutions that exhibit the same late-time asymptotics on $\mathcal I^+$ for each $\ell$: $r\phi_\ell|_{\mathcal I^+}\sim u^{-2}$ as $u\to\infty$.

## Submission history

From: Leonhard Kehrberger [view email]**[v1]**Mon, 31 May 2021 18:03:47 GMT (320kb,D)

**[v2]**Fri, 30 Jul 2021 17:43:58 GMT (322kb,D)

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