The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background
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arXiv
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| Formato: | Preprint |
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2021
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| _version_ | 1866916906356703232 |
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| author | Kehrberger, Lionor M. A. |
| author_facet | Kehrberger, Lionor M. A. |
| contents | In this paper, we derive the early-time asymptotics for fixed-frequency solutions $ϕ_\ell$ to the wave equation $\Box_g ϕ_\ell=0$ on a fixed Schwarzschild background ($M>0$) arising from the no incoming radiation condition on $\mathcal I^-$ and polynomially decaying data, $rϕ_\ell\sim t^{-1}$ 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 $\partial_v(rϕ_\ell)$ 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 $\partial_v(rϕ_\ell)$ 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 $\ell$: $rϕ_\ell|_{\mathcal I^+}\sim u^{-2}$ as $u\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_00035 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background Kehrberger, Lionor M. A. General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Analysis of PDEs In this paper, we derive the early-time asymptotics for fixed-frequency solutions $ϕ_\ell$ to the wave equation $\Box_g ϕ_\ell=0$ on a fixed Schwarzschild background ($M>0$) arising from the no incoming radiation condition on $\mathcal I^-$ and polynomially decaying data, $rϕ_\ell\sim t^{-1}$ 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 $\partial_v(rϕ_\ell)$ 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 $\partial_v(rϕ_\ell)$ 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 $\ell$: $rϕ_\ell|_{\mathcal I^+}\sim u^{-2}$ as $u\to\infty$. |
| title | The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2106.00035 |