{"paper":{"title":"The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\\ell$-Modes of Linear Waves on a Schwarzschild Background","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.AP","math.MP"],"primary_cat":"gr-qc","authors_text":"Lionor M. A. Kehrberger","submitted_at":"2021-05-31T18:03:47Z","abstract_excerpt":"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^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.00035","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.00035/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}