{"paper":{"title":"Static spheres in black hole spacetimes: pairing, energy conditions, and an upper bound on the innermost radius","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"gr-qc","authors_text":"Yong Song","submitted_at":"2026-07-23T02:22:38Z","abstract_excerpt":"In this work, we investigate the existence, relation to the energy conditions, and radial bounds of static spheres in general static, spherically symmetric, asymptotically flat black hole spacetimes. By analyzing the global behavior of a radial function constructed from the mass and radial pressure functions, we prove that a static sphere necessarily requires a negative radial pressure (tension). Furthermore, we show that non-degenerate static spheres must always appear in pairs: an inner unstable sphere and an outer stable one. Assuming the Weak Energy Condition (WEC) always holds, the inner "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20850","kind":"arxiv","version":1},"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/2607.20850/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"}