{"paper":{"title":"The Vector Volume and Black Holes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"gr-qc","authors_text":"Kayll Lake, William Ballik","submitted_at":"2013-10-07T20:06:49Z","abstract_excerpt":"By examining the rate of growth of an invariant volume $\\mathcal V$ of some spacetime region along a divergence-free vector field $v^\\alpha$, we introduce the concept of a \"vector volume\" $\\mathcal{V}_v$. This volume can be defined in various equivalent ways. For example, it can be given as $\\mathrm d \\mathcal V(\\mu) / \\mathrm d \\mu$, where $v^\\alpha \\partial_\\alpha = \\mathrm d / \\mathrm d \\mu$, and $\\mu$ is a parameter distance along the integral curve of $v$. Equivalently, it can be defined as $\\int v^\\alpha \\mathrm d \\Sigma_\\alpha$, where $\\mathrm d \\Sigma_\\alpha$ is the directed surface el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1935","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}