{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:5JUGKVLBGHNK3SSTYVLBUQSXMR","short_pith_number":"pith:5JUGKVLB","schema_version":"1.0","canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","source":{"kind":"arxiv","id":"1008.1605","version":2},"attestation_state":"computed","paper":{"title":"On Breakdown Criteria for Nonvacuum Einstein Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.AP","authors_text":"Arick Shao","submitted_at":"2010-08-09T23:43:46Z","abstract_excerpt":"The recent \"breakdown criterion\" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's Ph.D. thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1008.1605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-09T23:43:46Z","cross_cats_sorted":["gr-qc"],"title_canon_sha256":"f725871425171c94da7ecaead08a1e7764f3def7bab45f2c4707f3ad14ee4491","abstract_canon_sha256":"43d38d21a61c67bde472eabbd8dc0947c49798ece5142845b2bc41045262e3ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:46.210838Z","signature_b64":"irLMY4M33QJxFo3xF06/xBGCBgDM8FHgBrWMze+zG5sZDmUwYzbyco0Nf3pmKFnzqiqwTFE44baAMwHPO675Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","last_reissued_at":"2026-05-18T01:04:46.210417Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:46.210417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Breakdown Criteria for Nonvacuum Einstein Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.AP","authors_text":"Arick Shao","submitted_at":"2010-08-09T23:43:46Z","abstract_excerpt":"The recent \"breakdown criterion\" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's Ph.D. thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1605","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1008.1605","created_at":"2026-05-18T01:04:46.210478+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.1605v2","created_at":"2026-05-18T01:04:46.210478+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1605","created_at":"2026-05-18T01:04:46.210478+00:00"},{"alias_kind":"pith_short_12","alias_value":"5JUGKVLBGHNK","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"5JUGKVLBGHNK3SST","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"5JUGKVLB","created_at":"2026-05-18T12:26:04.259169+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR","json":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR.json","graph_json":"https://pith.science/api/pith-number/5JUGKVLBGHNK3SSTYVLBUQSXMR/graph.json","events_json":"https://pith.science/api/pith-number/5JUGKVLBGHNK3SSTYVLBUQSXMR/events.json","paper":"https://pith.science/paper/5JUGKVLB"},"agent_actions":{"view_html":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR","download_json":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR.json","view_paper":"https://pith.science/paper/5JUGKVLB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.1605&json=true","fetch_graph":"https://pith.science/api/pith-number/5JUGKVLBGHNK3SSTYVLBUQSXMR/graph.json","fetch_events":"https://pith.science/api/pith-number/5JUGKVLBGHNK3SSTYVLBUQSXMR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/action/storage_attestation","attest_author":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/action/author_attestation","sign_citation":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/action/citation_signature","submit_replication":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/action/replication_record"}},"created_at":"2026-05-18T01:04:46.210478+00:00","updated_at":"2026-05-18T01:04:46.210478+00:00"}