{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:QGY5KZQLLYD7JI4IGS3XYRZDH6","short_pith_number":"pith:QGY5KZQL","canonical_record":{"source":{"id":"0811.0354","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-11-03T18:09:16Z","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"title_canon_sha256":"7594a7f64b5f924ac818d3680a6f90be5e82644d4d5e8e4434a94d7fcb505f2d","abstract_canon_sha256":"e0bfd5273bfc2c31110b230adc82586adf6d393d3c977c18c77c4e0fdd80479c"},"schema_version":"1.0"},"canonical_sha256":"81b1d5660b5e07f4a38834b77c47233f9009cf6e66389cd44c6c3337d726767b","source":{"kind":"arxiv","id":"0811.0354","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.0354","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"0811.0354v1","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.0354","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"QGY5KZQLLYD7","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"QGY5KZQLLYD7JI4I","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"QGY5KZQL","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:QGY5KZQLLYD7JI4IGS3XYRZDH6","target":"record","payload":{"canonical_record":{"source":{"id":"0811.0354","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-11-03T18:09:16Z","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"title_canon_sha256":"7594a7f64b5f924ac818d3680a6f90be5e82644d4d5e8e4434a94d7fcb505f2d","abstract_canon_sha256":"e0bfd5273bfc2c31110b230adc82586adf6d393d3c977c18c77c4e0fdd80479c"},"schema_version":"1.0"},"canonical_sha256":"81b1d5660b5e07f4a38834b77c47233f9009cf6e66389cd44c6c3337d726767b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:55.526685Z","signature_b64":"XB5/ollra0Nsho4ApWQ6jy4uKdeiBQKcGxbxDMbfMAoc9lCSnPuetDh46RLyeapQENlJg1TQ40gXTeO+RErLDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81b1d5660b5e07f4a38834b77c47233f9009cf6e66389cd44c6c3337d726767b","last_reissued_at":"2026-05-18T03:05:55.525948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:55.525948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0811.0354","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"th6q7SgW6qNz/8fjpzuhLq1SJ4lPR9g12RSCYvnu0IKDefVOaTkT1SrT1Sv7xkFlFqdUZ0W4rawKzhy7jIk9BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:34:50.381438Z"},"content_sha256":"5e0f98f736af1bbc6cf79f5b45eaa09ae7147e004dc11839a7b75b386d80043a","schema_version":"1.0","event_id":"sha256:5e0f98f736af1bbc6cf79f5b45eaa09ae7147e004dc11839a7b75b386d80043a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:QGY5KZQLLYD7JI4IGS3XYRZDH6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lectures on black holes and linear waves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.DG","math.MP"],"primary_cat":"gr-qc","authors_text":"Igor Rodnianski, Mihalis Dafermos","submitted_at":"2008-11-03T18:09:16Z","abstract_excerpt":"These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spacetimes themselves as solutions to the Einstein equations, one of the central open problems of general relativity. After an introductory discussion of the Schwarzschild geometry and the black hole concept, the classical theorem of Kay and Wald on the boundedness of scalar waves o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.0354","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dkWVidCezg1nZI28xAHIFw3F3IqptRu0c2gYjuhiKdR/fNS2rg7uHAeNsqNxEkO5PwKxmJ4Sgvqas2Q9AzLcDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:34:50.381785Z"},"content_sha256":"1f30eb4a2b68a29c56f2f87bca8c9fc00f98d860d873ea7279ae6e4a42248018","schema_version":"1.0","event_id":"sha256:1f30eb4a2b68a29c56f2f87bca8c9fc00f98d860d873ea7279ae6e4a42248018"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6/bundle.json","state_url":"https://pith.science/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T13:34:50Z","links":{"resolver":"https://pith.science/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6","bundle":"https://pith.science/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6/bundle.json","state":"https://pith.science/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QGY5KZQLLYD7JI4IGS3XYRZDH6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:QGY5KZQLLYD7JI4IGS3XYRZDH6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e0bfd5273bfc2c31110b230adc82586adf6d393d3c977c18c77c4e0fdd80479c","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-11-03T18:09:16Z","title_canon_sha256":"7594a7f64b5f924ac818d3680a6f90be5e82644d4d5e8e4434a94d7fcb505f2d"},"schema_version":"1.0","source":{"id":"0811.0354","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.0354","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"0811.0354v1","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.0354","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"QGY5KZQLLYD7","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"QGY5KZQLLYD7JI4I","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"QGY5KZQL","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:1f30eb4a2b68a29c56f2f87bca8c9fc00f98d860d873ea7279ae6e4a42248018","target":"graph","created_at":"2026-05-18T03:05:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spacetimes themselves as solutions to the Einstein equations, one of the central open problems of general relativity. After an introductory discussion of the Schwarzschild geometry and the black hole concept, the classical theorem of Kay and Wald on the boundedness of scalar waves o","authors_text":"Igor Rodnianski, Mihalis Dafermos","cross_cats":["math-ph","math.AP","math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-11-03T18:09:16Z","title":"Lectures on black holes and linear waves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.0354","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5e0f98f736af1bbc6cf79f5b45eaa09ae7147e004dc11839a7b75b386d80043a","target":"record","created_at":"2026-05-18T03:05:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e0bfd5273bfc2c31110b230adc82586adf6d393d3c977c18c77c4e0fdd80479c","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2008-11-03T18:09:16Z","title_canon_sha256":"7594a7f64b5f924ac818d3680a6f90be5e82644d4d5e8e4434a94d7fcb505f2d"},"schema_version":"1.0","source":{"id":"0811.0354","kind":"arxiv","version":1}},"canonical_sha256":"81b1d5660b5e07f4a38834b77c47233f9009cf6e66389cd44c6c3337d726767b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81b1d5660b5e07f4a38834b77c47233f9009cf6e66389cd44c6c3337d726767b","first_computed_at":"2026-05-18T03:05:55.525948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:55.525948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XB5/ollra0Nsho4ApWQ6jy4uKdeiBQKcGxbxDMbfMAoc9lCSnPuetDh46RLyeapQENlJg1TQ40gXTeO+RErLDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:55.526685Z","signed_message":"canonical_sha256_bytes"},"source_id":"0811.0354","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e0f98f736af1bbc6cf79f5b45eaa09ae7147e004dc11839a7b75b386d80043a","sha256:1f30eb4a2b68a29c56f2f87bca8c9fc00f98d860d873ea7279ae6e4a42248018"],"state_sha256":"d84353b22bd7debd702c0a7d8a1ad5ae66a873570d05c07934ef63bbc6e935bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Azxetiz+LbR2lSzu1EnY9t/9ECIbQpJ3awPObwkSl3LpiErNoA9iKRrh5Gn5bi4C+S+SToAPDrQSgbhv8yhFBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T13:34:50.383649Z","bundle_sha256":"929ce1c478de2f40de8cfa3ec3960aa6dee99745f4744aaafda72b1b340289e0"}}