{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:KA5LAHZLBALTAUM2CBAQPJXY7P","short_pith_number":"pith:KA5LAHZL","canonical_record":{"source":{"id":"1103.0585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-03-03T00:25:55Z","cross_cats_sorted":["astro-ph.HE","math-ph","math.MP"],"title_canon_sha256":"1040786fd7d86cb58393ff5fc07a0e3303a9d05dc2ee33938a49701b928c94eb","abstract_canon_sha256":"8c0085e18af0b2667124f4e9ecc5e17f3d91ec323b034072796a65f1e10c3e8a"},"schema_version":"1.0"},"canonical_sha256":"503ab01f2b081730519a104107a6f8fbd7707d82379c3c701edeb43e3c2bcf63","source":{"kind":"arxiv","id":"1103.0585","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.0585","created_at":"2026-05-18T02:03:00Z"},{"alias_kind":"arxiv_version","alias_value":"1103.0585v1","created_at":"2026-05-18T02:03:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0585","created_at":"2026-05-18T02:03:00Z"},{"alias_kind":"pith_short_12","alias_value":"KA5LAHZLBALT","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"KA5LAHZLBALTAUM2","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"KA5LAHZL","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:KA5LAHZLBALTAUM2CBAQPJXY7P","target":"record","payload":{"canonical_record":{"source":{"id":"1103.0585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-03-03T00:25:55Z","cross_cats_sorted":["astro-ph.HE","math-ph","math.MP"],"title_canon_sha256":"1040786fd7d86cb58393ff5fc07a0e3303a9d05dc2ee33938a49701b928c94eb","abstract_canon_sha256":"8c0085e18af0b2667124f4e9ecc5e17f3d91ec323b034072796a65f1e10c3e8a"},"schema_version":"1.0"},"canonical_sha256":"503ab01f2b081730519a104107a6f8fbd7707d82379c3c701edeb43e3c2bcf63","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:00.659402Z","signature_b64":"WF+F9JPjWzHZY/KUhQhqf1b7lFHt11o0Wu+K1hI3VE7RKdPQ4rHF0iihVB8Bfc5xsoMlUDNKzkxiLWdQ1WRYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"503ab01f2b081730519a104107a6f8fbd7707d82379c3c701edeb43e3c2bcf63","last_reissued_at":"2026-05-18T02:03:00.658671Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:00.658671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.0585","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":"2026-05-18T02:03:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cUU+4UY1//Ea9MMsSxZpK3IJ5H90n5EKSAw5ng0zs/cMr1mlIWr98lYzKmpK8hOb0cKVZAAxeESzDByx7NSqAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:35:48.871681Z"},"content_sha256":"229b767c51e22d0d83b8396a434d6514fe29b20b36ce0029a0a3cfcffa9723a7","schema_version":"1.0","event_id":"sha256:229b767c51e22d0d83b8396a434d6514fe29b20b36ce0029a0a3cfcffa9723a7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:KA5LAHZLBALTAUM2CBAQPJXY7P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Dynamical Systems Approach to Schwarzschild Null Geodesics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph.HE","math-ph","math.MP"],"primary_cat":"gr-qc","authors_text":"Edward Belbruno, Frans Pretorius","submitted_at":"2011-03-03T00:25:55Z","abstract_excerpt":"The null geodesics of a Schwarzschild black hole are studied from a dynamical systems perspective. Written in terms of Kerr-Schild coordinates, the null geodesic equation takes on the simple form of a particle moving under the influence of a Newtonian central force with an inverse-cubic potential. We apply a McGehee transformation to these equations, which clearly elucidates the full phase space of solutions. All the null geodesics belong to one of four families of invariant manifolds and their limiting cases, further characterized by the angular momentum L of the orbit: for |L|>|L_c|, (1) the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0585","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":"2026-05-18T02:03:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tn2PSviZ/I/ysf0qkc/+04+ikDwT0MAy5mddnBq1/66MBDzl8sRu+2nHwjv1R2JlNhTbsDTjqhwmIP61SXZXAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:35:48.872212Z"},"content_sha256":"8e05c5f2813c6ffdd8aa03c497dbd862829f75fe5a60c8089dd63d3ae0928c49","schema_version":"1.0","event_id":"sha256:8e05c5f2813c6ffdd8aa03c497dbd862829f75fe5a60c8089dd63d3ae0928c49"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KA5LAHZLBALTAUM2CBAQPJXY7P/bundle.json","state_url":"https://pith.science/pith/KA5LAHZLBALTAUM2CBAQPJXY7P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KA5LAHZLBALTAUM2CBAQPJXY7P/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":"2026-05-26T09:35:48Z","links":{"resolver":"https://pith.science/pith/KA5LAHZLBALTAUM2CBAQPJXY7P","bundle":"https://pith.science/pith/KA5LAHZLBALTAUM2CBAQPJXY7P/bundle.json","state":"https://pith.science/pith/KA5LAHZLBALTAUM2CBAQPJXY7P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KA5LAHZLBALTAUM2CBAQPJXY7P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KA5LAHZLBALTAUM2CBAQPJXY7P","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":"8c0085e18af0b2667124f4e9ecc5e17f3d91ec323b034072796a65f1e10c3e8a","cross_cats_sorted":["astro-ph.HE","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-03-03T00:25:55Z","title_canon_sha256":"1040786fd7d86cb58393ff5fc07a0e3303a9d05dc2ee33938a49701b928c94eb"},"schema_version":"1.0","source":{"id":"1103.0585","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.0585","created_at":"2026-05-18T02:03:00Z"},{"alias_kind":"arxiv_version","alias_value":"1103.0585v1","created_at":"2026-05-18T02:03:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0585","created_at":"2026-05-18T02:03:00Z"},{"alias_kind":"pith_short_12","alias_value":"KA5LAHZLBALT","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"KA5LAHZLBALTAUM2","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"KA5LAHZL","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:8e05c5f2813c6ffdd8aa03c497dbd862829f75fe5a60c8089dd63d3ae0928c49","target":"graph","created_at":"2026-05-18T02:03:00Z","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":"The null geodesics of a Schwarzschild black hole are studied from a dynamical systems perspective. Written in terms of Kerr-Schild coordinates, the null geodesic equation takes on the simple form of a particle moving under the influence of a Newtonian central force with an inverse-cubic potential. We apply a McGehee transformation to these equations, which clearly elucidates the full phase space of solutions. All the null geodesics belong to one of four families of invariant manifolds and their limiting cases, further characterized by the angular momentum L of the orbit: for |L|>|L_c|, (1) the","authors_text":"Edward Belbruno, Frans Pretorius","cross_cats":["astro-ph.HE","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-03-03T00:25:55Z","title":"A Dynamical Systems Approach to Schwarzschild Null Geodesics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0585","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":"sha256:229b767c51e22d0d83b8396a434d6514fe29b20b36ce0029a0a3cfcffa9723a7","target":"record","created_at":"2026-05-18T02:03:00Z","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":"8c0085e18af0b2667124f4e9ecc5e17f3d91ec323b034072796a65f1e10c3e8a","cross_cats_sorted":["astro-ph.HE","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-03-03T00:25:55Z","title_canon_sha256":"1040786fd7d86cb58393ff5fc07a0e3303a9d05dc2ee33938a49701b928c94eb"},"schema_version":"1.0","source":{"id":"1103.0585","kind":"arxiv","version":1}},"canonical_sha256":"503ab01f2b081730519a104107a6f8fbd7707d82379c3c701edeb43e3c2bcf63","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"503ab01f2b081730519a104107a6f8fbd7707d82379c3c701edeb43e3c2bcf63","first_computed_at":"2026-05-18T02:03:00.658671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:00.658671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WF+F9JPjWzHZY/KUhQhqf1b7lFHt11o0Wu+K1hI3VE7RKdPQ4rHF0iihVB8Bfc5xsoMlUDNKzkxiLWdQ1WRYBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:00.659402Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.0585","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:229b767c51e22d0d83b8396a434d6514fe29b20b36ce0029a0a3cfcffa9723a7","sha256:8e05c5f2813c6ffdd8aa03c497dbd862829f75fe5a60c8089dd63d3ae0928c49"],"state_sha256":"d8bb21f1532af182c9a4ccc81557efb9146048b230cd3c5a86b73c9a0312adf2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"11kV0YL9JbVhxldbYQFAXYy4Uc/P0ZbEuAb3HyXr31oxW+VG0gEY/OSnl8wcDdny7aNOfZgV4PCVSPNaNkPABA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T09:35:48.876604Z","bundle_sha256":"c5f0a6938448f05036b96e90f9cbc1974773daf0f6634eade3a9277a9be3e661"}}