{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:FEZH5O5PZQQRMHVYM6CAE7E6AB","short_pith_number":"pith:FEZH5O5P","canonical_record":{"source":{"id":"1110.6508","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-10-29T09:04:07Z","cross_cats_sorted":["hep-th","math-ph","math.MP"],"title_canon_sha256":"28be33e1b763eb54b6bda0a783ad5775c2e153eff4643b33381e2183ea40e92e","abstract_canon_sha256":"534b3e517ff75fc6ea0a6ad94185b177582d3a3f64b8f672c0dc5d805f9caf46"},"schema_version":"1.0"},"canonical_sha256":"29327ebbafcc21161eb86784027c9e0070aa3226b04518ab774554489ae3cac1","source":{"kind":"arxiv","id":"1110.6508","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6508","created_at":"2026-05-18T01:59:50Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6508v2","created_at":"2026-05-18T01:59:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6508","created_at":"2026-05-18T01:59:50Z"},{"alias_kind":"pith_short_12","alias_value":"FEZH5O5PZQQR","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FEZH5O5PZQQRMHVY","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FEZH5O5P","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:FEZH5O5PZQQRMHVYM6CAE7E6AB","target":"record","payload":{"canonical_record":{"source":{"id":"1110.6508","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-10-29T09:04:07Z","cross_cats_sorted":["hep-th","math-ph","math.MP"],"title_canon_sha256":"28be33e1b763eb54b6bda0a783ad5775c2e153eff4643b33381e2183ea40e92e","abstract_canon_sha256":"534b3e517ff75fc6ea0a6ad94185b177582d3a3f64b8f672c0dc5d805f9caf46"},"schema_version":"1.0"},"canonical_sha256":"29327ebbafcc21161eb86784027c9e0070aa3226b04518ab774554489ae3cac1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:50.230378Z","signature_b64":"tMgGYFVjj82BEW1Ljkcn5z/4Vl2zfzy0DnAfIK31RzEWXrW3XOtL9aMnL5SEiqnb3iyXP5BK1suAL1k7Nrb3Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29327ebbafcc21161eb86784027c9e0070aa3226b04518ab774554489ae3cac1","last_reissued_at":"2026-05-18T01:59:50.229666Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:50.229666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.6508","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:59:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TJ3S5amz9Ezlaz6UF+i5WvzNCrWyNwn1wKy32zof9tpCDwwNHFprxaKlhUPMTvGCvz3IgMXYCxds2hpMPEm7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:43:29.158377Z"},"content_sha256":"c53986a6792ec21a02dff98a24cd047551e18f6c482e6569cc217819f516dc7e","schema_version":"1.0","event_id":"sha256:c53986a6792ec21a02dff98a24cd047551e18f6c482e6569cc217819f516dc7e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:FEZH5O5PZQQRMHVYM6CAE7E6AB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP"],"primary_cat":"gr-qc","authors_text":"G. W. Gibbons, M. Vyska","submitted_at":"2011-10-29T09:04:07Z","abstract_excerpt":"In this paper we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both $M/r_0$ and $M/b$ where $M$ is the mass of the black hole, $r_0$ the distance of closest approach of the light ray and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6508","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:59:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IKmYqM0nZvSxuyp0RtkNB7Nn53OIgOv2EylQs3OAiJ7wcjQW79C260BWoYJ4SxZ5wDIensRjoA2a9kJQS6YTCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:43:29.159128Z"},"content_sha256":"5785f40617e0f077c476678cfc4e815c52796878a0fb3173125ac820eb7102f2","schema_version":"1.0","event_id":"sha256:5785f40617e0f077c476678cfc4e815c52796878a0fb3173125ac820eb7102f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB/bundle.json","state_url":"https://pith.science/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB/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":"2026-06-09T06:43:29Z","links":{"resolver":"https://pith.science/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB","bundle":"https://pith.science/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB/bundle.json","state":"https://pith.science/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FEZH5O5PZQQRMHVYM6CAE7E6AB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:FEZH5O5PZQQRMHVYM6CAE7E6AB","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":"534b3e517ff75fc6ea0a6ad94185b177582d3a3f64b8f672c0dc5d805f9caf46","cross_cats_sorted":["hep-th","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-10-29T09:04:07Z","title_canon_sha256":"28be33e1b763eb54b6bda0a783ad5775c2e153eff4643b33381e2183ea40e92e"},"schema_version":"1.0","source":{"id":"1110.6508","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6508","created_at":"2026-05-18T01:59:50Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6508v2","created_at":"2026-05-18T01:59:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6508","created_at":"2026-05-18T01:59:50Z"},{"alias_kind":"pith_short_12","alias_value":"FEZH5O5PZQQR","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FEZH5O5PZQQRMHVY","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FEZH5O5P","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:5785f40617e0f077c476678cfc4e815c52796878a0fb3173125ac820eb7102f2","target":"graph","created_at":"2026-05-18T01:59:50Z","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":"In this paper we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both $M/r_0$ and $M/b$ where $M$ is the mass of the black hole, $r_0$ the distance of closest approach of the light ray and $","authors_text":"G. W. Gibbons, M. Vyska","cross_cats":["hep-th","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-10-29T09:04:07Z","title":"The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6508","kind":"arxiv","version":2},"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:c53986a6792ec21a02dff98a24cd047551e18f6c482e6569cc217819f516dc7e","target":"record","created_at":"2026-05-18T01:59:50Z","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":"534b3e517ff75fc6ea0a6ad94185b177582d3a3f64b8f672c0dc5d805f9caf46","cross_cats_sorted":["hep-th","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2011-10-29T09:04:07Z","title_canon_sha256":"28be33e1b763eb54b6bda0a783ad5775c2e153eff4643b33381e2183ea40e92e"},"schema_version":"1.0","source":{"id":"1110.6508","kind":"arxiv","version":2}},"canonical_sha256":"29327ebbafcc21161eb86784027c9e0070aa3226b04518ab774554489ae3cac1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29327ebbafcc21161eb86784027c9e0070aa3226b04518ab774554489ae3cac1","first_computed_at":"2026-05-18T01:59:50.229666Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:50.229666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tMgGYFVjj82BEW1Ljkcn5z/4Vl2zfzy0DnAfIK31RzEWXrW3XOtL9aMnL5SEiqnb3iyXP5BK1suAL1k7Nrb3Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:50.230378Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.6508","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c53986a6792ec21a02dff98a24cd047551e18f6c482e6569cc217819f516dc7e","sha256:5785f40617e0f077c476678cfc4e815c52796878a0fb3173125ac820eb7102f2"],"state_sha256":"97fc6212837b35c9eee167987821fa2ec90afa680784931abed9819508cd4da4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8cRhTQGcDq09mH35KNsXQ4YjaQls+h0Ht6M8qQtBxHgoNYy5ePdEfHo9PXz7I9pNiYKF20B9xSUKYDYuV3oTDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T06:43:29.162989Z","bundle_sha256":"668f2b51b69db77f0c58a10d94ab8d8bbe2a6ae9a6b0c441a8288820df4c20ac"}}