{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1996:JMRXAASH3Z77TJS3MES3RXCFUI","short_pith_number":"pith:JMRXAASH","canonical_record":{"source":{"id":"gr-qc/9609040","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"gr-qc","submitted_at":"1996-09-13T17:33:46Z","cross_cats_sorted":["astro-ph"],"title_canon_sha256":"78049f8b8ebed99c847ccb5bb2ff57be4ea253a767327e6891d5ebc74c31eb4a","abstract_canon_sha256":"8662e35b1585d352e603f047352ff98d52a556416a46f1052241133b81892060"},"schema_version":"1.0"},"canonical_sha256":"4b23700247de7ff9a65b6125b8dc45a213edd07593d9c333e9f7311b8f79aaf1","source":{"kind":"arxiv","id":"gr-qc/9609040","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"gr-qc/9609040","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"arxiv_version","alias_value":"gr-qc/9609040v2","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.gr-qc/9609040","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"pith_short_12","alias_value":"JMRXAASH3Z77","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"pith_short_16","alias_value":"JMRXAASH3Z77TJS3","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"pith_short_8","alias_value":"JMRXAASH","created_at":"2026-07-04T16:01:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1996:JMRXAASH3Z77TJS3MES3RXCFUI","target":"record","payload":{"canonical_record":{"source":{"id":"gr-qc/9609040","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"gr-qc","submitted_at":"1996-09-13T17:33:46Z","cross_cats_sorted":["astro-ph"],"title_canon_sha256":"78049f8b8ebed99c847ccb5bb2ff57be4ea253a767327e6891d5ebc74c31eb4a","abstract_canon_sha256":"8662e35b1585d352e603f047352ff98d52a556416a46f1052241133b81892060"},"schema_version":"1.0"},"canonical_sha256":"4b23700247de7ff9a65b6125b8dc45a213edd07593d9c333e9f7311b8f79aaf1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T16:01:18.424899Z","signature_b64":"2WzGBH/fweZ3Sw4WATG++Bjoyb4ZRB9DUn3xM53Iry5NCf/8UVHo8g0TF1jRrG7wkL4/VsnzKxuW4g9OIZAtBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b23700247de7ff9a65b6125b8dc45a213edd07593d9c333e9f7311b8f79aaf1","last_reissued_at":"2026-07-04T16:01:18.424434Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T16:01:18.424434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"gr-qc/9609040","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":"2026-07-04T16:01:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mx+bPIc6AfE+b7KIP4qIpxrRH0qdn8VFsQKIo2rV7XozVh5McffOT6MRu4u9KbtZyVJsqEs+H29eSCHa0sDOAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T15:06:20.893487Z"},"content_sha256":"8c229ea4254a89684ac081d396189bd840a37a73950f398769ab6ff4b9f55540","schema_version":"1.0","event_id":"sha256:8c229ea4254a89684ac081d396189bd840a37a73950f398769ab6ff4b9f55540"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1996:JMRXAASH3Z77TJS3MES3RXCFUI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond","license":"","headline":"","cross_cats":["astro-ph"],"primary_cat":"gr-qc","authors_text":"Marco Bruni, Sabino Matarrese, Sebastiano Sonego, Silvia Mollerach","submitted_at":"1996-09-13T17:33:46Z","abstract_excerpt":"We consider in detail the problem of gauge dependence that exists in relativistic perturbation theory, going beyond the linear approximation and treating second and higher order perturbations. We first derive some mathematical results concerning the Taylor expansion of tensor fields under the action of one-parameter families (not necessarily groups) of diffeomorphisms. Second, we define gauge invariance to an arbitrary order $n$. Finally, we give a generating formula for the gauge transformation to an arbitrary order and explicit rules to second and third order. This formalism can be used in a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"gr-qc/9609040","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/gr-qc/9609040/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T16:01:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+kvCtxpnJwPiq2/EFuZD5XMd+85Eb8q1iG+6vexe8zyEXHd2EIxRpSnfo74teycERU0yTpYZZDRYug8MbpagDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T15:06:20.894093Z"},"content_sha256":"f5acf34e4da1cc8bd893121432e22e6293f2b735bdd461655e6b76668bf9c4f2","schema_version":"1.0","event_id":"sha256:f5acf34e4da1cc8bd893121432e22e6293f2b735bdd461655e6b76668bf9c4f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JMRXAASH3Z77TJS3MES3RXCFUI/bundle.json","state_url":"https://pith.science/pith/JMRXAASH3Z77TJS3MES3RXCFUI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JMRXAASH3Z77TJS3MES3RXCFUI/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":"2026-08-14T15:06:20Z","links":{"resolver":"https://pith.science/pith/JMRXAASH3Z77TJS3MES3RXCFUI","bundle":"https://pith.science/pith/JMRXAASH3Z77TJS3MES3RXCFUI/bundle.json","state":"https://pith.science/pith/JMRXAASH3Z77TJS3MES3RXCFUI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JMRXAASH3Z77TJS3MES3RXCFUI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:JMRXAASH3Z77TJS3MES3RXCFUI","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":"8662e35b1585d352e603f047352ff98d52a556416a46f1052241133b81892060","cross_cats_sorted":["astro-ph"],"license":"","primary_cat":"gr-qc","submitted_at":"1996-09-13T17:33:46Z","title_canon_sha256":"78049f8b8ebed99c847ccb5bb2ff57be4ea253a767327e6891d5ebc74c31eb4a"},"schema_version":"1.0","source":{"id":"gr-qc/9609040","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"gr-qc/9609040","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"arxiv_version","alias_value":"gr-qc/9609040v2","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.gr-qc/9609040","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"pith_short_12","alias_value":"JMRXAASH3Z77","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"pith_short_16","alias_value":"JMRXAASH3Z77TJS3","created_at":"2026-07-04T16:01:18Z"},{"alias_kind":"pith_short_8","alias_value":"JMRXAASH","created_at":"2026-07-04T16:01:18Z"}],"graph_snapshots":[{"event_id":"sha256:f5acf34e4da1cc8bd893121432e22e6293f2b735bdd461655e6b76668bf9c4f2","target":"graph","created_at":"2026-07-04T16:01:18Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/gr-qc/9609040/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider in detail the problem of gauge dependence that exists in relativistic perturbation theory, going beyond the linear approximation and treating second and higher order perturbations. We first derive some mathematical results concerning the Taylor expansion of tensor fields under the action of one-parameter families (not necessarily groups) of diffeomorphisms. Second, we define gauge invariance to an arbitrary order $n$. Finally, we give a generating formula for the gauge transformation to an arbitrary order and explicit rules to second and third order. This formalism can be used in a","authors_text":"Marco Bruni, Sabino Matarrese, Sebastiano Sonego, Silvia Mollerach","cross_cats":["astro-ph"],"headline":"","license":"","primary_cat":"gr-qc","submitted_at":"1996-09-13T17:33:46Z","title":"Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"gr-qc/9609040","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":"sha256:8c229ea4254a89684ac081d396189bd840a37a73950f398769ab6ff4b9f55540","target":"record","created_at":"2026-07-04T16:01:18Z","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":"8662e35b1585d352e603f047352ff98d52a556416a46f1052241133b81892060","cross_cats_sorted":["astro-ph"],"license":"","primary_cat":"gr-qc","submitted_at":"1996-09-13T17:33:46Z","title_canon_sha256":"78049f8b8ebed99c847ccb5bb2ff57be4ea253a767327e6891d5ebc74c31eb4a"},"schema_version":"1.0","source":{"id":"gr-qc/9609040","kind":"arxiv","version":2}},"canonical_sha256":"4b23700247de7ff9a65b6125b8dc45a213edd07593d9c333e9f7311b8f79aaf1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b23700247de7ff9a65b6125b8dc45a213edd07593d9c333e9f7311b8f79aaf1","first_computed_at":"2026-07-04T16:01:18.424434Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T16:01:18.424434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2WzGBH/fweZ3Sw4WATG++Bjoyb4ZRB9DUn3xM53Iry5NCf/8UVHo8g0TF1jRrG7wkL4/VsnzKxuW4g9OIZAtBw==","signature_status":"signed_v1","signed_at":"2026-07-04T16:01:18.424899Z","signed_message":"canonical_sha256_bytes"},"source_id":"gr-qc/9609040","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c229ea4254a89684ac081d396189bd840a37a73950f398769ab6ff4b9f55540","sha256:f5acf34e4da1cc8bd893121432e22e6293f2b735bdd461655e6b76668bf9c4f2"],"state_sha256":"303aace62db569091f3381bdefc627cbbb54e3b6e989aadb76b36b603d4c7318"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lPTitZWU3C12qAmJ+GRoo+NjSNOWEWqRBBV0Rja+nMPIyep9eR2hUQsKPqiGveieSw9QqOjnGltbJqMNwz/9Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T15:06:20.955869Z","bundle_sha256":"70545d1533eb8e99cc4cf62f0299c2dd53f2d35465b1d054f7c90487f784b9cf"}}