{"paper":{"title":"Energy-Momentum Conservation as a Constraint to Restrict the Space of Viable Field Lagrangians: Vector Field versus Spin-Two Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Conservation of the symmetrized Belinfante tensor for a symmetric tensor field coupled to matter is equivalent to Feynman's consistency condition and fixes the field Lagrangian to the Einstein form.","cross_cats":["hep-th"],"primary_cat":"gr-qc","authors_text":"Antonio L\\'opez-Pinto, Satoshi Nakajima","submitted_at":"2026-04-19T20:21:00Z","abstract_excerpt":"We investigate whether the requirement of total energy-momentum conservation can act as a constraint on the family of admissible Lagrangian densities for an interaction field. The aim is not to give a mere field-theoretic derivation of general relativity, nor to introduce another gravitational energy-momentum pseudotensor starting from a prescribed gravitational Lagrangian. Rather, we ask the more general question of whether a conservation principle, once a definite field energy-momentum tensor has been specified, can determine the form of the field Lagrangian itself. For a Poincar\\'{e}-invari"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For a point particle coupled universally to the symmetric tensor field, we show that conservation of the total energy-momentum tensor is equivalent to Feynman's consistency condition. Assuming that the field Lagrangian is local, Lorentz invariant, and quadratic in first derivatives, this condition determines the Lagrangian, up to a total divergence, to be the Einstein Lagrangian density.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The field Lagrangian is local, Lorentz invariant, and quadratic in first derivatives; the coupling of the symmetric tensor field to the point particle is universal.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Conservation of the symmetrized Belinfante tensor is equivalent to Feynman's condition and determines the Einstein Lagrangian for a symmetric rank-two field on Minkowski space.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Conservation of the symmetrized Belinfante tensor for a symmetric tensor field coupled to matter is equivalent to Feynman's consistency condition and fixes the field Lagrangian to the Einstein form.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"312f4d9b7e8dade25954c0eba604822bf1c1ff5465995fd2bd2bcd4f76c4c444"},"source":{"id":"2604.17601","kind":"arxiv","version":3},"verdict":{"id":"96bda1d9-809f-48f4-a44e-6c70f09cf855","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T02:08:44.945683Z","strongest_claim":"For a point particle coupled universally to the symmetric tensor field, we show that conservation of the total energy-momentum tensor is equivalent to Feynman's consistency condition. Assuming that the field Lagrangian is local, Lorentz invariant, and quadratic in first derivatives, this condition determines the Lagrangian, up to a total divergence, to be the Einstein Lagrangian density.","one_line_summary":"Conservation of the symmetrized Belinfante tensor is equivalent to Feynman's condition and determines the Einstein Lagrangian for a symmetric rank-two field on Minkowski space.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The field Lagrangian is local, Lorentz invariant, and quadratic in first derivatives; the coupling of the symmetric tensor field to the point particle is universal.","pith_extraction_headline":"Conservation of the symmetrized Belinfante tensor for a symmetric tensor field coupled to matter is equivalent to Feynman's consistency condition and fixes the field Lagrangian to the Einstein form."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.17601/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"}