{"paper":{"title":"Global Hadamard form for the Green Function in Schwarzschild space-time","arxiv_id":"1606.03075","primary_cat":"gr-qc","submitted_at":"2016-06-09T19:30:25+00:00","version_compiled":"v4"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v4","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.57,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.87,"id":"s1","own":false,"role":"background","text":"The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.12,"method":0.0,"result":0.0,"background":0.87,"limitation":0.01,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.98,"id":"s2","own":true,"role":"contribution","text":"In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\\it complete} singularity structure beyond the normal neighbourhood.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.76,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}},{"i":3,"p":0.96,"id":"s3","own":true,"role":"interpretation","text":"We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.37,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.96}},{"i":4,"p":0.97,"id":"s4","own":true,"role":"method","text":"(The term non-smooth applies modulo the causality-generating step functions that must appear in the retarded Green function.) The singularity structure is determined using two independent approaches, one based on a Bessel function expansion of the Green function, and another that exploits a link between the Green functions of Schwarzschild spacetime and Pleba{\\'n}ski-Hacyan spacetime (the latter approach also yields another representation for the {\\it full} Schwarzschild Green function, not just for its non-smooth part).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.36,"role_probabilities":{"gap":0.0,"method":0.97,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.01}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"limitation","text":"Our representation is not valid in a neighbourhood of caustic points.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":1.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.84,"id":"s6","own":true,"role":"contribution","text":"We deal with these points by providing a separate representation for the Green function in Schwarzschild spacetime which makes explicit its (different) singularity structure at caustics of this spacetime.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.06,"result":0.05,"background":0.0,"limitation":0.05,"future_work":0.0,"contribution":0.84,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.99,"text":"In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\\it complete} singularity structure beyond the normal neighbourhood.","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":5896,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.07,"states_limitation":0.98,"human_participants":0.01,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.04,"compares_with_prior_results":0.81,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\\it complete} singularity structure beyond the normal neighbourhood.","location":"abstract s2"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v4"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":7,"sentences_in_abstract":7},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}