{"id":"https://openalex.org/W7204947774","doi":"https://doi.org/10.48550/arxiv.2608.29556","title":"Null Completeness and Directional Rigidity in Bianchi I Spacetimes","display_name":"Null Completeness and Directional Rigidity in Bianchi I Spacetimes","publication_year":2026,"publication_date":"2026-08-30","ids":{"openalex":"https://openalex.org/W7204947774","doi":"https://doi.org/10.48550/arxiv.2608.29556"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.29556","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29556","pdf_url":"https://arxiv.org/pdf/2608.29556","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.29556","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133379392","display_name":"Ye Zhou","orcid":"https://orcid.org/0009-0004-7050-7736","observed_orcids":["https://orcid.org/0009-0004-7050-7736"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Ye","raw_affiliation_strings":["Independent Researcher , Kunshan , Jiangsu , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher , Kunshan , Jiangsu , China","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5125954651","display_name":"Alan Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183974707","display_name":"Perimeter Institute","ror":"https://ror.org/013m0ej23","country_code":"CA","type":"facility","lineage":["https://openalex.org/I183974707"]},{"id":"https://openalex.org/I98251732","display_name":"McMaster University","ror":"https://ror.org/02fa3aq29","country_code":"CA","type":"education","lineage":["https://openalex.org/I98251732"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Zhang, Alan","raw_affiliation_strings":["Department of Physics & Astronomy , McMaster University , Hamilton , Ontario L8S 4M1 , Canada","Perimeter Institute for Theoretical Physics , Waterloo , Ontario N2L 2Y5 , Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics & Astronomy , McMaster University , Hamilton , Ontario L8S 4M1 , Canada","institution_ids":["https://openalex.org/I98251732"]},{"raw_affiliation_string":"Perimeter Institute for Theoretical Physics , Waterloo , Ontario N2L 2Y5 , Canada","institution_ids":["https://openalex.org/I183974707"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10095","display_name":"Cosmology and Gravitation Theories","score":0.75,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10095","display_name":"Cosmology and Gravitation Theories","score":0.75,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.1460999995470047,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11960","display_name":"Relativity and Gravitational Theory","score":0.06520000100135803,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bianchi-i-spacetime","display_name":"Bianchi I spacetime","score":0.9319999814033508},{"id":"https://openalex.org/keywords/null-energy-condition","display_name":"null energy condition","score":0.8289999961853027},{"id":"https://openalex.org/keywords/anisotropic-cosmological-models","display_name":"anisotropic cosmology","score":0.718999981880188}],"concepts":[{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.5575000047683716},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5342000126838684},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.5314000248908997},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.5081999897956848},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.48820000886917114},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.47999998927116394},{"id":"https://openalex.org/C130187892","wikidata":"https://www.wikidata.org/wiki/Q133327","display_name":"Spacetime","level":2,"score":0.4781999886035919},{"id":"https://openalex.org/C27156116","wikidata":"https://www.wikidata.org/wiki/Q1778098","display_name":"Pointwise convergence","level":3,"score":0.4368000030517578},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3774999976158142},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.3434999883174896},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3425000011920929},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.31859999895095825},{"id":"https://openalex.org/C147452769","wikidata":"https://www.wikidata.org/wiki/Q11452","display_name":"General relativity","level":2,"score":0.31630000472068787},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.31540000438690186},{"id":"https://openalex.org/C2780791080","wikidata":"https://www.wikidata.org/wiki/Q273711","display_name":"Einstein field equations","level":3,"score":0.3018999993801117},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.2996000051498413},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.296999990940094},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.28600001335144043},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C9136319","wikidata":"https://www.wikidata.org/wiki/Q362640","display_name":"Covariant transformation","level":2,"score":0.2818000018596649},{"id":"https://openalex.org/C79464548","wikidata":"https://www.wikidata.org/wiki/Q464794","display_name":"Minkowski space","level":2,"score":0.27070000767707825},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C118159210","wikidata":"https://www.wikidata.org/wiki/Q3555169","display_name":"Null vector","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C64999384","wikidata":"https://www.wikidata.org/wiki/Q3507967","display_name":"Symmetrization","level":2,"score":0.25780001282691956},{"id":"https://openalex.org/C139941754","wikidata":"https://www.wikidata.org/wiki/Q3058244","display_name":"Symmetric space","level":2,"score":0.25290000438690186},{"id":"https://openalex.org/C2778217198","wikidata":"https://www.wikidata.org/wiki/Q556977","display_name":"Cutoff","level":2,"score":0.25110000371932983}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.29556","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29556","pdf_url":"https://arxiv.org/pdf/2608.29556","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.29556","doi":"https://doi.org/10.48550/arxiv.2608.29556","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.29556","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.29556","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29556","pdf_url":"https://arxiv.org/pdf/2608.29556","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320311526","display_name":"McMaster University","ror":"https://ror.org/02fa3aq29"},{"id":"https://openalex.org/F4320319880","display_name":"Government of Canada","ror":"https://ror.org/010q4q527"},{"id":"https://openalex.org/F4320331462","display_name":"Institut P\u00e9rim\u00e8tre de physique th\u00e9orique","ror":"https://ror.org/013m0ej23"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204947774.pdf","grobid_xml":"https://content.openalex.org/works/W7204947774.grobid-xml"},"referenced_works_count":59,"referenced_works":["https://openalex.org/W1664590888","https://openalex.org/W1865847757","https://openalex.org/W1963548036","https://openalex.org/W1966550677","https://openalex.org/W1977185432","https://openalex.org/W1980184680","https://openalex.org/W1980851479","https://openalex.org/W1987493110","https://openalex.org/W1992541079","https://openalex.org/W1993103214","https://openalex.org/W1993993424","https://openalex.org/W1994171500","https://openalex.org/W2000914920","https://openalex.org/W2018320545","https://openalex.org/W2019151576","https://openalex.org/W2023781804","https://openalex.org/W2033317896","https://openalex.org/W2036814426","https://openalex.org/W2052045615","https://openalex.org/W2053550747","https://openalex.org/W2058872211","https://openalex.org/W2073489576","https://openalex.org/W2075040080","https://openalex.org/W2075411003","https://openalex.org/W2076932450","https://openalex.org/W2090970414","https://openalex.org/W2095374372","https://openalex.org/W2098001159","https://openalex.org/W2100880655","https://openalex.org/W2101799989","https://openalex.org/W2112677333","https://openalex.org/W2171797661","https://openalex.org/W2267350356","https://openalex.org/W2282901172","https://openalex.org/W2284436989","https://openalex.org/W2333328835","https://openalex.org/W2463487467","https://openalex.org/W2780684614","https://openalex.org/W2794823163","https://openalex.org/W2923950772","https://openalex.org/W3009127927","https://openalex.org/W3102720072","https://openalex.org/W3135288405","https://openalex.org/W3160194025","https://openalex.org/W3168294805","https://openalex.org/W4256734575","https://openalex.org/W4320485814","https://openalex.org/W4402597376","https://openalex.org/W4406279698","https://openalex.org/W4414414762","https://openalex.org/W4414430073","https://openalex.org/W4417296620","https://openalex.org/W7138903411","https://openalex.org/W7140744117","https://openalex.org/W7155401684","https://openalex.org/W7160323171","https://openalex.org/W7161764419","https://openalex.org/W7165136156","https://openalex.org/W7202143732"],"related_works":["https://openalex.org/W7202143732","https://openalex.org/W1985788767","https://openalex.org/W4316813256","https://openalex.org/W7222808302","https://openalex.org/W2113323904","https://openalex.org/W7213524587","https://openalex.org/W4317605783","https://openalex.org/W2979564715","https://openalex.org/W2034060744","https://openalex.org/W1677998785"],"abstract_inverted_index":{"In":[0,112],"Einstein":[1],"gravity,":[2],"every":[3,124],"null":[4,14,38,49,62,83,127,133],"geodesic":[5,39,102,128],"of":[6,173],"a":[7,31,36,113,174],"non-static":[8,114],"Bianchi":[9,26],"I":[10,27],"spacetime":[11],"satisfying":[12],"the":[13,25,48,72,101,170,183],"energy":[15,148],"condition":[16],"is":[17,51,107],"incomplete":[18],"in":[19,41,59],"at":[20,56,116,164,187],"least":[21],"one":[22,60],"direction":[23],"within":[24],"region.":[28],"We":[29],"prove":[30],"stronger":[32],"directional":[33,161],"rigidity":[34],"theorem:":[35],"single":[37],"complete":[40,126],"both":[42],"affine":[43,189],"directions":[44,120],"forces":[45],"staticity":[46],"if":[47],"curvature":[50,95,162],"nonnegative":[52],"along":[53,100,109],"it":[54],"and,":[55,103],"each":[57],"time,":[58],"transverse":[61],"direction.":[63],"The":[64,79,93],"spatial":[65,87],"metric":[66,157],"need":[67],"not":[68],"be":[69],"diagonal,":[70],"and":[71,136,145,168],"expansion":[73],"rates":[74],"may":[75],"have":[76],"either":[77],"sign.":[78],"proof":[80],"uses":[81],"caustic-free":[82],"congruences":[84],"whose":[85],"conserved":[86],"covectors":[88],"define":[89],"fixed":[90,165,188],"screen":[91],"hyperplanes.":[92],"two":[94],"conditions":[96],"admit":[97],"averaged":[98,132,184],"replacements":[99],"when":[104],"cosmic":[105,166],"time":[106],"complete,":[108],"comoving":[110],"worldlines.":[111],"metric,":[115],"most":[117],"four":[118],"oriented":[119],"are":[121],"optically":[122],"saturated;":[123],"other":[125],"has":[129],"strictly":[130],"negative":[131],"curvature.":[134],"Periodic":[135],"finite-pulse":[137],"examples":[138],"attain":[139],"this":[140],"bound":[141],"with":[142],"NEC-violating":[143],"sources":[144],"nonpositive":[146],"total":[147],"density.":[149],"For":[150],"finite":[151],"pulses,":[152],"we":[153],"derive":[154],"bounds":[155],"on":[156,169],"excursions":[158],"from":[159,182],"three":[160],"budgets":[163],"duration,":[167],"mode":[171],"mixing":[172],"free,":[175],"massless,":[176],"minimally":[177],"coupled":[178],"test":[179],"scalar":[180],"field":[181],"null-energy":[185],"deficit":[186],"duration.":[190]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-02T00:00:00"}