{"id":"https://openalex.org/W7214658264","doi":"https://doi.org/10.48550/arxiv.2609.30660","title":"Geometry as Thermodynamics:Entropy Stationarity and Horizon Residues","display_name":"Geometry as Thermodynamics:Entropy Stationarity and Horizon Residues","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214658264","doi":"https://doi.org/10.48550/arxiv.2609.30660"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30660","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30660","pdf_url":"https://arxiv.org/pdf/2609.30660","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/2609.30660","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153322950","display_name":"Wen-Xiang Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4405280161","display_name":"Guangzhou City University of Technology","ror":"https://ror.org/02hcyt080","country_code":"CN","type":"education","lineage":["https://openalex.org/I4405280161"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Chen, Wen-Xiang","raw_affiliation_strings":["School of Electronic Information , Guangzhou City University of Technology , Guangzhou , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electronic Information , Guangzhou City University of Technology , Guangzhou , China","institution_ids":["https://openalex.org/I4405280161"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5153322950"],"corresponding_institution_ids":["https://openalex.org/I4405280161"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T10025","display_name":"Black Holes and Theoretical Physics","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"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/T10025","display_name":"Black Holes and Theoretical Physics","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"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/T11960","display_name":"Relativity and Gravitational Theory","score":0.0010000000474974513,"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/T10095","display_name":"Cosmology and Gravitation Theories","score":0.00019999999494757503,"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/black-hole-thermodynamics","display_name":"black hole thermodynamics","score":0.7409999966621399},{"id":"https://openalex.org/keywords/surface-gravity","display_name":"surface gravity","score":0.7929999828338623},{"id":"https://openalex.org/keywords/hawking-temperature","display_name":"Hawking temperature","score":0.9269999861717224},{"id":"https://openalex.org/keywords/wald-entropy","display_name":"Wald entropy","score":0.722000002861023},{"id":"https://openalex.org/keywords/schwarzschild-black-hole","display_name":"Schwarzschild black hole","score":0.796999990940094},{"id":"https://openalex.org/keywords/smarr-relation","display_name":"Smarr relation","score":0.9739999771118164},{"id":"https://openalex.org/keywords/area-law","display_name":"area law","score":0.6269999742507935},{"id":"https://openalex.org/keywords/noether-charge","display_name":"Noether charge","score":0.6420000195503235},{"id":"https://openalex.org/keywords/contour-representation","display_name":"contour representation","score":0.37400001287460327},{"id":"https://openalex.org/keywords/spacetime-singularities","display_name":"spacetime singularities","score":0.7170000076293945}],"concepts":[{"id":"https://openalex.org/C115304011","wikidata":"https://www.wikidata.org/wiki/Q72755","display_name":"Schwarzschild radius","level":3,"score":0.589900016784668},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5669999718666077},{"id":"https://openalex.org/C33932047","wikidata":"https://www.wikidata.org/wiki/Q375553","display_name":"Boltzmann's entropy formula","level":3,"score":0.48840001225471497},{"id":"https://openalex.org/C124017977","wikidata":"https://www.wikidata.org/wiki/Q11412","display_name":"Gravitation","level":2,"score":0.4607999920845032},{"id":"https://openalex.org/C17777063","wikidata":"https://www.wikidata.org/wiki/Q4780925","display_name":"Apparent horizon","level":4,"score":0.4537000060081482},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.448199987411499},{"id":"https://openalex.org/C130187892","wikidata":"https://www.wikidata.org/wiki/Q133327","display_name":"Spacetime","level":2,"score":0.4406000077724457},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.4381999969482422},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4212000072002411},{"id":"https://openalex.org/C63745668","wikidata":"https://www.wikidata.org/wiki/Q911284","display_name":"Black hole thermodynamics","level":3,"score":0.3955000042915344},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3953999876976013},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3865000009536743},{"id":"https://openalex.org/C136125196","wikidata":"https://www.wikidata.org/wiki/Q742969","display_name":"Schwarzschild metric","level":3,"score":0.3813999891281128},{"id":"https://openalex.org/C166861157","wikidata":"https://www.wikidata.org/wiki/Q59151","display_name":"Cosmological constant","level":2,"score":0.37139999866485596},{"id":"https://openalex.org/C148292158","wikidata":"https://www.wikidata.org/wiki/Q181741","display_name":"Event horizon","level":3,"score":0.36899998784065247},{"id":"https://openalex.org/C2780791080","wikidata":"https://www.wikidata.org/wiki/Q273711","display_name":"Einstein field equations","level":3,"score":0.36399999260902405},{"id":"https://openalex.org/C147452769","wikidata":"https://www.wikidata.org/wiki/Q11452","display_name":"General relativity","level":2,"score":0.31700000166893005},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.31439998745918274},{"id":"https://openalex.org/C159176650","wikidata":"https://www.wikidata.org/wiki/Q43261","display_name":"Horizon","level":2,"score":0.2919999957084656},{"id":"https://openalex.org/C10784197","wikidata":"https://www.wikidata.org/wiki/Q233858","display_name":"Non-Euclidean geometry","level":3,"score":0.2903999984264374},{"id":"https://openalex.org/C72422203","wikidata":"https://www.wikidata.org/wiki/Q7392545","display_name":"SPHERES","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C188082385","wikidata":"https://www.wikidata.org/wiki/Q73792","display_name":"Charge (physics)","level":2,"score":0.2870999872684479},{"id":"https://openalex.org/C162241779","wikidata":"https://www.wikidata.org/wiki/Q1056595","display_name":"Proper time","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C15580451","wikidata":"https://www.wikidata.org/wiki/Q3077640","display_name":"Mass formula","level":2,"score":0.2759999930858612},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.2712000012397766},{"id":"https://openalex.org/C144951741","wikidata":"https://www.wikidata.org/wiki/Q131468","display_name":"White hole","level":3,"score":0.2653000056743622},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.26499998569488525},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.26339998841285706},{"id":"https://openalex.org/C50341732","wikidata":"https://www.wikidata.org/wiki/Q880302","display_name":"Black hole (networking)","level":5,"score":0.2563000023365021},{"id":"https://openalex.org/C80892491","wikidata":"https://www.wikidata.org/wiki/Q578555","display_name":"Noether's theorem","level":3,"score":0.25600001215934753},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.250900000333786}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30660","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30660","pdf_url":"https://arxiv.org/pdf/2609.30660","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.2609.30660","doi":"https://doi.org/10.48550/arxiv.2609.30660","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30660","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:2609.30660","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30660","pdf_url":"https://arxiv.org/pdf/2609.30660","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214658264.pdf","grobid_xml":"https://content.openalex.org/works/W7214658264.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W2042650304","https://openalex.org/W2045250792","https://openalex.org/W2052045615","https://openalex.org/W2052723426","https://openalex.org/W2062686508","https://openalex.org/W2090386790","https://openalex.org/W2096798206","https://openalex.org/W2097006314","https://openalex.org/W2104466725","https://openalex.org/W2108734736","https://openalex.org/W2118148695","https://openalex.org/W2144012213","https://openalex.org/W2146478425","https://openalex.org/W2154858141","https://openalex.org/W2898074472"],"related_works":["https://openalex.org/W4413636485","https://openalex.org/W7077857698","https://openalex.org/W2031231371","https://openalex.org/W2122464632","https://openalex.org/W2097203242","https://openalex.org/W2146637495","https://openalex.org/W2530346710","https://openalex.org/W2055220101","https://openalex.org/W2060381172","https://openalex.org/W2283737638"],"abstract_inverted_index":{"Thermodynamic":[0],"descriptions":[1],"of":[2,32,81,106,128,136,210],"gravity":[3],"involve":[4],"both":[5],"variational":[6],"conditions":[7],"for":[8,59,206],"spacetime":[9],"dynamics":[10],"and":[11,49,117,143,158,166,199],"analytic":[12,61],"structures":[13],"associated":[14,53],"with":[15,54,65,96],"horizons.":[16],"We":[17,132,175],"examine":[18],"their":[19,23],"relation":[20],"while":[21,160],"keeping":[22],"assumptions":[24],"distinct.":[25],"First,":[26],"we":[27,70],"present":[28],"an":[29,60,124,145,167,180],"explicit":[30,203],"derivation":[31,127],"the":[33,37,43,46,50,55,79,83,90,97,107,113,129,134,151,162],"Einstein":[34,164],"equation":[35],"from":[36],"established":[38],"null-vector":[39],"entropy":[40,182,189,195],"functional,":[41],"including":[42],"null":[44],"constraint,":[45],"matter":[47],"term,":[48],"integration":[51],"constant":[52],"cosmological":[56],"constant.":[57],"Second,":[58],"static":[62],"spherical":[63],"geometry":[64],"a":[66,72,102,137,172,187],"nondegenerate":[67],"Killing":[68],"horizon,":[69],"define":[71],"meromorphic":[73],"radial":[74],"one-form":[75],"whose":[76],"residue":[77,95],"is":[78],"inverse":[80],"twice":[82],"surface":[84,156],"gravity.":[85],"Euclidean":[86],"regularity":[87],"then":[88],"fixes":[89],"Hawking":[91],"temperature.":[92],"Combining":[93],"this":[94],"Einstein--Hilbert":[98],"Noether":[99],"charge":[100,200],"gives":[101],"normalized":[103],"contour":[104],"representation":[105],"Wald":[108],"entropy.":[109],"The":[110,191],"construction":[111],"reproduces":[112],"Schwarzschild":[114,152],"temperature,":[115],"entropy,":[116],"Smarr":[118],"relation,":[119],"but":[120],"does":[121,184],"not":[122,185],"constitute":[123],"independent":[125],"microscopic":[126],"area":[130],"law.":[131],"establish":[133],"limits":[135],"stronger":[138],"identification":[139],"between":[140],"pole":[141],"structure":[142],"dynamics:":[144],"asymptotically":[146],"flat":[147],"family":[148],"can":[149],"retain":[150],"horizon":[153,197],"residue,":[154],"area,":[155],"gravity,":[157],"mass":[159],"violating":[161],"vacuum":[163],"equation,":[165],"extremal":[168],"charged":[169],"solution":[170],"possesses":[171],"higher-order":[173],"pole.":[174],"also":[176],"show":[177],"why":[178],"exponentiating":[179],"unspecified":[181],"functional":[183],"produce":[186],"universal":[188],"residue.":[190],"resulting":[192],"framework":[193],"separates":[194],"stationarity,":[196],"analyticity,":[198],"normalization,":[201],"providing":[202],"consistency":[204],"tests":[205],"further":[207],"thermodynamic":[208],"interpretations":[209],"gravitational":[211],"singularities.":[212]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-29T00:00:00"}