[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-45815-en":3,"doc-seo-45815-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},45815,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Hawking dan Bardeen (lubang hitam)","Expressions are derived for the mass of a stationary axisymmetric spacetime solution of Einstein’s equations describing a black hole surrounded by matter, and for the mass difference between neighboring solutions. The event-horizon area A and the black-hole “surface gravity” k are identified with physical quantities analogous to entropy and temperature. This analogy leads to a set of four laws of black hole mechanics that correspond to, and in places extend, thermodynamic laws.","# The Four Laws of Black Hole Mechanics\n\nJ.M.Bardeen*  \nDepartment of Physics,Yale University,New Haven,Connecticut,USA  \nB.Carter and S.W.Hawking  \nInstitute of Astronomy.University of Cambridge,England  \nReceived January 24,1973  \nAbstract.Expressions are derived for the mass of a stationary axisymmetric solutionof the Einstein equations containing a black hole surrounded by matter and for thedifference in mass between two neighboring such solutions.Two of the quantities whichappear in these expressions,namely the area A of the event horizon and the \"surfacegravity\"k of the black hole,have a close analogy with entropy and temperature respectively.This analogy suggests the formulation of four laws of black hole mechanics which corre-spond to and in some ways transcend the four laws of thermodynamics.  \n## 1.Introduction\n\nIt is generally believed that a gravitationally collapsing body willgive rise to a black hole and that this black hole will settle down to astationary state.If the black hole is rotating,the stationary state mustbe axisymmetric [1](An improved version of this theorem involvingweaker assumptions is outlined in [2]and is given in detail in [3]).It has been shown that stationary axisymmetric black hole solutionswhich are empty outside the event horizon fall into discrete familieseach of which depends on only two parameters,the mass M and theangular momentum J[4-6].The Kerr solutions for M⁴>J²are onesuchfamily.It seems unlikely that there areany others.It also seems reason-able to suppose that the Newman-Kerr solutions for M⁴>J²+M²Q²,where Q is the electric charge,are the only stationary axisymmetric blackhole solutions which are empty outside the event horizon apart froman electromagnetic field.On the other hand there will be an infinitedimensional family of stationary axisymmetric solutions in whichthere are rings of matter orbiting the black hole.In Sections 2 and 3 ofthis paper we shall derive formulae for the mass of such a solution andfor the difference in mass of two nearby solutions.These formulae  \nJ.M.Bardeen er al.:  \ngeneralise the expressions found by Smarr [7]and Beckenstein [8]for the Kerr and Newman-Kerr solutions.We show that the quantitiesappearing in the formulae have well-defined physical interpretations.Of particular interest are the area A of the event horizon and the“surfacegravity”K,which appear together.These have strong analogies toentropy and temperature respectively.Pursuing this analogy we areled in Section 4 to formulate four laws of black hole mechanics whichare similar to,but distinct from,the four laws of thermodynamics.  \n## 2.The Integral Formula\n\nIn a stationary axisymmetric asymptotically flat space,there is aunique time translational Killing vector Kawhich is timelike near infinitywith K\"Ka=-1 and a unique rotational Killing vector Ka whoseorbits are closed curves with parameter length 2π.These Killing vectorsobey equations  \nKa;b=K[a;b],Ka;b=K1a;b],(1)  \nKa;bK=Ka:bK,  \n(2)  \nKa;b=-R“₆K,  \n(3)  \nKa;b=-R“,K,  \n(4)  \nwhere a semicolon denotes the covariant derivatives,square bracketsaround indices imply antisymmetrization and Rab=Racb with  \nfor any vector v⁴.Since Ka;b is antisymmetric,one can integrate Eq.(3)over a hypersurface S and transfer the volume on the left to an integralover a 2-surface aS bounding S:  \n(5)  \nwhere d2ab and d2aare the surface elements of aS and S respectively.We shall choose the surface to be spacelike,asymptotically flat,tangentto the rotation Killing vector K\",and to intersect the event horizon [1]in a 2-surface aB.The boundary as of S consists of CB and a 2-surfaceOS,at infinity.For an asymptotically flat space,the integral over aS。in equation(5)is equal to-4πM,where M is the mass as measured frominfinity.Thus  \n(6)  \nwhere  \nThe first integral on the right can be regarded as the contribution to thetotal mass of the matter outside the event horizon,and the secondintegral may be regarded as the mass of the black hole.One can integrateEq.(4)similarl","cbCaihpYtaGmgEhS","https://ap.wps.com/l/cbCaihpYtaGmgEhS","pdf",814348,4,1,10,"English","en",105,"# 1. Introduction\n# 2. The Integral Formula\n## Killing vectors and field equations\n## Mass and angular momentum integrals\n## Horizon generators and surface gravity","[{\"question\":\"Apa tujuan utama makalah ini?\",\"answer\":\"Makalah ini menurunkan ekspresi untuk massa solusi stasioner ber-simetris aksial yang berisi lubang hitam bermuatan/mengelilingi materi serta menurunkan beda massa untuk dua solusi yang berdekatan.\"},{\"question\":\"Apa hubungan analogi antara area cakrawala peristiwa dan surface gravity?\",\"answer\":\"Makalah menunjukkan bahwa area A cakrawala peristiwa dan “surface gravity” k memiliki analogi kuat dengan konsep entropi dan temperatur, masing-masing.\"},{\"question\":\"Bagaimana empat hukum mekanika lubang hitam diturunkan dan apa kaitannya dengan termodinamika?\",\"answer\":\"Melalui analogi A–entropi dan k–temperatur, makalah merumuskan empat hukum mekanika lubang hitam yang mirip dengan empat hukum termodinamika serta pada beberapa aspek melampauinya.\"}]",1783466627,25,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"hawking-and-bardeen-black-holes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/hawking-and-bardeen-black-holes/45815/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-13","2026-07-07",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Apa tujuan utama makalah ini?","Question",{"text":75,"@type":76},"Makalah ini menurunkan ekspresi untuk massa solusi stasioner ber-simetris aksial yang berisi lubang hitam bermuatan/mengelilingi materi serta menurunkan beda massa untuk dua solusi yang berdekatan.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Apa hubungan analogi antara area cakrawala peristiwa dan surface gravity?",{"text":80,"@type":76},"Makalah menunjukkan bahwa area A cakrawala peristiwa dan “surface gravity” k memiliki analogi kuat dengan konsep entropi dan temperatur, masing-masing.",{"name":82,"@type":73,"acceptedAnswer":83},"Bagaimana empat hukum mekanika lubang hitam diturunkan dan apa kaitannya dengan termodinamika?",{"text":84,"@type":76},"Melalui analogi A–entropi dan k–temperatur, makalah merumuskan empat hukum mekanika lubang hitam yang mirip dengan empat hukum termodinamika serta pada beberapa aspek melampauinya.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":22,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]