http://www(Updateof .livingrevlrr−2004−4)12, −r 2009−4 Line 1 h t t p : slash slash w ww froLmivinogpReenv.parenthesisRUepladtaivtietyo,fitoewLsivi(n2.g009R)or,eg/vlrp4eriod period l i v i n g r e v from lrr hyphen 2004 hyphen 4 closing parenthesis \[\begin{aligned} h t t p : / / w ww ˆ{ ( Update of } { Living Rev . } . l i v i n g to Relativity comma sub i to the power of 1 2 ew sub s to the power of comma hyphen from r to open parenthesis period sub o r to the power of 2009 sub g slash l to r e v ˆ{ lrr − 2004 − 4 ) } { Relativity , } { i }ˆ{ 1 } 2 { ew }ˆ{ , } { s } − ˆ{ r } { ( { . }ˆ{ 2009 } { o theQpouwearsoif c-loLsinogcpaarlentheEsisnceormgmya s-ubMr 4o2m0 0e9nhtyupmhen 4 Lainned2 hlineAngular Momentum in r }ˆ{ ) , } { g / l } { r } 4 } 20 0 9 − 4 \\ Quasi hyphen Local .. Energy hyphen Momentum .. and .. Angular .. Momentum .. in General Relativity \rule{3em}{0.4pt} \end{aligned}\] General .. Relativity L a´ sz l o´B . Szabados L a-acute sz l o-acute B period Szabados Wigner Research Centre for Physics WignerResearchCentreforPhysics \centerline{Quasi − Local \quad Energy − Momentum \quad and \quad Angular \quad Momentum \quad in } of the Hungarian Academy of Sciences oftheHungarianAcademyofSciences P period O period Box 49 P.O.Box49 \centerline{General \quad Relativity } H hyphen 1 525 Budapest 1 14 H-1525Budapest114 Hungary Hungary \ceenmtaeilr:lilbnseza{bLat$rm\kaicpuertieod{ak}f-k$subszi pemleraiiol$d: lhb\suzaacbu@ter{mok}i.$k fB−k.i. hSuzabados } h t tp : slash slash w w whptertipod:r/m/kwi pwewrio.d rk mf kkiip.erikodfhkiu.slahshuas/c˜liitbisldze subal bbs/z .. a b slash \centerline{Wigner Research Centre for Physics } Accepted on 24 March 2009 Acceptedon24March2009 Published on 1 9 June 2009 Publishedon19June2009 \centerline{of the Hungarian Academy of Sciences } open parenthesis Revised on 7 December 20(1Re2vicselodsoinng7pDaerceenmtbheers2is012) Abstract Abstract \ceTnhteeprrleisnenet{sPtaTt.uhesOporfe.tsheBnetoqsxutaats4ui9shoy}fpthheenqluoacsail-mloacsaslcmoamssm,aeneenregrygy- mhyompheenntummoamndenatnugmulaarn-dmaonmguelnatrumhyphen momentum constructioncsoninstrguecnteiornasl irnelgaetniveirtayl riselaretivviietwyeisdrpeveireiwodedF.irFsitrscto,mtmheagtehneergaelnideeraasl,idceoanscecpotms m, aandcosntrcaetpetgsiecso,mma and strategies comma \ceanstweerlllains eth{eaHsnwe−ceells1saas5rty2he5tonoeBlcsuestdsoaarcpyoentsosttorlsu1ctot1ac4onnds}tarnuactlyaznedtahneaqlyuzaestihheyqpuhaesni -loloccaallqquuaannttiittiieess ,coamremreacaalrleedr.ecalled period Then comma tThheevna,ritohuesvasprieocuisficspceocnifisctrcuocntsitornusctaionndstahnedirthperiorpperrotpieerstioepse(nbpoathresnutchceessissesbaontdhdsueficccieesnsceisesand deficiencies \ceanretdeirscliunssaeerd{eHdpiuesrcniuogsdaser.dy. .F}inalFlyinaclolmy ,ma .s.osmoemoefothfeth(eacotpueanl apnadrepnottheenstiisala)ctuaalpapnldicaptoiotnesntoifatlhceloqsuinasgip-alorceanlthesis .. applications of the quasi hyphen local concepts and specific constructcioonncsepatrseabnrdiesflpyecmifiecnctoionnsterducpteiorniosdare briefly mentioned . \centerline{email : lbszab $ @ $ rmki . k $ f−k { i . }$ hu } Line 1 hline Line 2 Attribution hyphen Non hyphen Commercial hyphen NoDerivs 3 period 0 Germany from h t tp : slash slash cr e a t i v e c o m mon s period o r g slash l i c e n se s slash by hyphen n c hyphen n d slash 3 to This review is licensed under a Creative Commons License period 0 slash d e slash to the power of p\ecreiondtAetrtlriinbuet{iohn−t Ntopn−: C/om/mwercwialw−.NorDemrivks3i.0G. ekrmfanykhittp.://hcreautivec$omm/ons.org\/tliecexnsteas/sbyc−inict−inldd/3eLice{nsel. }$ bs z \quad a b / } ThisreviewislicensedunderaCreativeCommons .0/de/ \centerline{Accepted on 24 March 2009 } \centerline{Published on 1 9 June 2009 } \centerline{( Revised on 7 December 20 1 2 ) } \centerline{Abstract } \centerline{The present status of the quasi − local mass , energy − momentum and angular − momentum } \centerline{constructions in general relativity is reviewed . First , the general ideas , concepts , and strategies , } \centerline{as well as the necessary tools to construct and analyze the quasi − local quantities , are recalled . } \centerline{Then , the various specific constructions and their properties ( both successes and deficiencies } \centerline{are discussed . \quad Finally , \quad some of the ( actual and potential ) \quad applications of the quasi − local } \centerline{concepts and specific constructions are briefly mentioned . } \[\begin{aligned} \rule{3em}{0.4pt} \\ Attribution − Non − Commercial − NoDerivs 3 . 0 Germany ˆ{ h t tp : / / cr e a t i v e c o m mon s . o r g / l i c e n se s / by − n c −{ n } d / 3 } { This review is licensed under a Creative Commons } License { . 0 / d e / }ˆ{ . }\end{aligned}\] Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access j ournal published by the Max Planck InstitIumteprfoinrtG..rasvlaitsahti.o.nTaelrPmhsy.s.icosf,UAsem M u¨ hlenberg 1 , 1 4476 Potsdam , Germany . ISSN 1 433 - 835 1 . \centerline{Imprint \quad / \quad Terms \quad of Use } ThisrLeivviienwg iRselviiceewnsseidnuRnedlaetrivaitCyreisataivpeeCerormevmieownsedAottpreibnuatciocnes-sNj oonur-nCalopmumbleirschieadl-bNyotDheerMivasx3P.la0nGckermanyLicense: http : /In/stcirtuetaetforiGvraevitcatoiomnmaol Pnhsysi.cs coormgm/alAmi Mc eu-dniesreesiss h/lebnyber-gn1 ccom-mnad1/44376.Po0ts/dadmec/om.ma FGiegrumreasnythpaetriod ISSN 1 433 hyphen 835 1 period \noindent Living Reviews in Relativity is a peer reviewed open access j ournal published by the Max Planck haveTbeheins rperveiveiwouisslylipcuenbsliesdheudndelesrewahCerreeamtivaeyCnootmbmeornespArotdturicbeudtiwonithhoyupthecnonNseonnthoyfpthheenoCrigoimnamlercial hyphen NoDerivs 3 period 0 copyrGigehrtmhaonlydeLrsic.ense : .. http : slash slash crea t i v e c ommo n s period org slash l i c e n se s slash by hyphen n c hyphen nd slash 3 period 0 slash de slash period \noindent Institute for Gravitational Physics , Am M $ \ddot{u} $ hlenberg 1 , 1 4476 Potsdam , Germany . 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Be..caFuisgeuraesLitvhiantg Reviews article can evolve over time , we recommend to cite the article as follows : have been previously published elsewhere may not be reproduced without consent of the original \noindent This review is li censed under a Creative Commons Attribution − Non − Commercial − NoDerivs 3 . 0 copyright holders period Germany License : \quad http : L/a´sz/lo´Bc.rSezaabatdois,v e c ommo n s . org / l i c e n se s / by − n c − nd / 3 . 0 / de / . \quad Figures that Because a Living Reviews article can evolve over time comma we recommend to cite the article as follows : Case 1 L a-acute szl o-acute B period Szabados comma Case 2 quotedblleft Quasi hyphen Local Energy Momentum an(cid:48)(cid:48)d Angular Momentum in General Case 1 \noindent have been previously p“Quubalsiis−heLdocaellEsneewrghyeMreommenatyumnaontdAbnegurlaerpMroomduencteudmiwnGitehnoeruatlRceolantsiveintyt of the original liqnueo−tedlibnleri−ghhtyCpahseen2−chomypmheanC−ashey3phLeinving Rev period Relativity comma 12 comma open parenthesis 2009 closing parenth,esis comma 4 period open square bracket Online Article closing square bracket : cited open square bracket less date greater closing square bracket comma Case 4 http : slash slash www period livingreviews \noindent copyright holders . 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Article Revisions Living Reviews supports two ways of keeping its articles up - to - date : FastA-rtticrlaec.k. Rreevviissiioonns A fast - track revision provides the author with the opportunity to add short \centerline{Article \quad Revisions } Living Reviews supports two wnaoytsicoefskoefecpuinrrgenittsraersteiaclrecshurpeshuyltpsh,entretnodhsyapnhdendedvaetleop:ments , or important publications to Fast hyphen t rack revisitohneAartfiacslteh.yphAenfatsrtac-ktrraecvkisiroenvispiroonviidserseftehreeeadutbhyorthweitrhespthoensoipbpleorstuubnjietcytteodiatdodr .shorItf an article \noindent Living Reviews supports two ways of keeping its articles up − to − date : notices of currenthraesseuanrcdherrgeosnueltas cfaosmtm-atrtarcekndresvaisniodnd,evaesloupmmmeanrtys coofmchmaangoersiwmipllobrteanlitstpeudbhliecraeti.ons to MajotrheuaprdtiaclteepeAriomda.j.orAufpadsattheywphilelnintcrlaucdkerseuvbissitoanntiisalrecfhearenegdesbayntdheadredsiptioonnssibalnedsuisbsjeucbtjeecdtittoor period .. If an article \noindent Fast − t rack revision A fast − track revision provides the author with the opportunity to add short has undergone a fast hfuylplheexntetrrnaaclkrerefevrieseioinngc.omItmisapausbulimshmedarwyiothf cahannegwespuwbilllicbaetiloinstendumhebreerp.eriod For dMetaaijloerdudpodcautmeeAntmataiojonroufpadnataertwicillel i’nscleuvdoelustuibonsta,nptlieaalscehraenfegrestoanthdeahdidsittoiroynsdaoncudmisenstubojfetchteto \hspace∗{\ fill }notices of current research results , trends and developments , or important publications to articlfeu’llseoxntelirnnealverersfeioreneiantg phertitopd I:t is/pu/bwlishwed wwith.a nlewi pvuibnlicgarteionvineumwbser.peroiordg / lrr - 2009 - 4 . 7 DeFcoermdbeetaril2ed01d2oc:umentation of an article quoteRriegchetntsdevevoelulotpiomnecnotms omfathpelefiaeseldreafreeritnocltuhdeedhi.stAoryfedwoscuubmseencttioonfsthaend more \hspace∗{\ fill }the article . \quad A fast − track revision is refereed by the responsible subject editor . \quad If an article thanafirfttiyclneeqwuoreteferrigenhtcessoanrleinaeddveerdsi,omnianto.r.ihmtptpro:vesmlaeshntsslaasnhdwcowrre..ctwionpseroifodthleitvexintgarreevmieadwesatperiod org slash lrr hyphen 2009 hyphen 4 period severa7lDpeocinetmsb,earn2d01th2e:b..i.b.liRogercaepnthydeisveuloppdmateendts. oTfhtehemfiaejlodrachreaningcelsuadreedapsefroiolldowAsf:ew subsections and more \centerline{has undergone a fast − track revision , a summary of changes will be listed here . } P−onenine−ag−colone tthharenefi−ftAynne−worneefeer−enpceersioadreonaed−dewd −comcommammaSin−oroinm−uptro−vebms−ehnet−seactn−dgcioorr−ecrtaio−nsno−f tvheitteaxttiaorenmaaldeeneatgyin N ewton a−i nthe or and in a several points comma and the briebllaitoigvriastpihcaylliys ucoprdraecteteddpNereiwodtoTnihaenmthaejoorrychisanadgedsedar.e as follows : \noindent Major update A major update will include substantial changes and additions and is subject to P-onPe s−ubtwnoineeig-hat−g-acgo−loconloenethreeA-Anenw-opnaerea-gprearpiohdononae-cwa-ncdoimdamteafSo-rotshuebtnot-ualtm-basssubofs-chloese-edcutn-igveirose-rsais-na-dvdietda.t i o n al ene gyin N ewton a-i .. nthe or and in a relativisticPall−y ficovrerzeecrtoe−daNg−ecwotloonneian tAhedoirsycuisssiaodndeodf tpheerimodonotonicity properties of the quasi - local mass expressions near \centerline{full external refereeing . It is published with a new publication number . } P-two sub eigshpta-atiagl-cinolfionnitey.,.mAontievwatpedarbaygrtahpehcoonmapactainbdiliidtyatweiftohrtthheeptoosttalNmewastsonoifacnlolsimedituinsivaedrdseeds i.s added period P-five subPze−rofi-avego-cnoel−oang−ec.o.loAnediscAussnieown opfartahgermapohnootnonthiceitiyncpormoppearttiibeisliotyf tohfethqeuamsionhyotpohneincitloycoafl tmhaesqsueaxspir-eslsoiocnasl mneaasrs \noindent For detailed documentation of an article ’ s evolution , please refer to the history document of the spatial infinity comma motivaetxedprbeyssitohnescaonmdptawtiobi‘listtyanwditahrdth’erepqousitreNmeewnttosniisanadldimedit.is added period P-five sub onPe-a−gfi-cvoelofinvee−a..g−AconloenweparAagdriaspphlayoendthfoerminucloamipsactoirbrielicttyedof,tahnedmtwonoorteofneirceintyceosfatrheeaqdudaesdi h.yphen lo cal mass \noindent article ’ s online version at \quad http : / / w w \quad w . l i vin gre vie w s . org / lrr − 2009 − 4 . expressions anPd t−wofivqeunoitneel−efatgs−tcaonlodnaerd qNueowterriegfhetrernecqeusiroenmtehnetsgeisneardadliezdatpioernioodf the Hawking energy and its monotonicity P-five sub five-a g-colon e .. Apdriospplearyteiedsf,oarmnduloanisthceorHreacwtekdincgoamnmdaGaenrdocthweonreerfgeyreonpceersaatorersaidndleodoppeqruioadntum gravity are \noindent 7 December 2012 : \hfill Recent developments of the field are included . A few subsections and more P-five sub nine-a g-colon e .. New references on the geandedraeldiz.ation of the Hawking energy and its monotonicity properties coPm−mnaiannednionne−tahge−cHoalownkeingAanndewGerreofecrhenecneertgoyaompeordaitfioerds icnonlosotrpuqctuiaonntubymZghraanvigty, arehaving better asymptotic \noindent than fifty new references are added , minor improvements and corrections of the text are made at added period behaviour at spatial infinity , is given . P-nine subPn−inoen-aezge-rcoo−loang−etw..ocAolonne−we refNereewncreefteoreancmesodoinfietdhecoinnisttiarulcbtoiounndbayryZhvaanlugecpormobmleam..ahnadviintsgpboettetnertiaaslycmonpnteoctticion \noindent several points , and the bibliography is updated . The major changes are as follows : behaviour at spatial infinity cwomithmathiesqguivaesni -pleoricoadl Hamiltonian approach are added . P-onPe−suobnzeezreor-oa−ga-gt−wtohrteoectohloen−peoweTrhoef cmoolorne-dee.t.aNileedwdriesfceursesnicoensoofnththeeroinleitoiafltbhoeuanrdeaar2y-vafoluremp(roabsleampaarntdoifttshpeobteonutniadl-connection \centerline{ $ P−one { nine−a g−colon e } three−A n−one e−period one−w−comma S−o { n−u } t−b { s−h e−e with the quasi hyphen lo cal Haarmyiclotonndiiatinonaspp)riosagcihveanreaandddneedwpreerfioerdences are added . c t−g }$ i $ o−r a−n−v $ it a t i o n al ene gyin N ewton $ a−i $ \quad nthe or and in a } P−onezero−ag−ninecolon−e A−Po-onnepeesruiobdz−erno-−a go-ntehtrhereeeto−tehtehrpeoew−erpoefricoodlo−n-weS..−Tiuhbe−maodr−esde−etdac−ileedd−dtiiscou−ssiwonnotfhthe trohle ofetshuegagreeat−2 hsyipnhefnof−orrmthoeperenfepraernecnetchoesnisgas aupaatritonof the bound hyphen o \centerline{relativiasntdicaadlelfiyniticoonrfroercthteedquaNseiw- ltooncaialnquatnhteiotiry ibsy Naedsdteerd, C.he}n , Liu and Sun . Several new ary conditions closing parenthesis is given and new references are addeedsperiod P-one sub zero-a g-nine to the power of colon-erAef-eornenecpeesraiorde-and-odneedt.hree-e three-period-w S-i sub u b-a sub d-s e-d c-e d-t i o-w n t h .. t h .. e su gge t-s i n f o-r \centerline{ $ P−two { eight−a g−colon e }$ \quad A new paragraph on a candidate for the total mass of closed universes is added . } the referPen−ceconoen g .. u ation A more detailed discussion of the properties of the Komar integral is given . one−ag−threecolon−e P−one and a dSefionnieti−onefco−r tthweoqpueraisoidh−ypthen lo cal quarnetitoiRrowg1aoizRedowby2aedsdqnuganttwitoimNesterrescobm−muasCecheionncso.mmOanLeiiu anodnTSuonl mpeariqoudo.t.erSiegvhetr−alsnew one−ag−fourcolon−e i−threeoi−n−s \hspace∗{\ fill } $ P−five { zero−a g−colon e }$ \quad A discussion of the monotonicity properties of the quasi − local mass expressions near references are added perieondergy expression , the other is on the gravitational energy in the post - Newtonian limit of GR . P-one sPub−oonnee-a g-three to the powSeervoefraclolnoenw-er.e.feArenmcoerseodnetgaeiolemdedtriisccuinsseiqounaoliftitehsefoprrobplaecrktiehseloefstahreeKadomdeadr.integral is given period one−ag−sixcolon−e \centerline{spatial infinity , motivated by the compatibility with the post Newtonian limit is added . } P-one sPub−oonne-ea g-four to the power oAf cnoelwonp-ear.a.gSraopnhe-oenc(-tawnodptehreiorde-fterseunbcei-ttohr)eeaoreic-enn-ts rreefoormru.l.atgiaonizaenddbyitasdpdronogftowfo m .. res b-u sec ions period .. Onei .. one−ag−eightcolon−e onT ol ma quoteright-sThorne ’ s hoop conjecture for spherically symmetric configurations is given . \hspace∗{\ fill } $ P−five { one−a g−colon e }$ \quad A new paragraph on the incompatibility of the monotonicity of the quasi − lo cal mass P−one A−oneenpeerrgioyde−xpnre−ssitohnreceofomumr−a tehtheroeteh−erpiesroiondt−hewgSra−viotationbalae−nesrgeyci−n nthee−ptost hyphenntNrewtoonpiaynbloimuitdoof G−Rf rpuenricood−la−l p s−e d b odi sisa dded . two−ag−threecolon−e n−u i−won−e P-one sub one-a g-six to the power of colon-e .. Several new references on geometric inequalities for black heles are added period \centerline{expressions and two ‘ standard ’ requirements is added . } P-one sub one-a g-eight to the power of colon-e .. A new paragraph on open parenthesis and the reference to closing parenthesis a recent reformulation and it s proof of \centerline{ $ P−five { five−a g−colon e }$ \quad A displayed formula is corrected , and two references are added . } Thorne quoteright s hoop conjecture for spherically symmetric configurations is given period P-one sub two-a g-three to the power of colon-e A-one period-n-three four-e three-period-w S-o sub n-u b a-s e c-n e-t sub i-w o n-e n t r .. o p y bo u d o-f runc \hspace∗{\ fill } $ P−five { nine−a g−colon e }$ \quad New references on the generalization of the Hawking energy and its monotonicity o-l a-l p s-e d .. b odi sisa dded period \hspace∗{\ fill }properties , and on the Hawking and Geroch energy operators in loop quantum gravity are \centerline{added . } \hspace∗{\ fill } $ P−nine { nine−a g−colon e }$ \quad A new reference to a modified construction by Zhang , \quad having better asymptotic \centerline{behaviour at spatial infinity , is given . } \hspace∗{\ fill } $ P−one { zero−a g−two ˆ{ colon−e }}$ \quad New references on the initial boundary value problem and its potential connection \centerline{with the quasi − lo cal Hamiltonian approach are added . } \hspace∗{\ fill } $ P−one { zero−a g−three ˆ{ colon−e }}$ \quad The more detailed discussion of the role of the area 2 − form ( as a part of the bound − \centerline{ary conditions ) is given and new references are added . } \centerline{ $ P−one { zero−a g−nine ˆ{ colon−e }} A−one period−n−one three−e three−period−w S−i { u } b−a { d−s e−d c−e d−t }$ i $ o−w $ n t h \quad t h \quad e su gge $ t−s $ i n f $ o−r $ the referencec o n g \quad u ation } \hspace∗{\ fill }and a definition for the quasi − lo cal $\left . quantiti\begin{array}{c} o \\ es \end{array}by\right.$ Nester , Chen , Liu and Sun . \quad Several new \centerline{references are added . } \centerline{ $ P−one { one−a g−three ˆ{ colon−e }}$ \quad A more detailed discussion of the properties of the Komar integral is given . } \centerline{ $ P−one { one−a g−four ˆ{ colon−e }}$ \quad S $ one−e c−two period−t { i−three o i−n−s }$ r e o r \quad ga ized by add ng two m \quad res $ b−u $ sec ions . \quad Onei \quad onT ol ma $ quoteright−s $ } \hspace∗{\ fill }energy expression , the other is on the gravitational energy in the post − Newtonian limit of GR . \centerline{ $ P−one { one−a g−six ˆ{ colon−e }}$ \quad Several new references on geometric inequalities for black heles are added . } \hspace∗{\ fill } $ P−one { one−a g−eight ˆ{ colon−e }}$ \quad A new paragraph on ( and the reference to ) a recent reformulation and it s proof of \centerline{Thorne ’ s hoop conjecture for spherically symmetric configurations is given . } \centerline{ $ P−one { two−a g−three ˆ{ colon−e }} A−one period−n−three four−e three−period−w S−o { n−u }$ b $ a−s $ e $ c−n e−t { i−w o n−e }$ n t r \quad o p y bo u d $ o−f $ runc $ o−l a−l $ p $ s−e $ d \quad b odi sisa dded . } Contents 1 I n t r o d u c t i o n 9 2 ECnoneterntgs y - M o m e n t u m a n d A n g u l a r M o m e n t u m o f M a t t e r F i e l d s 1 1 \noindent Contents 21. 1.... IEn tnr oe rdg...y. -u cmt ioo nm....e9n t u m a n d a n g u l a r - m o m e n t u m d e n s i t y o f m a t t e r 2 .fi...eEl d...s. n e. r g......y hy.phe.n M. .....o m. ..... e ..1..1n2t.u1.....1m aTnh..e.. ds...y. mAm.... ne..t..rgiucl aern...e. rMg....yo-..m.. mo...m. e net .... u m .... o f .... M .... a t t e r .... F \noindent 1 \hfill I n t r o d \hfill u c t i o n \hfill 9 niteulmd s ..t..e1n1s o r . . . . . . . . . . . . . . . . . . . 1 1 2 . 1 . 2 T h e c a n o n i2c paelriNod 1o..eEt h.. neerr g yc uhyrprheen ..nmto ... m.e n.t u.m ...a n... d. .. a. n g. .. .u l a. r h.yph.en ... m.o ...m e.n t.u m. .. .d e .n s i t y .. o f m .. a t t e r .. fi e l d s .. \noindent 2 \hfill E \hfill n e r g \hfill y − M \hfill o m \hfill e \hfill n t u \hfill m a n \hfill d \hfill A \hfill n \hfill g u l a r \hfill M \hfill o \hfill m \hfill e n t \hfill u m \hfill o f \hfill M \hfill a t t e r \hfill F i e l d s \hfill 1 1 . pe.riod... p.erio1d1..2p.er2iodQ..upeariso−di..lpoecriaodl .. epenrieodr g.. peyri-omd .. poermiod .e. npetriuod .m. 1 1a n d a n g u l a r m o m e n t u 2mperioodf 1 ptehrieod 1m..aTt theer.. sfiyeml dms.. e. t r.i c... e.n e1r3g2...y2h.y1phenTm h..eo md..eefin tnui mt i.o. nt e nosf o rq.u. paesriio-dl.. period .. period .. period .. period .. 2 . 1 \quad E \quad n e r g y − \quad m o \quad m e n t u m \quad a n \quad d \quad a n g \quad u l a r − \quad m o \quad m e n t u m \quad d e n s i t y \quad o f m \quad a t t e r \quad opceariold .q. pueariodn..tpietriiode s.. pe.riod. .. p.erio.d ... per.iod... p.erio.d ...peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p1e3rio2d..2. p.e2riod .. 1 1 $ fi $ e l d s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 H a2mperioidl1t openrioida2n.. Ti n..threoc adnu..cotni oi nc aolfNt h.. oee t hq .u. e ra.s. ic-ulrorcea..l n tq.. pueraiondt..i pteireiosd ...peri.od ... pe.riod. .. period .. period .. period .. period .. 2 . 1 . 1 \quad T h e \quad s y m m \quad e t r i c \quad e n e r g \quad y − m \quad o m \quad e n t u m \quad t e n s o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 . pe.riod... p.erio.d ...per.iod1..5pe2ri.o2d...3periPodr.o. ppeerirodt i..epserioodf.. pterhioed ..qpueraiosdi..-ploercioadl.. pqeruioad ..npetriiotdi.e. pserio.d ... per.iod .. period .. period .. period .. 1 1 2 . 1 . 2 \quad T \quad h e c a n \quad o n i c a l N \quad o e t h \quad e r \quad c u r r e \quad n t \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 . .2 p.erio.d 2... Q.u a.s h.yphe.n i.l o c. a l... e.n e.r g... y.hyp.hen.m ..1o6m2 ...2e.n4t uG..lmo.b. aa nl de..na ne..rgguyl-amr mo.. omme.. e n t u .. m .. o f .. t h e .. m a t t e r 2 . 2 \quad Q u a s $ − { i }$ l o c a l \quad e n e r g \quad y − m \quad o m \quad e n t u \quad m \quad a n d \quad a n \quad g u l a r m \quad o m \quad e n t u \quad m \quad o f \quad t h e \quad m a t t e r \quad n.. fit ae l dasn..dperaiodn .g. peuriold .a. premriodo.. pemrieodn.t.a1 3. . . . . . . . . . . . . . . . 1 7 $ fi $ e l d s \quad . \quad . \quad . \quad . \quad 1 3 2 . 2 . 5 Q u a2spie-rlioodc2apleriorda1 ..dTi.a. thiev..ed emfio..dneist i oan ..nodf ..aq ucalsaishyspihcenallo cvaelr..sqi ouna .. no ft i ttiheesh.. peoriloodg..rpaerpiod .h. yperiofdo.r. period .. period .. period 2 . 2 . 1 \quad T \quad h e \quad d e $ fi $ \quad n i t i o n \quad o f \quad q u a s i − l o c a l \quad q u a \quad n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 3 .. permiod .a. tpetreiord ..fipeerilod d.. speri.od ... pe.riod... p.erio.d ...per.iod... pe.riod. .. .peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.eriod .. period .. 1 3 2 . 2 . 2 \quad H \quad a m \quad i l t o n i a n \quad i n t r o \quad d u c t i o n o f t h \quad e \quad q u \quad a s i − l o c a l \quad q \quad u a n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5 . .2 p.erio.d 2.peri.od 2... H. ...a m... i.l t o1n7i a n .. i n t r o .. d u c t i o n o f t h .. e .. q u .. a s i hyphen l o c a l .. q .. u a n t i t i e s .. period .. period .. period 2 . 2 . 3 \quad P r o p e r t i e s \quad o f \quad t h e \quad q u a s i − lo c a l \quad q u a \quad n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6 3..Openriotdh.. peerEiodn..eperriogdy..-peMrioodm.. peerinodt.u. pmerioadn.. pderAiodn..gpeuriolda..rpMeriodo.m. 1e5n t u m o f G r a v i t a t i 2 . 2 . 4 \quad G l o b a l \quad e n \quad e r g y − m \quad o \quad m e n \quad t a \quad a n d \quad a n g \quad u l \quad a r m \quad o \quad me n t a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7 n 2 period 2 period 3 .. Pgr o p e r t i e s .. o f .. t hSey..sqtueams i hyphen lo c a l .. q u a ..sn t i t i e s .. period .. per1io9d .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6 − 3 . 1 O n t h e g\rcaevnitteraltinioen{2a l. 2e n.e5r g\qyu-amd Qo\mquade nuta us im− la no dc aa nl g\quuadl arr ma \qouadmde nita ut mi vdee \qnusadi tmy :o dTe hs e\qudadi a \quad n d \quad a \quad c l a s s i c a l \quad v e r s i o n \quad o f \quad t h e h \quad o l o g r a p \quad h y \quad f o r } 2 period 2 period 4 .. G l o b a l .. e n .. e r g y hyphen m .. o .. m e n .. t a .. a n d .. a n g .. u l .. a r m .. o .. me n t a .. period .. period .. peffiriod .. period .. pecriuodl t..ipeersiod... p.eriod. .. .peri.od ... per.iod... p.eriod. .. .peri.od ... per.iod... p.eriod. .. .peri.od ... per.iod... p.eriod. .. .1 7 . . . \hspace∗{\ fill }m \quad a t t e r \quad $ fi $ e l \quad d s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7 . .2 p.erio.d 2.peri.od 5. .. Q. ...u a.s i h.yph.en l.o c.a l... r1a9..3d.i1a.t1i v eT.. mhoed erso..oat.. nodf ..ta ..hcel a sdsiiffic a lc..uvl teires i o n .. o f .. t h e h .. o l o g r a p .. h s :y ..Gf orra v i t a t i o n a l e n e r g y i n N e w t o n ’ s t h e o r y . . . . 1 9 \noindent 3 \hfill O n \hfill t h \hfill e \hfill E n \hfill e r \hfill g y − \hfill M o m \hfill e n t u m \hfill a n \hfill d \hfill A n g \hfill u l a r \hfill M \hfill o m \hfill e n t u \hfill m \hfill o f \hfill G r \hfill a v i t \hfill a t i n \hfill g \hfill S y s t e m \hfill s \hfill 19 3 . 1 . 2 T h em .r. oaot ttoe fr ..tfi ehl .e. dds i.. pffieriocdu.l.tpiereiosd:.. pGerrioadv..iptearitodi o..nperiaodl e..npeerriogdy..-pmeriodo m.. peenriotdu.m. periiond ..Eperiiond s..tpeeriiond’.s. period .. period .. period ..tphereioodr..yper.iod... p.erio.d ...per.iod... p.eriod. ...peri.od ... pe.riod. .. .peri.od ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period .. \centerline{3 . 1 \quad O \quad n \quad t h \quad e \quad g r a v i t a t io n a l \quad e n e r g y − m \quad o m \quad e n t \quad u \quad m \quad a n d \quad a n g \quad u l a r m \quad o \quad m e n t \quad u m \quad d e \quad n s i t y : \quad T \quad h e \quad d . pe.riod... p.erio.d ...per.iod... pe.riod. .. 21073 . 1 . 3 P s e u d o t e n s o r s . . . . . . . . . . . . . $\left . i\begin{aligned} & − \\ . .3 ..... O. n ..... t.h ..... e..... E. n..... e. r ..... g.y h.yphe.n ..... M. o m. ..... e n. t u.m ..... a2n1.3.... d1....4. ASntgr..a..tuelgaire.s... Mt o.... oam .... e n t u .... m .... o f .... G r .... a & ffi \end{aligned}\right.$ } vvoiitd.p...saet iund...o. tge..n..sSoyrsstIe:mB ....asc....k19g r o u n d m e t r i c s / c o n n e c t i o n s . . . . 22 33. 1pe.ri5od 1S..t Or a..tne.g. itehs.. et..og raavvoiitda t iposneauld.. oetneenrsgoyrhsypIhIen: mT.. ohme.. etnett.r. au ..dmf.o. ar mn d ..aal insg ..mu.l a r m .. o .. m e n t .. u m .. d e .. n s c u l t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 9 . i t.y :... T... h. e ... d C. ase. 1 h.yph2e2n3C.as1e.26ffi S t r a t e g i e s t o a v o i d p s e u d o t e n s o r s I I I : H i g h 3 . 1 . 1 \quad T \quad h e \quad r o o t \quad o f \quad t \quad h e \quad d i $ ffi $ \quad c u l t i e s : \quad G r a v i t a t i o n a l \quad e \quad n e r g y i n \quad N \quad e w t o \quad n ’ s t \quad h e o r y \quad . \quad . \quad . \quad . \quad 1 9 e r cduelrtiiveast..ipveeriocd ..uprerrioedn..tpserio.d ... per.iod... p.erio.d ..2p3eriod .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 3 . 2 O n t h e g.l. opebriaodl ..epneriode.r.gpyer-iodm.. poeriomde..npteruiodm.. peariond ..dperaiodn g..upelraiord ..mpeorimod ..eperniotdu.. pemriodo..f pegriorda..vpietraiodt i..npgeriosdy..spteeriodm..sp:eriod .. period .. \centerline{3 . 1 . 2 \quad T \quad h e \quad r o o t o f \quad t \quad h e \quad d i \quad $ ffi $ c u l t i e s : \quad G r a v i t a t i o n \quad a l e n e r g y − m \quad o m e n t u m \quad i n \quad E \quad i n s t e i n ’ s } peTriodh..epersioudc..cpeersiosde.s. pe.riod... p.erio.d ...per.iod... pe.riod. .. .peri.od ... pe.riod... p.erio.d ...1 9. . . . . . . . . .3 p.erio.d 1 p.erio.d 1... T. .. h. e ... r o.o t... o.f ...t ..2h4e3... d2i.ffi1 .. Sc upl tai et si a: .l.iGnr afivniittayt:i o nEanle..reg..yn-e rmg yoimn .. N .. e w t o .. n quoteright s t .. h e t h e o r y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 20 e o rnyt.u. permiod ... pe.riod... p.erio.d ...per.iod ... 1.9 . . . . . . . . . . . . . 24 3 . 2 . 2 S p a t i 3 . 1 . 3 \quad P s e \quad u d o t e n s o r s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 2 1 a l i n3fiperiondi1tpyer:iodA2 .. Tn g..uhlea..rr omo t oofm.. t ..ehne ..tdui ..mffi c.u l t.i e.s : ... G.r a v. i t.a t i.o n... a.l e.n e r. g y.hyp.hen.m .. o m e n t u m .. i n .. E .. i n s t e 3 . 1 . 4 \quad S t r a t e gi e s \quad t o \quad a v o i d p s e \quad u d o t e n s o r s I : B \quad a c \quad k g r o \quad u n \quad d \quad m \quad e t r i c s / c o n n \quad e c t i o n \quad s \quad . \quad . \quad . \quad . \quad 22 . i n.quo.teri.ght.s . . 26 3t. h2 e. 3o r yN.. puerlioldi .n. fiperiondi.t. ype:riEod ..npeerriogdy..- mperioodm.. peeriondt..upermiod ... pe.riod... p.erio.d ...per.iod... pe.riod. .. .peri.od .. period .. period .. period .. period 3 . 1 . 5 \quad S t r a t e g i e s \quad t o \quad a v o i d \quad p s e u d o t e n s o r s I I : \quad T \quad h e \quad t e t r a \quad d f o r m \quad a l i s \quad m . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 22 . .. .perio.d ... per.iod... p.erio.d ...peri.od ... pe.riod27..3p.er2io.d4.. pNeriodu..l lpeirniofid .. npeirtioyd:.. pAerinodg.. puerliaodr.. pmeroiomd .. peeriondt.. period .. period .. period .. period .. 3 . 1 . 6 \quad S t r a t e g i e s \quad t o \quad a v o i d \quad p s e \quad u d o t e n s o r s I I I : \quad H i g h e r \quad d e r i v a t i v e c \quad u r r e n t s \quad . \quad . \quad . \quad . \quad . \quad . \quad 23 upermiod ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio.d ...20. . . . . . . . . . 29 3 . 3 T h e n e c e s s i3tpyeriodo f1 peqriuoda3s i..-Pl oscea..lui tdyoftoern soobrsseprervioadb..l epesriiond ..gpeerniod .e.rpaerliorde.l.apteriiovdi.t. yperi.od ... pe.riod... p.erio.d .. period .. period .. period .. period \centerline{3 . 2 \quad O \quad n \quad t h e \quad g l o b a l \quad e n \quad e r g y − \quad m o \quad m e n t u \quad m \quad a n \quad d \quad a n g u l a r \quad m o m \quad e \quad n t u \quad m \quad o f \quad g r a v i t a t i n g \quad s y s t e \quad m s : } . .. .peri.od .3. 1period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. pe3ri.od3 ...1periNod ..opernioldo..cpaerlioidt y.. peorifod .t.hpeerigodr a..v2i1t a t i o n a l e n e r g y - m o m e n t u m a n d a n T \quad h e \quad s u c c e s s e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 24 g u l3apreriomdo1 pemrioedn4t..uSmt r a3t1e3g.i3e.s2.. t oD.. aovmo i dapi nsse .. fuodro tqeun s oarssi -Il:oBc..aalc ..qkugarnot.i.tuine.s. d ... m... e.t r i c s slash c o n n .. e c t i o n .. s .. 3 . 2 . 1 \quad S p \quad a t i a l i n \quad $ fi $ n i t y : \quad E n e r g y − \quad m o m \quad e \quad n t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 24 . pe.riod... p.erio.d ...per.iod... pe.riod. .. 2.2 . . . . . . . . . 32 3 . 2 . 2 \quad S p a t i a l i n $ fi $ \quad n i t y : \quad A \quad n g u l a r \quad m \quad o m \quad e n \quad t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 26 33. 3pe.ri3od 1Spterrioadt5e.g. iSetsr attoecgoi e sn..sttor.u. catv oqi du.a. spis-eluod octaeln sqour s aI In:t..i Tt ie..sh e... t.e t r. a ... d f. o r.m ... a l i s .. m period .. period .. period .. . pe.riod... p.erio.d ...per.iod... pe.riod. .. p32eriod .. period .. period .. 22 3 . 2 . 3 \quad N \quad u l l i n $ fi $ \quad n i t y : E \quad n e r g y − m \quad o m \quad e n t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 27 4 T3 poeroiolds1 pteroiodC6 .o. Snt rsattreugci et sa.. tno d.. aAv oni da..lpyszee.. uQduo taesnis-oLr soIcIaI :l..QH iug hae rn..tdi te ri ei vsa t35i v e c .. u r r e n t s .. period .. period .. 3 . 2 . 4 \quad N \quad u l l i n $ fi $ \quad n i t y : \quad A n g \quad u l a r \quad m o m \quad e \quad n t \quad u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 29 pe4ri.od1 .. Tperiohde.. pgereioodm.. peeriotdr.y. 23o f s p a c e li k e t w o - s u r f a c e s . . . . . . . . . . . 3 . 3 \quad T \quad h e \quad n e c e s s i t y \quad o f \quad q u a s i − l o c a l i t y f o r \quad o b s e r v a b l e s i n \quad g e n \quad e r a l r e l a t i v i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 3 1 . .3 p.erio.d 2... O... n. .. t. h e... g. l o.b a.l .. .e n... e3r5g4y.h1yp.h1en ..Tm oh..emLe notrue .n. tmz.i. aann .. vde..catnogrubl aur n.. mdolme.. e .. n t u .. m .. o f .. g r a v i t a t i . n g. .. .s y s. t e... m. s :. . . . . . . . . . . . . . . . . . . . 35 4 . 1 . 2 C o n n e 3 . 3 . 1 \quad N \quad o \quad n l o c a l i t y \quad o f \quad t h e g r a v i t a t i o \quad n a l e \quad n e r g y − \quad m o \quad m e n t u m \quad a n d \quad a n g u l a r \quad m o \quad m e n t u m \quad 3 1 c t i oT ..nhse ... s u.c c.e s.s e .s .. .peri.od ... pe.riod... p.erio.d ... per.iod... p.erio.d ...per.iod ... pe.riod. .. .peri.od ... pe.riod... p.erio.d .. period .. period .. period .. period 3 . 3 . 2 \quad D \quad o m \quad a i n s \quad f o r \quad q u \quad a s i − l o c a l \quad q u a n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 32 . .. .peri.od ... pe.riod... p.erio3d5.4. p.e1ri.od3.. pEermiod .b. epedridodi n.. gpesrioda.n. dpecrioodn.v. epexriiodt y.. pecrioodn.. dpiertioido.n. pserio.d ... per.iod... period .. period .. period .. period .. . pe.riod... p.erio.d ...per.iod... pe.riod. .. p.erio.d ... per.iod... p.erio.d ..3p6eriod .. 24 \hspace∗{\ fill }3 . 3 . 3 \quad S t r a t e g i e s \quad t o c o \quad n s t r u c t \quad q u a s i − l o \quad c a l \quad q u \quad a n t i t ie s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 32 43. p1e.ri4od 2Tperihode1s p..iSnpo..r ba tuinadllien ...fi n.i t .y : ... E.n e.r g.y hy.phe.n ...m o. m... e... n.t u... m. .. .peri.od ... pe.riod... period .. period .. period .. period .. . pe.riod... p.erio.d ...per.iod... pe.riod. .. p.erio3d7 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 24 \noindent 4 \hfill T \hfill o o l s \hfill t o \hfill C o n \hfill s t r u c t \hfill a \hfill n d \hfill A n \hfill a l y z e \hfill Q u \hfill a s i − L o c a l \hfill Q \hfill u a \hfill n t i t i e s \hfill 35 43. 1pe.ri5od 2Cperiuodr 2v.a. tSupraet i ai dl iennfit.i.tniietsy : ... A... n.g u.l a.r ...m ... o m. ...e n... t.u ...m ... per.iod... p.erio.d ...peri.od .. period .. period .. period .. period . .. .perio.d ... per.iod... p.eriod. .. .perio.d ... per.iod3.8. p4e.ri1od. .6. peTriohde.. peGriodH..Pperiofdo.r. pemrioadl..isperimod ...per.iod... pe.riod. .. p.eriod .. period .. 26 4 . 1 \quad T \quad h e \quad g e o m \quad e t r y \quad o f \quad s p a c e li k e t w \quad o − s \quad u r f a c e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35 . .3 p.erio.d 2.peri.od 3. .. N. ...u l.l i n.fi ... n i.t y.: E... n.e r.g y.hyp.hen.m ... o m. ...e n.t u... m3.8.4pe.r1iod. 7.. peIrriorde..dpueriodc .. period .. period .. period .. period .. 4 . 1 . 1 \quad T \quad h e \quad L o r e n t z i a n \quad v e c t o r b \quad u n \quad d l e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35 i bpelreiod .p. aperritods .. opefriodt h.. eperidode.r. ipveriaod .t. ipvereiod .o. ppeeriroda .t.operrsiod... pe.riod. .. p.erio.d ...per.iod... p.eriod. .. .peri.od ... per.iod .. period .. period .. period .. 27 4 . 1 . 2 \quad C o n n e c t i o \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 35 . .3 p.erio.d 23p9eriod 4 .. N .. u l l i n fi .. n i t y : .. A n g .. u l a r .. m o m .. e .. n t .. u .. m .. period .. period .. period .. period .. period .. period .. period .. 4 . 1 . 3 \quad E m \quad b e d d i n g s \quad a n d c o n v e x i t y \quad c o n \quad di t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 36 pe4r.i1o.d8 ..SperOio(d1,..1)p−ercioodn.. penrieodc t..ipoerniodo.. penrieod- f..oprermiod .v. eperrsiouds.. pearniod .h. poelrioodn.o. pemrioidc.i. tpyeriod. .. p.erio.d ... per.iod .. period .. period .. 29 . .3 p.erio.d 3... T... h. e ... n e39c4e.s2s i tSyt..aonfd..aqru adssiihtyphuenaltoicoanlsi t yt of o re..voabl us e ravtaeb lte shi ne.. gqeuna.s. ie−rla l r e l a t i v i t y .. period .. period .. \hspace∗{\ fill }4 . 1 . 4 \quad T \quad h e s p i n o r b u n d l e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 37 opcearilod .q. puerioadn.. pteriiotdi.e. speri.od ... pe.riod... p.erio.d ...per.iod... 3.1 . . . . 39 4 . 2 . 1 R o u n d s p h e r e s 3 p.erio.d 3.peri.od 1. .. N. ...o ...n l.o c a. l i.t y ... o.f .. t. h e.g r.a v.i t a. t i.o ...n a.l e ... n e. r g.y h.yph.en ... m.o .. .m e.n t.u m .. a n d .. a n g u l a r .. m o .. m e 4 . 1 . 5 \quad C \quad u r v a t u r e \quad i d e n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 38 . n t. u m. ..43014 . 2 . 2 S m a l l s u r f a c e s . . . . . . . . . . . . . . . . . . . 4 . 1 . 6 \quad T h e \quad G \quad H P \quad f o r \quad m a l is \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 38 . .3 p.erio.d 3.peri.od 2. .. D. ...o m... a.i n.s ...f o.r .. .q u... a4s 1i h4yp.h2en. 3l o cLaalr..gqeu asnpt i thieerse..spnerioeda..rpserpioadt..i paelriiod .. period .. period .. period .. period 4 . 1 . 7 \quad I r r e d u \quad c i b l e \quad p a r t s \quad o f \quad t h e \quad d e r i v a \quad t i v e \quad o p e r a t o r s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39 n.fi. penrioidt.y. pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio.d ... per.iod... p.erio.d ...peri.od ... pe.riod. .. p.erio.d ..44period .. period .. period .. period .. period .. 32 3 period 3 period 3 .. S t r a t e g i e s .. t o c o .. n s t r u c t .. q u a s i hyphen l o .. c a l .. q u .. a n t i t ie s .. period .. period .. period .. period .. period .. $ 4 . 1 . 8 S O ( 1 , 1 ) − $ c o n \quad n e c t i o n \quad o \quad n e − f o r m \quad v e r s u s \quad a n \quad h o l o n o \quad m i c i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 32 4 . 2 \quad S t a n d a r \quad d s i t \quad u a t i o n s \quad t o \quad e v a l u \quad a t e \quad t \quad h e \quad q u a s i 4 .... T .... o o l s .... t o .... C o n .... s t r u c t .... a .... n d .... A n .... a l y z e .... Q u .... a s i hyphen L o c a l .... Q .... u a .... n t i t i e s .... 35 $−{ l }$ o c a l \quad q u \quad a n \quad t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 39 4 period 1 .. T .. h e .. g e o m .. e t r y .. o f .. s p a c e li k e t w .. o hyphen s .. u r f a c e s period .. period .. period .. period .. period .. period .. period 4 . 2 . 1 \quad R \quad o u n \quad d s p \quad h e r e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 40 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 4 . 2 . 2 \quad S m \quad a l l s \quad u r f a c e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 4 1 period .. 35 4 . 2 . 3 \quad L a r g e \quad s p \quad h e r e s n \quad e a r s p a t i a l i n $ fi $ \quad n i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 44 4 period 1 period 1 .. T .. h e .. L o r e n t z i a n .. v e c t o r b .. u n .. d l e .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 35 4 period 1 period 2 .. C o n n e c t i o .. n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 35 4 period 1 period 3 .. E m .. b e d d i n g s .. a n d c o n v e x i t y .. c o n .. di t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 36 4 period 1 period 4 .. T .. h e s p i n o r b u n d l e .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 37 4 period 1 period 5 .. C .. u r v a t u r e .. i d e n t i t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 38 4 period 1 period 6 .. T h e .. G .. H P .. f o r .. m a l is .. m .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 38 4 period 1 period 7 .. I r r e d u .. c i b l e .. p a r t s .. o f .. t h e .. d e r i v a .. t i v e .. o p e r a t o r s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 39 4 period 1 period 8 S O open parenthesis 1 comma 1 closing parenthesis hyphen c o n .. n e c t i o n .. o .. n e hyphen f o r m .. v e r s u s .. a n .. h o l o n o .. m i c i t y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 39 4 period 2 .. S t a n d a r .. d s i t .. u a t i o n s .. t o .. e v a l u .. a t e .. t .. h e .. q u a s i hyphen l o c a l .. q u .. a n .. t i t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 39 4 period 2 period 1 .. R .. o u n .. d s p .. h e r e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 40 4 period 2 period 2 .. S m .. a l l s .. u r f a c e s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 4 1 4 period 2 period 3 .. L a r g e .. s p .. h e r e s n .. e a r s p a t i a l i n fi .. n i t y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 44 4 . 2 . 4 L a r g e s p h e r e s n e a r n u l l i n fi n i t y . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 . 2 . 5 O t h e r s p e c i a l s i t u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 . 3 O n l i s t s o f c r i t e r i a o f r e a s o n a b l e n e s s o f t h e q u a s i - l o c a l q u a n t i t i e s . . . . . . . . . . 48 4 . 3 . 1 G e n e r a l e x p e c t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 . 3 . 2 P r a g m a t i c c r i t e r i a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 . 3 . 3 I n c o m p a t i b i l i t y o f c e r t a i n ‘ n a t u r a l ’ e x p e c t a t i o n s . . . . . . . . . . . . . . . 5 1 5 T he B a r t n i k M a s s a n d i t s M o d i fi c a t i o n s 53 5. 1 T he B artnik m ass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535. 1. 1 Them ain idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535. 1. 2 Th e m a i n p r o p e r t i e s o f m B ( Σ) . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 . 1 . 3 T h e c o m p u t a b i l i t y o f t h e B a r t n i k m a s s . . . . . . . . . . . . . . . . . . . . 55 5 . 2 B r a y ’ s m o d i fi c a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 T h e H a w k i n g E n e r g y a n d i t s M o d i fi c a t i o n s 57 6 . 1 T h e H a w k i n g e n e r g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6 . 1 . 1 T h e d e fi n i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6 . 1 . 2 H a w kin g e n e r g y f o r s p h e r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6 . 1 . 3 P o s i t i v i t y a n d m o n o t o n i c i t y p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . 58 6 . 1 . 4 T w o g e n e r a l i z a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6 . 2 T h e G e r o c h e n e r g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6 . 2 . 1 T h e d e fi n i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6 . 2 . 2 M o n o t o n i c i t y p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 . 3 T h e H a y w a r d e n e r g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 7 P e n r o s e ’ s Q u a s i - L o c a l E n e r g y −M o m e n t u m a n d A n g u l a r M o m e n t u m 62 7 . 1 M o t i v a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7 . 1 . 1 H o w d o t h e t w is t o r s e m e r g e ? . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7 . 1 . 2 T w i s t o r s p a c e a n d t h e k i n e m a t i c a l t w i s t o r . . . . . . . . . . . . . . . . . . . 63 7 . 2 T h e o r i g i n a l c o n s t r u c t i o n f o r c u r v e d s p a c e t i m e s . . . . . . . . . . . . . . . . . . . . 64 7 . 2 . 1 T w o - s u r f a c e t w i s t o r s a n d t h e k i n e m a t i c a l t w i s t o r . . . . . . . . . . . . . . . 64 7 . 2 . 2 T h e H a m i l t o n i a n i n t e r p r e t a t i o n o f t h e k i n e m a t i c a l t w i s t o r . . . . . . . . . . 65 7 . 2 . 3 T h e H e r m i t i a n s c a l a r p r o d u c t a n d t h e i n fi n i t y t w i s t o r . . . . . . . . . . . . 66 7 . 2 . 4 T h e v a r i o u s l i m i t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7 . 2 . 5 T h e q u a s i - lo c a l m a s s o f s p e c i fi c t w o - s u r f a c e s . . . . . . . . . . . . . . . . . . 68 7 . 2 . 6 S m a l l s u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7 . 3 T h e m o d i fi e d c o n s t r u c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7 . 3 . 1 T h e ‘ i m p r o v e d ’ c o n s t r u c t i o n w i t h t h e d e t e r m i n a n t . . . . . . . . . . . . . . 70 7 . 3 . 2 M o d i fi c a t i o n t h r o u g h T o d ’ s e x p r e s s i o n . . . . . . . . . . . . . . . . . . . . . 7 1 7 . 3 . 3 M a s o n ’ s s u g g e s t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 8 A p p r o a c h e s B a s e d o n t h e N e s t e r – W i t t e n 2 - F o r m 72 8 . 1 T h e L u d v i g s e n – V i c k e r s c o n s t r u c t io n . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8 . 1 . 1 T h e d e fi n i t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8 . 1 . 2 R e m a r k s o n t h e v a l i d i t y o f t h e c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . 74 8 . 1 . 3 M o n o t o n i c i t y , m a s s - p o s i t i v i t y a n d t h e v a r i o u s l i m i t s . . . . . . . . . . . . . 74 8 . 2 T h e D o u g a n – M a s o n c o n s t r u c t io n s . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8 . 2 . 1 H o l o m o r p h i c / a n t i h o l o m o r p h i c s p i n o r fi e l d s . . . . . . . . . . . . . . . . . . 75 8 . 2 . 2 T h e g e n e r i c i t y o f t h e g e n e r i c t w o - s u r f a c e s . . . . . . . . . . . . . . . . . . . 76 8 . 2 . 3 P o s i t i v i t y p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8 . 2 . 4 T h e v a r i o u s l i m i t s . . . . . . . . . .4 p.erio.d 2.peri.od 4. .. .L ...a r... g.e s.p h... e.r e.s ...n e.a r.n ...u l.l i n.fi n. i t.y ...per7io8d .. period .. period .. period .. period .. period .. period .. period .. \hspace∗{\ fill }4 . 2 . 4 \quad L \quad a r \quad g e s p h \quad e r e s \quad n e a r n \quad u l l i n $ fi $ n i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 45 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 45 4 period 2 period 5 .. O t h e r s p .. e c i a l s i t u a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 4 . 2 . 5 \quad O t h e r s p \quad e c i a l s i t u a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 47 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 4 . 3 \quad O n \quad l i s t s \quad o f \quad c r i t e r i a \quad o f r e a s o \quad n a b l e n \quad e s s \quad o f t h \quad e \quad q u a s i − l o c a l \quad q u a \quad n t i t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 48 .. period .. 47 4 period 3 .. O n .. l i s t s .. o f .. c r i t e r i a .. o f r e a s o .. n a b l e n .. e s s .. o f t h .. e .. q u a s i hyphen l o c a l .. q u a .. n t i t i e s .. period .. 4 . 3 . 1 \quad G e \quad n e r a l e x p e c t a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 48 period .. period .. period .. period .. period .. period .. period .. period .. period .. 48 4 . 3 . 2 \quad P r a g m \quad a t i c \quad c r i t e r i a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 49 4 period 3 period 1 .. G e .. n e r a l e x p e c t a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 4 . 3 . 3 \quad I \quad n c o \quad m p \quad a t i b i l i t y o f c e r t a i n ‘ n a t u r a l ’ e \quad x p e c t a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 5 1 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 48 \noindent 5 \hfill T \hfill he \hfill B \hfill a r t n i k \hfill M \hfill a s s \hfill a n \hfill d \hfill i t s \hfill M \hfill o \hfill d i \hfill 4 period 3 period 2 .. P r a g m .. a t i c .. c r i t e r i a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period $ fi $ c a t i o n s \hfill 53 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 49 5 . 1 \quad T \quad h e \quad B \quad a r t n i k \quad m \quad a s s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 53 4 period 3 period 3 .. I .. n c o .. m p .. a t i b i l i t y o f c e r t a i n quoteleft n a t u r a l quoteright e .. x p e c t a t i o n s .. period .. period .. period .. 5 . 1 . 1 \quad T h e m \quad a i n \quad i d e a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 53 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 5 1 5 . 1 . 2 \quad T h \quad e \quad m \quad a i n \quad p r o p e r t i e s \quad o f \quad $ m $ B $ ( \Sigma ) 5 .... T .... he .... B .... a r t n i k .... M .... a s s .... a n .... d .... i t s .... M .... o .... d i .... fi c a t i o n s .... 53 . . . . . . . . . . . . . . . . . . . . . . . . . 54 $ 5 period 1 .. T .. h e .. B .. a r t n i k .. m .. a s s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. \hspace∗{\ fill }5 . 1 . 3 \quad T h \quad e \quad c o \quad m p \quad u t a \quad b i l i t y \quad o f t h \quad e B \quad a r t n i k \quad m a s s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 55 period .. period .. period .. period .. period .. period .. period .. period .. 53 5 period 1 period 1 .. T h e m .. a i n .. i d e a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period \hspace∗{\ fill }5 . 2 \quad B r a y \quad ’ s \quad m o d i $ fi $ c a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 55 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 53 \noindent 6 \hfill T \hfill h e \hfill H \hfill a \hfill w k i n g \hfill E \hfill n e r g y \hfill a n d \hfill i t s \hfill M o \hfill d i \hfill 5 period 1 period 2 .. T h .. e .. m .. a i n .. p r o p e r t i e s .. o f .. m B open parenthesis Capital Sigma closing parenthesis period period period period period $ fi $ c a t i o n s \hfill 57 period period period period period period period period period period period period period period period period period period period period 54 5 period 1 period 3 .. T h .. e .. c o .. m p .. u t a .. b i l i t y .. o f t h .. e B .. a r t n i k .. m a s s .. period .. period .. period .. period .. period .. period .. \hspace∗{\ fill }6 . 1 \quad T h e \quad H a w k i n g e n e r g \quad y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 57 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 55 5 period 2 .. B r a y .. quoteright s .. m o d i fi c a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 6 . 1 . 1 \quad T \quad h e \quad d e \quad $ fi $ n i t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 57 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 6 . 1 . 2 \quad H \quad a w \quad kin g \quad e n e r g y f o r s p \quad h e r e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 57 period .. period .. period .. period .. period .. period .. period .. 55 6 .... T .... h e .... H .... a .... w k i n g .... E .... n e r g y .... a n d .... i t s .... M o .... d i .... fi c a t i o n s .... 57 \hspace∗{\ fill }6 . 1 . 3 \quad P \quad o s i t i v i t y \quad a n \quad d \quad m o \quad n o t o n i c i t y \quad p r o p e r t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 58 6 period 1 .. T h e .. H a w k i n g e n e r g .. y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 6 . 1 . 4 \quad T \quad w \quad o g \quad e n \quad e r a l i z a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 59 period .. period .. period .. period .. period .. period .. 57 6 . 2 \quad T h e \quad G e r o c \quad h e n e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 59 6 period 1 period 1 .. T .. h e .. d e .. fi n i t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 6 . 2 . 1 \quad T \quad h e \quad d e $ fi $ n i t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 59 period .. period .. period .. period .. period .. period .. 57 6 . 2 . 2 \quad M \quad o n o t o n i c i t y \quad p r o p \quad e r t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 60 6 period 1 period 2 .. H .. a w .. kin g .. e n e r g y f o r s p .. h e r e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period \hspace∗{\ fill }6 . 3 \quad T h e \quad H \quad a y \quad w a r d \quad e n e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 6 1 .. 57 6 period 1 period 3 .. P .. o s i t i v i t y .. a n .. d .. m o .. n o t o n i c i t y .. p r o p e r t i e s .. period .. period .. period .. period .. period .. period .. \noindent 7 \hfill P e \hfill n r o s \hfill e ’ s \hfill Q u a s i − L \hfill o c a l E \hfill n e r g \hfill y $−{ M }$ period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 58 \hfill o m \hfill e n t \hfill u m \hfill a n d \hfill A n \hfill g u l a r \hfill M \hfill o \hfill m e n \hfill t \hfill u \hfill m \hfill 62 6 period 1 period 4 .. T .. w .. o g .. e n .. e r a l i z a t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 . 1 \quad M o t i v a t i o n \quad s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 62 period .. period .. period .. period .. 59 7 . 1 . 1 \quad H o \quad w \quad d o \quad t \quad h e \quad t w is t o r s \quad e \quad m e r g \quad e $ ? . . 6 period 2 .. T h e .. G e r o c .. h e n e r g y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period . . . . . . . . . . . . . . . . . . . . . . . . 62 $ .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 . 1 . 2 \quad T \quad w i s t o r \quad s p a c e \quad a n \quad d \quad t h e \quad k i n e \quad m a t i c a l \quad t w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 63 period .. period .. period .. period .. period .. period .. period .. 59 7 . 2 \quad T \quad h e \quad o r i g i n a l \quad c o n s t r \quad u c t i o n \quad f o r \quad c u r v e d \quad s p a c e t i m \quad e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 64 6 period 2 period 1 .. T .. h e .. d e fi n i t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 7 . 2 . 1 \quad T \quad w o − s u r f a c e \quad t w i s t o r s \quad a n \quad d \quad t h e \quad k i n e \quad m a t i c a l \quad t w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 64 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 . 2 . 2 \quad T h \quad e H \quad a m i l t o n i a n \quad i n t e r p r e t a t i o n \quad o f \quad t h e \quad k i n e m \quad a t i c a l t \quad w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 65 period .. period .. period .. period .. period .. period .. 59 7 . 2 . 3 \quad T h \quad e H \quad e r m \quad i t i a n \quad s c a l a r \quad p r o d u c t \quad a \quad n d t h e i n 6 period 2 period 2 .. M .. o n o t o n i c i t y .. p r o p .. e r t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period $ fi $ n i t y \quad t w i s t o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 66 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 . 2 . 4 \quad T \quad h e v a r i o u s \quad l i \quad m i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 66 period .. period .. 60 7 . 2 . 5 \quad T \quad h e \quad q u a s i − lo c a l m \quad a s s \quad o f \quad s p e c i $ fi $ \quad c \quad t w o − s u r f a c e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 68 6 period 3 .. T h e .. H .. a y .. w a r d .. e n e r g y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 . 2 . 6 \quad S m \quad a l l s u r f a c e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 70 period .. period .. period .. period .. period .. period .. period .. 6 1 7 . 3 \quad T h \quad e m \quad o \quad d i $ fi $ e d \quad c o n s t r \quad u c t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 70 7 .... P e .... n r o s .... e quoteright s .... Q u a s i hyphen L .... o c a l E .... n e r g .... y hyphen M .... o m .... e n t .... u m .... a n d .... A n .... g u l a r .... M 7 . 3 . 1 \quad T \quad h e ‘ i m p \quad r o \quad v e d ’ c o n s t r \quad u c t i o n \quad w i t h \quad t h e \quad d e t e r m \quad i n a n t \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 70 .... o .... m e n .... t .... u .... m .... 62 7 period 1 .. M o t i v a t i o n .. s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period \hspace∗{\ fill }7 . 3 . 2 \quad M \quad o \quad d i $ fi $ c a t i o n \quad t h r o u g h \quad T o \quad d ’ s \quad e x p r e s s i o \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 7 1 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 62 \hspace∗{\ fill }7 . 3 . 3 \quad M a s o \quad n ’ s s u \quad g g \quad e s t i o n \quad s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 7 1 7 period 1 period 1 .. H o .. w .. d o .. t .. h e .. t w is t o r s .. e .. m e r g .. e ? period period period period period period period period period period period period period period period period period period period period period period period period period period 62 \noindent 8 \hfill A \hfill p p r o \hfill a c h e s \hfill B \hfill a s e d \hfill o n \hfill t h \hfill e N \hfill e s t e r \hfill −− \hfill W i t t e n \hfill 2 − F o r \hfill m \hfill 72 7 period 1 period 2 .. T .. w i s t o r .. s p a c e .. a n .. d .. t h e .. k i n e .. m a t i c a l .. t w i s t o r .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 63 \hspace∗{\ fill }8 . 1 \quad T h e L u d v i g s e n \quad −− V i c k e r s \quad c o \quad n s t r u c t io \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 73 7 period 2 .. T .. h e .. o r i g i n a l .. c o n s t r .. u c t i o n .. f o r .. c u r v e d .. s p a c e t i m .. e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 64 8 . 1 . 1 \quad T h e d e $ fi $ n i t io \quad n . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 73 7 period 2 period 1 .. T .. w o hyphen s u r f a c e .. t w i s t o r s .. a n .. d .. t h e .. k i n e .. m a t i c a l .. t w i s t o r .. period .. period .. period .. period .. 8 . 1 . 2 \quad R e m \quad a r k s \quad o n \quad t h e \quad v a l i d i t y \quad o f \quad t h e \quad c o n s t r u c t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 74 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 64 8 . 1 . 3 \quad M o n o t o n i c i t y , m \quad a s s − p o s i t i v i t y \quad a n \quad d t h \quad e \quad v a r i o u s l i m i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 74 7 period 2 period 2 .. T h .. e H .. a m i l t o n i a n .. i n t e r p r e t a t i o n .. o f .. t h e .. k i n e m .. a t i c a l t .. w i s t o r .. period .. period .. period .. 8 . 2 \quad T \quad h e D \quad o \quad u g a n \quad −− M \quad a s o n \quad c o n s t r \quad u c t io n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 75 period .. period .. period .. period .. period .. period .. period .. 65 7 period 2 period 3 .. T h .. e H .. e r m .. i t i a n .. s c a l a r .. p r o d u c t .. a .. n d t h e i n fi n i t y .. t w i s t o r .. period .. period .. period .. period .. 8 . 2 . 1 \quad H \quad o l o m \quad o r p h i c / \quad a n t i h \quad o l o m \quad o r p h i c \quad s p i n \quad o r period .. period .. period .. period .. period .. period .. period .. period .. 66 $ fi $ e l d s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 75 7 period 2 period 4 .. T .. h e v a r i o u s .. l i .. m i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 8 . 2 . 2 \quad T h \quad e \quad g e n \quad e r i c i t y \quad o f \quad t h e \quad g e \quad n e r i c \quad t w o − s u r f a c e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 76 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 66 8 . 2 . 3 \quad P o s i t i v i t y \quad p r o \quad p e r t i e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 77 7 period 2 period 5 .. T .. h e .. q u a s i hyphen lo c a l m .. a s s .. o f .. s p e c i fi .. c .. t w o hyphen s u r f a c e s period .. period .. period .. period .. 8 . 2 . 4 \quad T \quad h e v a r i o u s \quad l i \quad m i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 78 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 68 7 period 2 period 6 .. S m .. a l l s u r f a c e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 70 7 period 3 .. T h .. e m .. o .. d i fi e d .. c o n s t r .. u c t i o n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 70 7 period 3 period 1 .. T .. h e quoteleft i m p .. r o .. v e d quoteright c o n s t r .. u c t i o n .. w i t h .. t h e .. d e t e r m .. i n a n t .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 70 7 period 3 period 2 .. M .. o .. d i fi c a t i o n .. t h r o u g h .. T o .. d quoteright s .. e x p r e s s i o .. n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 1 7 period 3 period 3 .. M a s o .. n quoteright s s u .. g g .. e s t i o n .. s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 7 1 8 .... A .... p p r o .... a c h e s .... B .... a s e d .... o n .... t h .... e N .... e s t e r .... endash .... W i t t e n .... 2 hyphen F o r .... m .... 72 8 period 1 .. T h e L u d v i g s e n .. endash V i c k e r s .. c o .. n s t r u c t io .. n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 73 8 period 1 period 1 .. T h e d e fi n i t io .. n period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 73 8 period 1 period 2 .. R e m .. a r k s .. o n .. t h e .. v a l i d i t y .. o f .. t h e .. c o n s t r u c t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 74 8 period 1 period 3 .. M o n o t o n i c i t y comma m .. a s s hyphen p o s i t i v i t y .. a n .. d t h .. e .. v a r i o u s l i m i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 74 8 period 2 .. T .. h e D .. o .. u g a n .. endash M .. a s o n .. c o n s t r .. u c t io n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 75 8 period 2 period 1 .. H .. o l o m .. o r p h i c slash .. a n t i h .. o l o m .. o r p h i c .. s p i n .. o r fi e l d s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 75 8 period 2 period 2 .. T h .. e .. g e n .. e r i c i t y .. o f .. t h e .. g e .. n e r i c .. t w o hyphen s u r f a c e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 76 8 period 2 period 3 .. P o s i t i v i t y .. p r o .. p e r t i e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 77 8 period 2 period 4 .. T .. h e v a r i o u s .. l i .. m i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 78 8 . 3 A s p e c i fi c c o n s t r u c t i o n f o r t h e K e r r s p a c e t i m e . . . . . . . . . . . . . . . . . . . . . 79 9 Q u a s i - L o c a l S p i n A n g u l a r M o m e n t u m 80 9 . 1 T h e L u d v i g s e n – V i c k e r s a n g u l a r mo m e n t u m . . . . . . . . . . . . . . . . . . . . . . 80 9 . 2 H o l o m o r p h i c / a n t i h o l o m o r p h i c s p i n a n g u l a r m o m e n t a . . . . . . . . . . . . . . . . . 8 1 9 . 3 A s p e c i fi c c o n s t r u c t i o n f o r t h e K e r r s p a c e t i m e . . . . . . . . . . . . . . . . . . . . . 82 1 0 T h e H a m i l t o n – J a c o b i M e t h o d 83 1 0 . 1 T h e B r o w n – Y o r k e x p r e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1 0 . 1 . 1 T h e m a i n i d e a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1 0 . 1 . 2 T h e v a r i a t i o n o f t h e a c t i o n a n d t h e s u r f a c e s t r e s s - e n e r g y t e n s o r . . . . . . . 84 1 0 . 1 . 3 T h e g e n e r a l f o r m o f t h e B r o w n – Y o r k q u a s i - l o c a l e n e r g y . . . . . . . . . . . 85 1 0 . 1 . 4 F u r t h e r p r o p e r t ie s o f t h e g e n e r a l e x p r e s s i o n s . . . . . . . . . . . . . . . . . 86 1 0 . 1 . 5 T h e H a m i l t o n i a n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 1 0 . 1 . 6 T h e fl a t s p a c e a n d l i g h t c o n e r e f e r e n c e s . . . . . . . . . . . . . . . . . . . . 89 1 0 . 1 . 7 F u r t h e r p r o p e r t i e s a n d t h e v a r io u s l i m i t s . . . . . . . . . . . . . . . . . . . . 90 1 0 . 1 . 8 O t h e r p r e s c r i p t i o n s f o r t h e r e f e r e n c e c o n fi g u r a t i o n . . . . . . . . . . . . . . 93 1 0 . 2 K i j o w s k i ’ s a p p r o a c h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 1 0 . 2 . 1 T h e r o l e o f t h e b o u n d a r y c o n d i t io n s . . . . . . . . . . . . . . . . . . . . . . 93 1 0 . 2 . 2 T h e a n a l y s i s o f t h e H i l b e r t a c t i o n a n d t h e q u a s i - l o c a l i n t e r n a l a n d f r e e e n e r g i e s 94 1 0 . 3 E p p ’ s e x p r e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1 0 . 3 . 1 T h e g e n e r a l f o r m o f E p p ’ s e x p r e s s i o n . . . . . . . . . . . . . . . . . . . . . . 95 1 0 . 3 . 2 T h e d e fi n i t i o n o f t h e r e f e r e n c e c o n fi g u r a t i o n . . . . . . . . . . . . . . . . . . 96 1 0 . 3 . 3 T h e v a r i o u s l i m i t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 1 0 . 4 T h e e x p r e s s i o n o f L i u a n d Y a u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 1 0 . 4 . 1 T h e L i u – Y a u d e fi n i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 1 0 . 4 . 2 T h e m a i n p r o p e r t i e s o f E KLY (S) . . . . . . . . . . . . . . . . . . . . . . . . 97 1 0 . 4 . 3 G e n e r a l i z a t i o n s o f t h e o r i gi n a l c o n s t r u c t io n . . . . . . . . . . . . . . . . . . 98 1 0 . 5 T h e e x p r e s s i o n o f W a n g a n d Y a u . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 1 1 T o w a r d s a F u l l H a m i l t o n i a n A p p r o a c h 102 1 1 . 1 T h e 3 + 1 a p p r o a c h e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 1 1 . 1 . 1 T h e q u a s i −l o c a l c o n s t r a i n t a l g e b r a a n d t h e b a s i c H a m i l t o n i a n . . . . . . . . 1 3 1 1 . 1 . 2 T h e t w o - s u r f a c e o b s e r v a b l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 1 1 . 2 A p p r o a c h e s b a s e d o n t h e d o u b l e - n u l l f o l i a t i o n s . . . . . . . . . . . . . . . . . . . . 1 5 1 1 . 2 . 1 T h e 2 +2 d e c o m p o s i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 1 1 . 2 . 2 T h e 2 +2 q u a s i - l o c a l i z a t i o n o f t h e B o n d i – S a c h s m a s − lo s s . . . . . . . . . . 1 6 1 1 . 3 T s h e c o v a r i a n t a p p r o a c h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 1 1 . 3 . 1 T h e c o v a r i a n t p h a s e s p a c e m e t h o d s . . . . . . . . . . . . . . . . . . . . . . . 1 6 1 1 . 3 . 2 T h e g e n e r a l e x p r e s s i o n s o f C h e n , N e s t e r a n d T u n g : C o v a r i a n t q u a s i - l o c a l H a m i l t o n i a n s w i t h e x p l i c i t r e f e r e n c e c o n fi g u r a t i o n s . . . . . . . . . . . . . . 1 7 1 1 . 3 . 3 T h e r e f e r e n c e c o n fi g u r a t i o n o f N e s t e r , C h e n , L i u a n d S u n . . . . . . . . . . 1 9 1 1 . 3 . 4 C o v a r i a n t q u a s i - l o c a l H a m i l t o n i a n s w i t h g e n e r a l r e f e r e n c e t e r m s . . . . . . 1 1 0 1 1 . 3 . 5 P s e u d o t e n s o r s a n d q u a s i - l o c a l q u a n t i t ie s . . . . . . . . . . . . . . . . . . . . 1 1 1 1 2 C o n s t r u c t i o n s f o r S p e c i a l S p a c e t i m e s 1 13 1 2 . 1 T h e K o m a r i n t e g r a l f o r s p a c e t i m e s w i t h K i l l i n g v e c t o r s . . . . . . . . . . . . . . . . 1 1 3 1 2 . 2 T h e e ff e c t i v e m a s s o f K u l k a r n i , C h e l l a t h u r a i , a n d D a d h i c h f o r t h e K e r r s p a c e t i m e . 1 1 3 1 2 . 3 E x p r e s s i o n s i n s t a t i c s p a c e t i m e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 1 2 . 3 . 1 T o l m a n ’ s e n e r g y f o r s t a t i c s p a c e t i m e s . . . . . . . . . . . . . . . . . . . . . 1 1 4 1 2 . 3 . 2 T h e K a t z – L y n d e n - B e l l – I s r a e l e n e r g y f o r s t a t i c s p a c e t i m e s . . . . . . . . . . 1 1 4 8 period 3 .. A s p e c i fi c c o n s t r .. u c t i o .. n f o r .. t h e .. K e r r s p a c e t i m .. e .. period .. period .. period .. period .. period .. period .. period .. \hspace∗{\ fill }8 . 3 \quad A s p e c i $ fi $ c c o n s t r \quad u c t i o \quad n f o r \quad t h e \quad K e r r s p a c e t i m \quad e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 79 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 79 9 .... Q u a s i hyphen L .... o c .... a l .... S .... p i .... n .... A .... n g u l .... a r .... M .... o .... m e .... n t u .... m .... 80 \noindent 9 \hfill Q u a s i − L \hfill o c \hfill a l \hfill S \hfill p i \hfill n \hfill A \hfill n g u l \hfill a r \hfill M \hfill o \hfill m e \hfill n t u \hfill m \hfill 80 9 period 1 .. T h e .. L .. u d v i g s e n endash V i c k e r s .. a n g u l a r .. mo m .. e .. n t u .. m .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 80 \hspace∗{\ fill }9 . 1 \quad T h e \quad L \quad u d v i g s e n −− V i c k e r s \quad a n g u l a r \quad mo m \quad e \quad n t u \quad m \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 80 9 period 2 .. H .. o l o m .. o r p h i c slash a n t i h o l o .. m o r p .. h i c .. s p i n .. a n .. g u l a r .. m o m .. e n t a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 8 1 9 . 2 \quad H \quad o l o m \quad o r p h i c / a n t i h o l o \quad m o r p \quad h i c \quad s p i n \quad a n \quad g u l a r \quad m o m \quad e n t a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 8 1 9 period 3 .. A .. s p .. e c i fi .. c .. c o n s t r u .. c t i o n .. f o r t .. h e .. K .. e r r s p a c e t i m .. e period .. period .. period .. period .. period .. period .. 9 . 3 \quad A \quad s p \quad e c i $ fi $ \quad c \quad c o n s t r u \quad c t i o n \quad f o r t \quad h e \quad K \quad e r r s p a c e t i m \quad e . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 82 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 82 1 .... 0 .... T h .... e H .... a .... m i l t o n .... endash J a c o b i M .... e t .... h o .... d .... 83 \noindent 1 \hfill 0 \hfill T h \hfill e H \hfill a \hfill m i l t o n \hfill −− J a c o b i M \hfill e t \hfill h o \hfill d \hfill 83 1 0 period 1 .. T .. h e B .. r o w n .. endash Y o r k e .. x p r e s s i o .. n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 0 . 1 \quad T \quad h e B \quad r o w n \quad −− Y o r k e \quad x p r e s s i o \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 83 period .. period .. period .. period .. period .. 83 1 0 . 1 . 1 \quad T \quad h e \quad m \quad a i n \quad i d e a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 83 1 0 period 1 period 1 .. T .. h e .. m .. a i n .. i d e a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 1 0 . 1 . 2 \quad T h e \quad v a r i a t i o n \quad o f t h \quad e \quad a c t i o n \quad a n d \quad t h e s \quad u r f a c e \quad s t r e s s − e \quad n e r g \quad y \quad t e n \quad s o r \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 84 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 0 . 1 . 3 \quad T \quad h e \quad g e n \quad e r a l f o r m \quad o f \quad t h \quad e \quad B r o w n \quad −− Y o r k \quad q u a s i − l o c a l \quad e n \quad e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 85 period .. period .. period .. period .. period .. period .. 83 1 0 . 1 . 4 \quad F \quad u r t h e r p r o p e r t ie s \quad o f \quad t h e \quad g e n \quad e r a l e \quad x p r e s s i o \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 86 1 0 period 1 period 2 .. T h e .. v a r i a t i o n .. o f t h .. e .. a c t i o n .. a n d .. t h e s .. u r f a c e .. s t r e s s hyphen e .. n e r g .. y .. t e n .. s o r .. period .. period .. period .. period .. period .. period .. period .. 84 1 0 . 1 . 5 \quad T \quad h e \quad H \quad a m \quad i l t o n i a n \quad s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 88 1 0 period 1 period 3 .. T .. h e .. g e n .. e r a l f o r m .. o f .. t h .. e .. B r o w n .. endash Y o r k .. q u a s i hyphen l o c a l .. e n .. e r g y .. period .. 1 0 . 1 . 6 \quad T h e \quad $ fl $ a t s p a c e \quad a n d \quad l i g h t \quad c o \quad n e \quad r e f e r e n c e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 89 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 85 1 0 period 1 period 4 .. F .. u r t h e r p r o p e r t ie s .. o f .. t h e .. g e n .. e r a l e .. x p r e s s i o .. n s .. period .. period .. period .. period .. period .. \hspace∗{\ fill }1 0 . 1 . 7 \quad F \quad u r t h e r p r o p e r t i e s \quad a \quad n d \quad t h \quad e \quad v a r io \quad u s \quad l i m i t s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 90 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 86 1 0 period 1 period 5 .. T .. h e .. H .. a m .. i l t o n i a n .. s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period \hspace∗{\ fill }1 0 . 1 . 8 \quad O t h e r \quad p r e s c r i p t i o n s \quad f o r \quad t h e \quad r e f e r e n c e \quad c o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. $ fi $ g u r a t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 93 period .. period .. period .. period .. period .. 88 1 0 period 1 period 6 .. T h e .. fl a t s p a c e .. a n d .. l i g h t .. c o .. n e .. r e f e r e n c e s .. period .. period .. period .. period .. period .. period .. period \hspace∗{\ fill }1 0 . 2 \quad K i j o w \quad s k i ’ s \quad a p p r o a c h \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 93 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 89 1 0 period 1 period 7 .. F .. u r t h e r p r o p e r t i e s .. a .. n d .. t h .. e .. v a r io .. u s .. l i m i t s period .. period .. period .. period .. period .. period .. 1 0 . 2 . 1 \quad T h e \quad r o l e \quad o f t h e \quad b o \quad u n d a r y \quad c o n d i t io \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 93 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 90 1 0 . 2 . 2 \quad T \quad h e \quad a n a l y s i s o f t h e \quad H i l b e r t a c t i o n \quad a n d \quad t \quad h e \quad q u a s i − l o c a l i n t e r n a l a n d f r e e \quad e n \quad e r g i e s \quad 94 1 0 period 1 period 8 .. O t h e r .. p r e s c r i p t i o n s .. f o r .. t h e .. r e f e r e n c e .. c o n fi g u r a t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 93 1 0 . 3 \quad E p p ’ s e \quad x p r e s s i o \quad n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 95 1 0 period 2 .. K i j o w .. s k i quoteright s .. a p p r o a c h .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 0 . 3 . 1 \quad T h e \quad g e n e r a l \quad f o r m \quad o f \quad E p p ’ s \quad e x p r e s s i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 95 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 1 0 . 3 . 2 \quad T \quad h e \quad d e $ fi $ n i t i o n \quad o f \quad t \quad h e \quad r e f e r e n \quad c e c o n .. period .. period .. period .. period .. period .. period .. period .. 93 $ fi $ g \quad u r a t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 96 1 0 period 2 period 1 .. T h e .. r o l e .. o f t h e .. b o .. u n d a r y .. c o n d i t io .. n s .. period .. period .. period .. period .. period .. period .. period .. 1 0 . 3 . 3 \quad T \quad h e \quad v a r i o \quad u s \quad l i m \quad i t s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 96 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 93 1 0 . 4 \quad T \quad h e e x p r e s s i o n o f L \quad i u \quad a n \quad d \quad Y a u \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 97 1 0 period 2 period 2 .. T .. h e .. a n a l y s i s o f t h e .. H i l b e r t a c t i o n .. a n d .. t .. h e .. q u a s i hyphen l o c a l i n t e r n a l a n d f r e e .. e n .. e 1 0 . 4 . 1 \quad T \quad h e \quad L i u −− Y \quad a u \quad d e $ fi $ n i t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 97 r g i e s .. 94 1 0 . 4 . 2 \quad T \quad h e \quad m \quad a i n \quad p r o p \quad e r t i e s \quad o f \quad $ E $ K L Y $ ( S 1 0 period 3 .. E p p quoteright s e .. x p r e s s i o .. n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ) . . . . . . . . . . . . . . . . . . . . . . . . 97 $ .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 95 \hspace∗{\ fill }1 0 . 4 . 3 \quad G e n \quad e r a l i z a t i o \quad n s o f \quad t h e o r i gi n \quad a l c o \quad n s t r u c t io n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 98 1 0 period 3 period 1 .. T h e .. g e n e r a l .. f o r m .. o f .. E p p quoteright s .. e x p r e s s i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 95 \hspace∗{\ fill }1 0 . 5 \quad T h e e x p r e s s i o \quad n o f \quad W \quad a n g \quad a n d \quad Y a \quad u \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 99 1 0 period 3 period 2 .. T .. h e .. d e fi n i t i o n .. o f .. t .. h e .. r e f e r e n .. c e c o n fi g .. u r a t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 96 \noindent 1 \hfill 1 \hfill T \hfill o \hfill w a r \hfill d s \hfill a F \hfill u l l \hfill H \hfill a m \hfill i l t o n \hfill i a n \hfill A \hfill p p r o a c h \hfill 102 1 0 period 3 period 3 .. T .. h e .. v a r i o .. u s .. l i m .. i t s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. \hspace∗{\ fill }1 1 . 1 \quad T \quad h e \quad $ 3 + 1 $ \quad a p p \quad r o a c h \quad e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 2 period .. period .. period .. period .. period .. 96 1 0 period 4 .. T .. h e e x p r e s s i o n o f L .. i u .. a n .. d .. Y a u .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 . 1 . 1 \quad T \quad h e \quad q u a s i $−{ l }$ o c a l \quad c o n s t r a i n t \quad a l g e b r a \quad a n \quad d \quad t h e \quad b \quad a s i c \quad H a \quad m i l t o n i a n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 3 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 1 \quad 1 . 1 . 2 \quad T \quad h e \quad t w o − s u r f a c e \quad o b s e r v a b l e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 4 .. period .. period .. 97 1 \quad 1 . 2 \quad A p p r o a c h e s \quad b \quad a s e \quad d o n \quad t h \quad e \quad d o u b l e − n \quad u l l f o l i a t i o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5 1 0 period 4 period 1 .. T .. h e .. L i u endash Y .. a u .. d e fi n i t i o n .. period .. period .. period .. period .. period .. period .. period .. period .. period 1 1 . 2 . 1 \quad T \quad h e \quad $ 2 + 2 $ \quad d e c o m \quad p o s i t i o \quad n . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 5 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 . 2 . 2 \quad T h e \quad $ 2 + 2 $ \quad q u a s i − l o c a l i z a t i o n \quad o f \quad t h e \quad B o \quad n d i −− S a c h s m \quad a s period .. period .. period .. 97 $ − { s }$ lo s s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6 1 0 period 4 period 2 .. T .. h e .. m .. a i n .. p r o p .. e r t i e s .. o f .. E K L Y open parenthesis S closing parenthesis period period period period period 1 1 . 3 \quad T h e c o v a r i a n t a p p r o a c h \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6 period period period period period period period period period period period period period period period period period period period 97 1 1 . 3 . 1 \quad T \quad h e \quad c o v a r i a n t \quad p \quad h a s e \quad s p \quad a c e m \quad e t h \quad o \quad d s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 6 1 0 period 4 period 3 .. G e n .. e r a l i z a t i o .. n s o f .. t h e o r i gi n .. a l c o .. n s t r u c t io n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 98 \centerline{1 1 . 3 . 2 \quad T h e \quad g \quad e n e r \quad a l e x p r e s s i o n s o f \quad C h e n , N \quad e s t e r \quad a n d \quad T \quad u n g : \quad C o \quad v a r i a n t \quad q u \quad a s i − l o c a l } 1 0 period 5 .. T h e e x p r e s s i o .. n o f .. W .. a n g .. a n d .. Y a .. u .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. H \quad a m \quad i l t o n i a n s \quad w i t h \quad e \quad x p l i c i t \quad r e f e r e n \quad c e \quad c o n period .. period .. 99 $ fi $ \quad g u r a t i o \quad n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 7 1 .... 1 .... T .... o .... w a r .... d s .... a F .... u l l .... H .... a m .... i l t o n .... i a n .... A .... p p r o a c h .... 102 1 \quad 1 . 3 . 3 \quad T \quad h e r e f e r e \quad n c e \quad c o n \quad $ fi $ g u r a t i o n o f N \quad e s t e r , \quad C \quad h e n , \quad L i u a n d S \quad u n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 9 1 1 period 1 .. T .. h e .. 3 plus 1 .. a p p .. r o a c h .. e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. \hspace∗{\ fill }1 \quad 1 . 3 . 4 \quad C o \quad v a r i a \quad n t \quad q u a s i − l o c a l \quad H \quad a m \quad i l t o n i a n s \quad w i t h \quad g e n \quad e r a l r e f e r e n c e t e r \quad m s \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 0 period .. period .. period .. period .. period .. period .. period .. 1 2 1 1 period 1 period 1 .. T .. h e .. q u a s i hyphen l o c a l .. c o n s t r a i n t .. a l g e b r a .. a n .. d .. t h e .. b .. a s i c .. H a .. m i l t o n i a n .. period .. \hspace∗{\ fill }1 1 . 3 . 5 \quad P s e u d o t e n \quad s o r s \quad a n \quad d \quad q \quad u a s i − l o c a l \quad q u a n t i t ie s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 1 period .. period .. period .. period .. period .. period .. period .. 1 3 1 .. 1 period 1 period 2 .. T .. h e .. t w o hyphen s u r f a c e .. o b s e r v a b l e s .. period .. period .. period .. period .. period .. period .. period .. period \noindent 1 2 \hfill C o n \hfill s t r \hfill u c t i o n s \hfill f o r \hfill S p e c i a l S \hfill p \hfill a c e t i m \hfill e s \hfill 1 13 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 4 1 2 . 1 \quad T \quad h e K \quad o \quad m a r \quad i n t e g r \quad a l f o r s p a c e t i m e s \quad w i t h \quad K \quad i l l i n g \quad v e c t o r s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 3 1 .. 1 period 2 .. A p p r o a c h e s .. b .. a s e .. d o n .. t h .. e .. d o u b l e hyphen n .. u l l f o l i a t i o n s .. period .. period .. period .. period .. period .. 1 2 . 2 \quad T h \quad e \quad e $ ff $ \quad e c t i v e \quad m a s s \quad o f \quad K u l k \quad a r n i , \quad C h e l l a t h \quad u r a i , \quad a n \quad d D \quad a d h i c h \quad f o r \quad t h e \quad K \quad e r r \quad s p a c e t i m \quad e . \quad 1 1 3 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 5 1 2 . 3 \quad E \quad x p r e s s i o n s i n s t a t i c s p a c e t i \quad m e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 4 1 1 period 2 period 1 .. T .. h e .. 2 plus 2 .. d e c o m .. p o s i t i o .. n period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 \quad 2 . 3 . 1 \quad T \quad o l \quad m a n ’ s \quad e n \quad e r g y \quad f o r s t a t i c s p a c e t i \quad m e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 4 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period 1 2 . 3 . 2 \quad T h \quad e \quad K a t z −− \quad L y n \quad d e n − B \quad e l l −− \quad I s r a e l \quad e n e r g y f o r \quad s t a t i c \quad s p a c e t i \quad m e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 4 .. period .. period .. 1 5 1 1 period 2 period 2 .. T h e .. 2 plus 2 .. q u a s i hyphen l o c a l i z a t i o n .. o f .. t h e .. B o .. n d i endash S a c h s m .. a s hyphen s lo s s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6 1 1 period 3 .. T h e c o v a r i a n t a p p r o a c h .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6 1 1 period 3 period 1 .. T .. h e .. c o v a r i a n t .. p .. h a s e .. s p .. a c e m .. e t h .. o .. d s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 6 1 1 period 3 period 2 .. T h e .. g .. e n e r .. a l e x p r e s s i o n s o f .. C h e n comma N .. e s t e r .. a n d .. T .. u n g : .. C o .. v a r i a n t .. q u .. a s i hyphen l o c a l H .. a m .. i l t o n i a n s .. w i t h .. e .. x p l i c i t .. r e f e r e n .. c e .. c o n fi .. g u r a t i o .. n s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 7 1 .. 1 period 3 period 3 .. T .. h e r e f e r e .. n c e .. c o n .. fi g u r a t i o n o f N .. e s t e r comma .. C .. h e n comma .. L i u a n d S .. u n .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 9 1 .. 1 period 3 period 4 .. C o .. v a r i a .. n t .. q u a s i hyphen l o c a l .. H .. a m .. i l t o n i a n s .. w i t h .. g e n .. e r a l r e f e r e n c e t e r .. m s .. period .. period .. period .. period .. period .. period .. 1 1 0 1 1 period 3 period 5 .. P s e u d o t e n .. s o r s .. a n .. d .. q .. u a s i hyphen l o c a l .. q u a n t i t ie s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 1 1 2 .... C o n .... s t r .... u c t i o n s .... f o r .... S p e c i a l S .... p .... a c e t i m .... e s .... 1 13 1 2 period 1 .. T .. h e K .. o .. m a r .. i n t e g r .. a l f o r s p a c e t i m e s .. w i t h .. K .. i l l i n g .. v e c t o r s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 3 1 2 period 2 .. T h .. e .. e ff .. e c t i v e .. m a s s .. o f .. K u l k .. a r n i comma .. C h e l l a t h .. u r a i comma .. a n .. d D .. a d h i c h .. f o r .. t h e .. K .. e r r .. s p a c e t i m .. e period .. 1 1 3 1 2 period 3 .. E .. x p r e s s i o n s i n s t a t i c s p a c e t i .. m e s period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 4 1 .. 2 period 3 period 1 .. T .. o l .. m a n quoteright s .. e n .. e r g y .. f o r s t a t i c s p a c e t i .. m e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 4 1 2 period 3 period 2 .. T h .. e .. K a t z endash .. L y n .. d e n hyphen B .. e l l endash .. I s r a e l .. e n e r g y f o r .. s t a t i c .. s p a c e t i .. m e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 1 4 1 2 . 3 . 3 S t a t i c s p a c e t i m e s a n d p o s t - N e w t o n i a n a p p r o x i m a t i o n . . . . . . . . . . . . 1 1 5 1 3 1 2 peAriod 3 pperipodl3i .c. aS tt ai ot i c s pn.. sa c e t iimn.. e s ..Ga n .. depno s t hyephrena.l. RN e .. wet ol an itainv..i ta pyp r o x1i1m6.. a t i o n .. period .. period .. period .. \hspace∗{\ fill }1 2 . 3 . 3 \quad S t a t i c s p \quad a c e t i m \quad e s \quad a n \quad d p o s t − \quad N e \quad w t o n i a n \quad a p p r o x i m \quad a t i o n \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 5 pe1ri3od. 1.. peCrioadl.c. peuriolda.t.ipoeriodn.o.fpteriiodda..l hpeeriaodt.i.npgeriod. .. .peri.od ... pe.riod... 1.1 5. . . . . . . . . . . .1 3......A ..... p.p l.i c a. t i.o ..... n.s ..... i n......G .1...1e6n1.3....e2r aGl R .e...oeml a teitvriitcy .i..n. 1e1q6u a l i t i e s f o r b l a c \noindent 1 3 \hfill A \hfill p p l i c a t i o \hfill n s \hfill i n \hfill G \hfill e n \hfill e r a l R \hfill e l a t i v i t y \hfill 1 16 k h 1o3lpeesri.od 1. .. .C a.l c ... u.l a t. i o... n. o f. t i.d a.l h.e a.t i n. g ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per1io1d6..1p3e.rio2d .. period .. period .. period .. period . 1.. peOriond ..tphereiodP..epneriodr o..speerioidn.. peerqiouda..l pietryiod ... pe.riod. .. .peri.od ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period .. 1 3 . 1 \quad C a l c \quad u l a t i o \quad n o f t i d a l h e a t i n g \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 6 . pe.riod... p.erio.d ...per.iod... pe.riod. .. 1111661 3 . 2 . 2 O n t h e h o o p c o n j e c t u r e . . . . . . 1 3 . 2 \quad G \quad e o m \quad e t r i c \quad i n e q u a l i t i e s \quad f o r b l a c k h \quad o l e s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 6 . .1 3.per.iod.2 ...G ... e o. m ... e.t r i.c ...i n.e q.u a.l i t.i e.s ...f o .r b l.a c.k h... o. l e1s1pe8ri1od3...2pe.r3iod .O. perniotdh.. peerDiod .. period .. period .. period .. period 1 3 . 2 . 1 \quad O n \quad t h e \quad P e n \quad r o s e \quad i n \quad e q u a l i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 6 a.i.nperiiond .. epeqriuoda.l.i tpeyriod... p.erio.d ...per.iod... p.eriod. .. .peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. .perio.d ... per.iod... period .. period .. period .. period .. 1 \quad 3 . 2 . 2 \quad O \quad n t h \quad e \quad h o o p \quad c o n j e c t u r e \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 8 . pe.riod... 1.1 61 1 9 1 3 . 3 Q u a s −i l o c a l l a w s o f b l a c k h o l e d y n a m i c s . . . . 1 3 . 2 . 3 \quad O \quad n t h \quad e D \quad a i n \quad i n \quad e q u a li t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 1 9 . .1 3.per.iod.2 pe.riod.1 ... O .n ...t h.e .. .P e.n ...r o.s e ... i n. .. .e q u. a l.i t1y2..0p1e3rio.d3... p1erioQdu..apseriio−dl.o. pcearilod .t. hpeerirod .. period .. period .. period .. period 1 3 . 3 \quad Q \quad u a s $ − { i }$ l o c a l \quad l a w \quad s \quad o f \quad b l a c k h \quad o l e \quad d y n a \quad m i c s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 20 m..operdioyd ..npearimod ..i cpesriodo.f.bpelraiocdk.. pheroiodl e..sper.iod... p.erio.d ...per.iod... p.eriod. ...peri.od ... pe.riod. .. .peri.od ... pe.riod1.. period .. period .. period .. period .. 1 3 . 3 . 1 \quad Q u a s i $−{ l }$ o c a l \quad t h e r \quad m o \quad d y \quad n a m \quad i c s \quad o f b l a c k \quad h o l e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 20 20p1er3iod. 3...p2eriodO..n1 1i6s o l a t e d a n d d y n a m i c h o r i z o n s . . . . . . . . . . . . 1 3 . 3 . 2 \quad O n \quad i s o l a t e d \quad a n d \quad d y \quad n a m \quad i c h o r i z o n s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 2 1 . .1 ... 3 p.erio.d 2.per.iod.2 ...O ... n t. h .1. 2e1..1h3o.o4p ..Ec onntjreocptyubr e .o. peurinodd..s .peri.od ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period .. 1 3 . 4 \quad E \quad n t r o p y b \quad o \quad u n d s . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 22 . pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio.d ...per.iod... p.eriod. .. .peri.od ... pe.riod... p.erio.d ...peri.od ... pe.riod1.2.2period .. period .. period .. period .. period .. period ..1p3er.io4d...1perOiodn.. 1B1e8k e n s t e i n (cid:48)s b o u n d s f o r t h e e n t r o p y . . . . . . . . . \hspace∗{\ fill }1 3 . 4 . 1 \quad O n \quad B e k e n s t e i n \quad $’{ s }$ \quad b o \quad u n d s \quad f o r \quad t \quad h e e n t r o p y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 22 . .1 3.per.iod.2 pe.riod.3 ... O ... n.t h ... e1D2.2. a i n .. i n .. e q u a li t y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period ..1p3er.io4d...2perOiodn.. pteriohde.. pheroioldo..gprearipodh.i. cperihody.p. peoriotdh..epseirsiod ... pe.riod. .. .peri.od ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period .. 1 3 . 4 . 2 \quad O n \quad t \quad h e \quad h o l o g r a p h i c \quad h y p \quad o t h e s i s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 22 . pe.riod... p.erio.d ...1 1.9 . . . . . 1 22 1 3 . 4 . 3 E n t r o p y b o u n d s o f A b r e u a n d 1 3 . 4 . 3 \quad E n t r o p y \quad b \quad o u n d s \quad o f A \quad b r e \quad u \quad a n \quad d \quad V i s s e r \quad f o r \quad u n \quad c o l l a p s e d b o \quad di e s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 23 V i s1s3e rperiofdo3r .. uQn.. ucaoslhlyaphpesnei dl obcoa ld..i el as w... s... o.f ...b l.a c.k h... o.l e ... d1y2n3a1.3. .m5i cQs ..upaesriiod- l..opceraiold .. period .. period .. period .. period 1 3 . 5 \quad Q u a s i − l o c a l \quad r a d i a t i v e m \quad o \quad d e s \quad o f \quad g e n e r a l \quad r e l a t i v i t y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 23 r a..dpeiraiotdi.v. epemriodo.. pdereiosd ..opferiogde..npeerrioadl.. preeriloadt..ipveiritody .. .peri.od ... pe.riod. .. p.erio.d ... pe.riod... p.erio.d ... per.iod... period .. period .. period .. period .. 1 3 . 6 \quad P \quad o t e n t i a l a \quad p p l i c a t i o n \quad s i n c o s \quad m o l o g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 24 . pe.riod... 1.20. 1 23 1 3 . 6 P o t e n t i a l a p p l i c a t i o n s i n c o s m o l o g y . . . . . . . . .1 3.per.iod.3 pe.riod. 1 ... Q.u a.s i h.yph.en l.o c.a l ... t h. e r... m. o... d.y ..1n2a4m .. i c s .. o f b l a c k .. h o l e s .. period .. period .. period .. period .. period .. \noindent 1 4 \hfill S \hfill u m \hfill m \hfill a r y : \hfill A c \hfill h i e v e \hfill m e n t s , \hfill D \hfill i 1p4erioSd .u. pmeriomd .. paerrioyd:.. pAericodh.. ipeerivode..mpereiond .t.spe,rioDd ..ipeffiriodc .u. pletriioeds..,peariond ..dpOeripode.. nperiIods.s. upeeriosd .1.21520 $ ffi $ \hfill c u l t ie s , \hfill a n \hfill d O p e \hfill n \hfill I s s u e s \hfill 125 1143. 1perioOd 3npetrihode2 .B. O na.r.tins oi kl a tme ads..sa nadn..dd yH..anwa mk..iincgh oerni zeor gnys .. p.erio.d ... per.iod... p.erio.d ...peri.od .. period .. period .. period .. period . .. .peri.od ... pe.riod... p.erio.d ...per.iod ... pe.riod. .. p.erio1d2.5. 1pe4ri.od2 .. Operniodt.. pherieod .P. peenriordo.s. eperiomd ..apesrsiod ... pe.riod. .. p.eriod .. period .. 1 2 1 1 4 . 1 \quad O \quad n t h e \quad B \quad a r t n i k \quad m a s s \quad a n d \quad H a w \quad k i n g \quad e n e r g y \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 25 . .1 3.per.iod.4 ...E ... n .t r o. p y. b ... o... u.n d. s p.erio.d ...per.iod... p.eriod. ...peri.od ... pe.riod. .. .peri.od ... pe.riod... p1e2r6iod .. period .. period .. period .. period 1 4 . 2 \quad O n \quad t \quad h e \quad P e n r o s e \quad m \quad a s s \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 26 1 4 . 3 O n t h e D o u..gperiaodn.. p–eMriod a..spoeriodn.e.nperieodr g..ype-riodm..opmerioedn..tpaerioadn.. pedritod .h. peehriodo..l operiomdo..rpperhioidc./. peraiond ..tpiehrioodl.o. mperioodr..pperhioidc..s ppeirinod .. period .. pearinodg.. puerliaodr.. pmeroiod m.. peenriotda.. p.erio.d ...per.iod... pe.riod. .. .peri.od ... pe.riod... 1.22. . . . . . . . . . \centerline{1 4 . 3 \quad O \quad n t h e D \quad o u g \quad a n \quad −− M \quad a s o \quad n e n \quad e r g y − \quad m o m \quad e n t a \quad a n \quad d t \quad h e h \quad o l o \quad m o r p h i c / \quad a n \quad t i h o l o m \quad o r p \quad h i c s p i n } . .1 3.per.iod.4 pe.riod. 1 ... O.n ...B e. k e.n s.t e.i n1..2q6uoteright s .. b o .. u n d s .. f o r .. t .. h e e n t r o p y .. period .. period .. period .. period .. period .. pe1riod4... p4erioOdn.. petrihode.. pBerrioodw.. penri–odY..oprekrio–dt.y. ppeerioedx.p. preerisods i..opnesriod... p.erio.d ...per.iod... pe.riod. .. p.erio.d ... per.iod .. period .. 1 22 \hspace∗{\ fill }a n g \quad u l a r \quad m o \quad m e n t a \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 26 . .1 3.per.iod.4 pe.riod. 2 ... O.n ...t ...h e... h. o l1o2g7r a p h i c .. h y p .. o t h e s i s .. period .. period .. period .. period .. period .. period .. period .. period .. 1period .. pe5riod .. perioAd .. period ..cpekrinod .. periodo.. wperiod .. perilode.d. pgeriod .. perimod .. period .e. pnertiosd .. period .1.3p0eriod .. period .. period .. period .. 1 22 \hspace∗{\ fill }1 \quad 4 . 4 \quad O n \quad t h e \quad B r o w \quad n −− Y o r k −− t y p e e x p r e s s i o ns \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad . \quad 1 27 R 1 3 period 4 period 3 .. E n t r o p y .. b .. o uenfdesr..eonf Ac .e. sb r e .. u .. a n .. d .. V i s s e r .. f o r .. u n .. c o1l3l1a p s e d b o .. di e s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 23 \noindent 1 \hfill 5 \hfill A \hfill c k n \hfill o w \hfill l e d g \hfill m \hfill e n t s \hfill 130 1 3 period 5 .. Q u a s i hyphen l o c a l .. r a d i a t i v e m .. o .. d e s .. o f .. g e n e r a l .. r e l a t i v i t y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 23 \noindent R \hfill e f e r e n c e s \hfill 13 1 1 3 period 6 .. P .. o t e n t i a l a .. p p l i c a t i o n .. s i n c o s .. m o l o g y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 24 1 4 .... S .... u m .... m .... a r y : .... A c .... h i e v e .... m e n t s comma .... D .... i ffi .... c u l t ie s comma .... a n .... d O p e .... n .... I s s u e s .... 125 1 4 period 1 .. O .. n t h e .. B .. a r t n i k .. m a s s .. a n d .. H a w .. k i n g .. e n e r g y .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 25 1 4 period 2 .. O n .. t .. h e .. P e n r o s e .. m .. a s s .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 26 1 4 period 3 .. O .. n t h e D .. o u g .. a n .. endash M .. a s o .. n e n .. e r g y hyphen .. m o m .. e n t a .. a n .. d t .. h e h .. o l o .. m o r p h i c slash .. a n .. t i h o l o m .. o r p .. h i c s p i n a n g .. u l a r .. m o .. m e n t a .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 26 1 .. 4 period 4 .. O n .. t h e .. B r o w .. n endash Y o r k endash t y p e e x p r e s s i o ns .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 27 1 .... 5 .... A .... c k n .... o w .... l e d g .... m .... e n t s .... 130 R .... e f e r e n c e s .... 13 1 Quasi - Local Energy - Momentum and Angular Momentum in General Relativity 9 Quasi hyphen Local Energy hyphen Momentum and Angular Momentum in General Relativity .... 9 \noindent Quasi − Local Energy − Momentum and Angular Momentum in General Relativity \hfill 9 1 hliInnetroduction 1 .. Introduction Ov\e[r t\hreullaest{33e5my}e{a0rs.4,potn}e o\f]the greatest achievements in classical general relativity has certainly been the proof of the Over the last 35 years comma one of the greatest achievements in classical general relativity has certainly positivityofthetotal gravitationalenergy,bothatspatialandnullinfinity. Itispreciselyitspositivitythatmakesthisnotion been the proof of the positivity of the total gravitational energy comma both at spatial and null infinity period not only important ( because of its theoretical It is precisely its positivity that makes this notion not only important open parenthesis because of its theoretical sig\nnifiocianndceen)t, b1ut\aqlusoada uIsnetfurlotdouolctiinotnhe everyday practice of working relativists . This success significance closing parenthesis comma but also a us eful tool in the everyday practice of working relativists period .... This success inspired the more ambitious claim to associate energy ( or rather energy - momentum and , ultimately , inspired the more ambitious claim to associate energy open parenthesis or rather energy hyphen momentum and comma ultimately comma an\gnuolairnmdeomntentOuvmerastwheell )latostext3en5deydea,rbsut,finoitnee ,ospfactehteimegdroemataiensst, ia.cehi.e,vaetmtheentqusasiin- locclaalslseivcela.l Ogbevnioeurslayl, trheelativity has certainly angular momentum as well closing parenthesis to extended comma but finite comma spacetime domains comma i period e period comma at the quasi hyphen local quabsie-enlo ctahlequpanrtoitoifes ocofultdhperovpidoesiatmivoiretydetoafiledthcehartaocttearilzatgiornavofitthaetisotanteaslofetnheerggryavi,tatbioontahl ‘afiteldsp’ tahtainatlheangldobanlull infinity . level period onesIt, soisthepyr(ectoigseetlhyer wititsh mpoorseigteinveirtalyqutahsia-tlomcaakles this notion not only important ( because of its theoretical Obviously comma the quasi hyphen lo cal quantities could provide a more detailed characterization of the states observables ) would be interesting in their own right . of the gravitational quoteleft field quoteright than the global ones comma so they open parenthesis together with more general quasi hyphen lo cal \nMooirnedoveenrt, fisnidginngiafnicaapnprcoepri)ate,nobtuiotnaoflsenoeragyu-smoemfuenltutmoaonldianngutlhaermeovmeernytduamywopurladcbteicimepoorftanwtofrrokmintgherpeolinattivists . \hfill This success observables closing parenthesis would be interesting in their own right period of view of applications as well . For example , they may play a cen - tral role in the proof of the full Penrose inequality ( as Moreover comma finding an appropriate notion of energy hyphen momentum and angular momentum would th\eynohianvdeeanlrteadiynspplaiyreeddintthheepmrooorfeofambitious claim to associate energy ( or rather energy − momentum and , ultimately , be important from the point of view of applications as well period .. For example comma they may play a cen hyphen the Riemannian version of this inequality ) . The correct , ultimate formulation of black hole ther - modynamics should tral role in the proof of the full Penrose inequality open parenthesis as they have already played in the proof of pr\onbaobinlydbeenbtasaedngounlqaurasmi -omloencatullmy daefisnewdeinllter)naltoeneerxgyte,nednetrdopy, ,bauntgulfairnmitome en,tusmpa,ceetcti.meIndonummaeirniscal,cailcu.lateion.s , at the quasi − local level . the Riemannian version of this inequality closing parenthesis period .. The correct comma ultimate formulation of black hole ther hyphen , coOnsbervvieoduqsulyant,itietsh(eorquatalseias−t thloosecfoarlwqhiuchanbatlit-ies could provide a more detailed characterization of the states modynamics should probably be based on quasi hyphen lo cally defined internal energy comma entropy comma angular ancoefequthateiongsrcaavnibteatdieorinveadl )‘arefiueselddto’cotnhtaronl tthheeerrgolrosb.al Hoonweesver,, isnosutchhecyalc(ulattoiognestahlelrthewdiotmhaminsoraeregfienniteer,ail. quasi − lo cal momentum comma etc period .. In numerical calculations comma conserved quantities open parenthesis or at least those for which bal hyphen e . , quasi - local . Therefore , a solid theoretical foundation of the quasi - lo cal conserved quantities is needed . ance equations can be derived closing parenthesis are used to control the errors period .. However comma in such calculations all \nHooiwnedveern,t ocbosnetrravraybtloesthe)hwigohuelxdpebcetatiionntseorfetshtein1g98i0ns t,hfienidringowann arpipgrhotpria.te quasi - lo cal notion of energy - the domains are finite comma i period e period comma quasi hyphen local period Therefore comma a solid theoretical foundation of the quasi hyphen lo cal momentum has proven to be surprisingly difficult . Nowadays , the state of the art is typically postmodern : although there conserved quantities is needed period arMe soerveeoravlerpro,mifsiinngdianndg uasnefual psupgrgoesptrioinaste, wneontoitononloyfhaevneenrogyult−immatoem,engteunmeralalyndaccaenpgteudlaerxpmreossmioenntfuomr thweoeunldergy - However comma .. contrary to the high expectations of the 1 980 s comma finding an appropriate quasi hyphen lo cal mombeentiummpaonrdtaensptecfiraollmy fotrhtehepanogiunltaromfomveinetwumo,fbuatptphelriecaistniootnesvenasa cwoneslelns.us\inquthaedreFlaotrivietyxacmompmleun,itytohneygenmearayl play a cen − notion of energy hyphen momentum has proven to be surprisingly difficult period Nowadays comma the state of the art is quetstrioanls (rfoolreexianmptlehe, wphartoodof woefmtehane byfuelnlergPye-nmroosmeenitunmeq?ualjiutsyt a(geanseratlheexyprehsasivoen caonlrtaeiandinyg aprblaityraerdy fiunncttiohnes proof of typically postmodern : although there are several promising and useful suggestions comma we not only have , or rather a definite one , free of any ambiguities , even of additive constants ) , or on the list of the criteria of reasonableness no ultimate comma generally accepted expression for the energy hyphen momentum and especially for the angular of\snucohinedxpernetssiotnhse. RiTemheavnanriioauns svueggressitoionnsoafre btahsiesd oinndeiqffueraelnittpyhil)oso.ph\iqesu/adapTprhoeachceosrarnedcgtive, duiffletriemntarteesulftosrinmtuhleation of black hole ther − momentum comma but there is not even a consensus in the relativity community on general questions open parenthesis for sammeosdityunataimonic.sAsphpoaruelndtlyp,rtohbeaidbelays abned sbuacsceedssesonof oqnueacsoins−trulcotioncahlalvye vdereyfilintetdle ininfluteenrcneaoln oetnheerrgcyons,truecntitornosp.y , angular example comma what do we mean by energy hyphen momentum ? .. j ust a general expression containing arbitrary mTohmeeanimtumofth,eeptrecsen.t\pqapueardisI,nthenreufmoreer,itcwaolfolcda.lcuFliarsttio,tnosco,llecctoannsderrevveiedwqthueavnartiiotuisesspec(ifiocrsugagtestlieonasst, tahnods,e for which bal − functions comma or rather a definite one comma free of any ambiguities comma even of additive constants closing parenthesis comma or on the second , to st imulate the interaction between the different approaches by clarifying the general , potentially - common points list of the criteria of reasonableness of such expressions period .. The various suggestions are based on , \nisosiunedseanndt qaunescteionesq.uatTiohnuss ,canwebweandteedrinvoetdon)lyatorewruitseeda ‘twohoc-onditdro-lwhtahte’ reervireowrs, bu.t\tqoucaodnceHnotrwateeveorn t,hein such calculations all different philosophies slash approaches and give different results in the same situation period Apparently comma the undtehrsetanddoimngaionfsthearbeasifciqnuietsteion,s (isu.cheas.wh,yqshuoausldi t−helgoracvaitlat.ionTalheenreerfgoyr-emo,maentsuomliadndtahnegoulraertmicomalentfuomun,doart,ion of the quasi − lo cal ideas and successes of one construction have very lit tle influence on other constructions period morceognesneerravlelyd, quaanyntoibtsieervsabilesofnteheedegrdavi.tational ‘ field ’ , be necessarily quasi - lo cal ) and ideas behind the various The aim of the present paper is comma therefore comma twofold period .. First comma to collect and review the various specific constructions . Consequently , one third of the present review is devoted to these general questions . We review specific suggestions comma .. and comma second comma to st imulate the interaction between the different approaches thHeoswpeecvifiecr co,n\stqruucatdioncsoanntdrathreyirtporoptehretieshiognhly ienxtpheecsteacotniodnpsartof, thaned1in9t8h0e thsird, pfairntdwiengdisacunssavperpyrobrpierfliaytseomqeuasi − lo cal by clarifying the general comma .. potentially hyphen common points comma .. issues and questions period .. Thus comma .. we wanted ( pontoenttiioanl )oapfpleicnaetriognys o−f tmheomquenatsuim- lohcaasl qpuraonvtietinest.o bAelthsouurghprtihsisinpgaplyer isdaiftfhiecaurtlta re.viNewowoafdkanyoswn,antdhepusbtliashteed of the art is not only to write a quoteleft who hyphen did hyphen what quoteright review comma but to concentrate on the understanding of the basic resutltysp,iwcealbleylievpeotshtamtoitdecornntai:nsasletvheoraulgnhewtehleemreentasr,eobsseervvaetrioanls ,psruogmgeisstiionngs eatncd. useful suggestions , we not only have questions open parenthesis such as why should the gravitational energy hyphen momentum and angular momentum comma or comma nSourpurilstiinmglyateenou,ghg,emnoesrtaolflytheaidcecaespatneddcoenxcepprtesstshiaotnappfeoarrinthceonneencetirognyw−ithmthoemgernatvuim-taantidonaelsepneercgiya-llmyomfeonrtumthe angular more generally comma .. any observable of the gravitational quoteleft field quoteright comma be necessarily quasi hyphen lo cal closing parenthesis .. and ideas andmaonmgeunlaturmmom, ebnututmtchaenrbee iinstrondoutcedevinen( aandchoennsceencasnusbeiunndtehre- relativity community on general questions ( for behind the various specific constructions period Consequently comma one third of the present review is devoted stweox−amtopnel−eo−,pewriohdao−tcodmomawwe−mdefa−nerbry−ome−nee−rvgiy−−parmenormigehntteu−mth$−w?eh$−t\eq−uahdeoryjofumstattaerfigeeldnseirnaMl inekxopwrseksisspioacneticmoentaining arbitrary to these general questions period .. We review the specific constructions and their properties only in the . Tfhuusnescc,otniindoSpneascrtti,ocnomoBrm−arta.h.traheneedcrominmatah−deeetlfhiiirn−diitpne−arnttow−neefdr−i,sacou−fsrnste−vedeer−youcfberi−aenflRyyo−saotmhm−ebseoi−pgeenunfi−tpiaenreseont−,heltseiv−sepdnoot−eonftpira−alndcold−oosifitn−igcvpeedarucernoetnhtshetasiatsnwatpespli)cat,ionosrofotnhethe willlaipsptlyotof gtrahveitycirniSteecrtiioan qoufasire-alosocanlaebnelregnye-sms oomfenstuumchanedxapnrgeuslasriomnosme.ntu\mquoafdthTehmeatvtearrfiieoludssansdugdigsceusstsiothnesir are based on quasi hyphen lo cal quantities period .. Although this paper is at heart a review of known and published results comma propdi-fefretireesn. t Tphheiplohsiloospohphieysof/quaaspip-rlooaccalhiteysinagnednergailvreeladtivifitfyewreilnl bte dreemsounlsttsratiend inthMeinskaomwsekissiptauceattimioenwh.ereAtpheparently , the we believe that it contains several new elements comma observations comma suggestions etc period enerigdye-ams oamnedntusmucacnedsasnegsulaorfmoonmeenctuomnsotfrtuhectmioatnterhfiaevldes avreertyrealtietd qtulaesi -ilnofclaulleyn.cTeheonnwoetthurenrtocothnesdtirffiuccutltiioesnosf . Surprisingly enough comma most of the ideas and concepts that appear in connection with the gravi hyphen gravitationalenergy-momentumandangularmo-mentum,andweclarifywhythegravitationalobservablesshouldnecessarily t ational energy hyphen momentum and angular momentum can be introduced in open parenthesis and hence can be under hyphen beTqhueasaii-mloocafl .theThperesent paper is , therefore , twofold . \quad First , to collect and review the various s two-t sub one-o-period o-comma w-d f-e sub r r-o m-e-v i-parenright e-t h-w sub e h-t e-h sub eory of matter fields in Minkowski spacetime period Thus comma specific suggestions , \quad and , second , to st imulate the interaction between the different approaches in Section by clarifying the general , \quad potentially − common points , \quad issues and questions . \quad Thus , \quad we wanted B-three sub comma-e l i-i sub n-n t-f sub r-a o-n t-d sub e-u c e-R sub o-t h-s e-e n f-n sub e o-l t-d o-p to the power of r-n o-o f-c edure that we will apply to not only to write a ‘ who − did − what ’ review , but to concentrate on the understanding of the basic gravity in Section Living Reviews in Relativity questions ( such as why should the gravitational energy − momentum and angular momentum , or , quasi hyphen lo cal energy hyphen momhtenttpum:an/d/anwguwlar mwom.entlumiovfithnegmreavtteirefiewldssan.d doirsgcus/slthrerir-pr2o0p09hy-ph4en erties period .. The philosophy of quasi hyphen lo cality in general relativity will be demonstrated in Minkowski \noindent more generally , \quad any observable of the gravitational ‘ field ’ , be necessarily quasi − lo cal ) \quad and ideas spacetime where the energy hyphen momentum and angular momentum of the matter fields are treated behind the various specific constructions . Consequently , one third of the present review is devoted quasi hyphen lo cally period Then we turn to the difficulties of gravitational energy hyphen momentum and angular mo hyphen to these general questions . \quad We review the specific constructions and their properties only in the mentum comma and we clarify why the gravitational observables should necessarily be quasi hyphen lo cal period .. The second part , \quad and in the third part we discuss very briefly some ( potential ) applications of the hline quasi − lo cal quantities . \quad Although this paper is at heart a review of known and published results , Living Reviews in Relativity we believe that it contains several new elements , observations , suggestions etc . ht tp : slash slash w w .. w period l i vi ngrev i e w s period org slash lrr hyphen 2009 hyphen 4 Surprisingly enough , most of the ideas and concepts that appear in connection with the gravi − t ational energy − momentum and angular momentum can be introduced in ( and hence can be under − \noindent s $ two−t { one−o−period o−comma } w−d f−e { r r−o m−e−v } i−parenright e−t h−w { e } h−t e−h { eory }$ of matter fields in Minkowski spacetime . Thus , in Section $ B−three { comma−e }$ l $ i−i { n−n } t−f { r−a o−n t−d { e−u c }} e−R { o−t h−s e−e n } f−n { e } o−l t−d o−p ˆ{ r−n o−o f−c }$ edure that we will apply to gravity in Section quasi − lo cal energy − momentum and angular momentum of the matter fields and discuss their prop − erties . \quad The philosophy of quasi − lo cality in general relativity will be demonstrated in Minkowski spacetime where the energy − momentum and angular momentum of the matter fields are treated quasi − lo cally . Then we turn to the difficulties of gravitational energy − momentum and angular mo − mentum , and we clarify why the gravitational observables should necessarily be quasi − lo cal . \quad The \[ \rule{3em}{0.4pt} \] \hspace∗{\ fill }Living Reviews in Relativity \hspace∗{\ fill }ht tp : / / w w \quad w . l i vi ngrev i e w s . org / lrr − 2009 − 4