General covariance and general relativity 791 General covariance and the foundations of general relativity: eight decades of dispute John D Norton hep—rtment of ristory —nd €hilosophyD …niversity of €itts˜urghD €itts˜urghD €e ISPTHD …ƒe Abstract iinstein oered the prin™iple of gener—l ™ov—ri—n™e —s the fund—ment—l physi™—l prin™iple of his gener—l theory of rel—tivityD —nd —s responsi˜le for extending the prin™iple of rel—tivity to —™™eler—ted motionF „his view w—s disputed —lmost immedi—tely with the ™ounterE™l—im th—t the prin™iple w—s no rel—tivity prin™iple —nd w—s physi™—lly v—™uousF „he dis—greeE ment persists tod—yF „his —rti™le reviews the development of iinstein9s thought on gener—l ™ov—ri—n™eD its rel—tion to the found—tions of gener—l rel—tivity —nd the evolution of the ™ontinuing de˜—te over his viewpointF „his review w—s re™eived in m—r™h IWWQF 792 J D Norton Contents IF sntrodu™tion UWR PF „he ˜—™kground of spe™i—l rel—tivity UWT PFIF vorentz ™ov—ri—n™e —nd the rel—tivity of inerti—l motion UWT PFPF winkowski9s introdu™tion of geometri™—l methods UWT PFQF gov—ri—n™e versus inv—ri—n™e in spe™i—l rel—tivity UWU QF iinstein9s development of gener—l rel—tivity UWU QFIF „he e—rly ye—rs IWHUEIWIPX prin™iple of equiv—len™e —nd the rel—tivity of inerti— UWV QFPF „he –intwurf9 theory IWIPEIWISX gener—l ™ov—ri—n™e g—ined —nd lost UWW QFQF „he hole —rgumentX gener—l ™ov—ri—n™e ™ondemned VHI QFRF iinstein9s IWIT —™™ount of the found—tions of gener—l rel—tivityX gener—l ™ov—ri—n™e reg—ined VHP QFSF „he pointE™oin™iden™e —rgument VHR QFTF „he qottingen defense of gener—l ™ov—ri—n™e VHS QFUF iinstein9s three prin™iples of IWIV VHT QFVF w—™h9s prin™iple fors—ken VHV QFWF iinstein9s o˜je™tion to —˜solutes VHW RF „he f—vour—˜le textE˜ook —ssimil—tion of iinstein9s viewX fr—gment—tion —nd mut—tion VII RFIF iinstein9s prin™iple of equiv—len™e —s — ™ov—ri—n™e prin™iple —nd its l—ter misrepresent—tion VIP RFPF „he e—rly ye—rsX IWITEIWQH VIR RFQF „he le—n ye—rsX IWQHEIWTH VIS RFRF ‚e˜irthX IWTHEIWVH VIS RFSF ‚e™ent ye—rs sin™e IWVH VIT SF ss gener—l ™ov—ri—n™e physi™—lly v—™uousc VIU SFIF urets™hm—nn9s o˜je™tionX the pointE™oin™iden™e —rgument turned —g—inst iinstein VIU SFPF iinstein9s reply VIW SFQF qener—lly ™ov—ri—nt formul—tions of xewtoni—n me™h—ni™s VPH SFRF eutom—ti™ gener—l ™ov—ri—n™eX ™oordin—te free formul—tion VPI SFSF v—ter responses to urets™hm—nn9s o˜je™tion VPS TF ss the requirement of gener—l ™ov—ri—n™e — rel—tivity prin™iplec VPW TFIF his—n—logies with the prin™iple of rel—tivity of spe™i—l rel—tivity VPW TFPF ‚el—tivity prin™iples —s symmetry prin™iples VQI TFQF goordin—te systems versus fr—mes of referen™e VQS TFRF ‚el—tivity prin™iples —nd the equiv—len™e of fr—mes VQU UF qener—l rel—tivity without prin™iples VRI UFIF qener—l rel—tivity without gener—l rel—tivity VRI UFPF „he prin™iple of equiv—len™e —s the fund—ment—l prin™iple VRP UFQF gh—llenges to the prin™iple of equiv—len™e VRP General covariance and general relativity 793 VF ilimin—ting the —˜solute VRQ VFIF enderson9s —˜solute —nd dyn—mi™—l o˜je™ts VRQ VFPF ‚esponses to enderson9s viewpoint VRT VFQF xo gr—vit—tion—l eldEno sp—™etime points VRU VFRF ‡h—t —re —˜solute o˜je™ts —nd why should we despise themc VRU WF found—ries —nd puzzles VRW WFIF ss gener—l ™ov—ri—n™e too gener—lc yr not gener—l enoughc VRW WFPF „he iinstein puzzle VSH IHF gon™lusion VSP 794 J D Norton 1. Introduction sn xovem˜er IWISD iinstein ™ompleted his gener—l theory of rel—tivityF elmost eight de™—des l—terD we univers—lly —™™l—im his dis™overy —s one of the most su˜lime —™ts of hum—n spe™E ul—tive thoughtF roweverD the question of pre™isely wh—t iinstein dis™overed rem—ins un—nsweredD for we h—ve no ™onsensus over the ex—™t n—ture of the theory9s found—tionsF ss this the theory th—t extends the rel—tivity of motion from inerti—l motion to —™™eler—ted motionD —s iinstein ™ontendedc yr is it just — theory th—t tre—ts gr—vit—tion geometri™—lly in the sp—™etime settingc ‡hen iinstein ™ompleted his theoryD his own —™™ount of the found—tions of the theory w—s —dopted ne—rly univers—llyF roweverD —mong the voi™es welE ™oming the new theory were sm—ll murmurs of dissentF yver the ˜rief moments of history th—t followedD these murmurs grew until they —re now some of the loudest voi™es of the ™ontinuing de˜—teF sn —ny logi™—l systemD we h—ve gre—t freedom to ex™h—nge theorem —nd —xiom without —ltering the system9s ™ontentF „hus we need no longer formul—te iu™lide—n geometry with ex—™tly the denitions —nd postul—tes of iu™lid or use pre™isely xewton9s three l—ws of motion —s the found—tions of ™l—ssi™—l me™h—ni™sF roweverD some two millenni— —fter iu™lid —nd three ™enturies —fter xewtonD we still nd their postul—tes —nd l—ws within our systemsD now possi˜ly —s theorems —nd sometimes even in — wording rem—rk—˜ly ™lose to the origin—lF „he ™ontinuing dis—greement over the found—tions of iinstein9s theory extends well ˜eyond su™h —n orderly exp—nsion of our underst—nding of — theory9s found—tionsF st is f—r more th—n — squ—˜˜le over the most perspi™—™ious w—y to reorg—nize postul—te —nd theorem or to ™l—rify ˜rief moments of v—guenessF „he voi™es of dissent pro™l—im th—t iinstein w—s mist—ken over the fund—ment—l ide—s of his own theory —nd th—t the ˜—si™ prin™iples iinstein proposed —re simply in™omp—ti˜le with his theoryF w—ny newer texts m—ke no mention of the prin™iples iinstein listed —s fund—ment—l to his theoryY they —ppe—r —s neither —xiom nor theoremF et ˜estD they —re re™—lled —s ide—s of purely histori™—l import—n™e in the theory9s form—tionF „he very n—me –gener—l rel—tivity9 is now routinely ™ondemned —s — misnomer —nd its use often ze—lously —voided in f—vour ofD s—yD ’iinstein9s theory of gr—vit—tion4F ‡h—t h—s ™ompli™—ted —n e—sy resolution of the de˜—te —re the —lter—tions of iinstein9s own position on the found—tions of his theoryF et dierent times of his lifeD he sought these found—tions in three prin™iples —nd with v—rying emph—sisF „hey were the prin™iple of equiv—len™eD w—™h9s prin™iple —nd the prin™iple of rel—tivityF fy his own —dmission @iinstein IWIVAD he did not —lw—ys distinguish ™le—rly ˜etween the l—st twoF eg—inD he lost ™ompletely his enthusi—sm for w—™h9s prin™ipleD —˜—ndoning it unequivo™—lly in his l—ter lifeF „he re™eption —n development of iinstein9s —™™ount in the liter—ture h—s ˜een —nything ˜ut — gr—™eful evolutionF st h—s ˜een more — pro™ess of un™ontrolled mut—tionD fr—gmenE t—tion —nd even disintegr—tionF „he prin™iple of equiv—len™e took root in so m—ny v—ri—nt forms th—t enderson —nd q—utre—u @IWTWD pITSTA eventu—lly l—mented th—t there —re –—lE most —s m—ny formul—tions of the prin™iple —s there —re —uthors writing —˜out itF9 „his dissip—tion is —t le—st p—rti—lly fuelled ˜y skepti™—l —tt—™ks on the prin™iple su™h —s ƒynge9s @IWTHD p ixA f—mous ™ompl—int th—t he h—s never ˜een —˜le to nd — version of the prin™iple th—t is not f—lse or trivi—lF „he lo™us of gre—test ™ontroversy h—s ˜een —t the ™ore of iinstein9s interpret—tionD the prin™iple of rel—tivityF hoes the gener—l theory extend the prin™iple of rel—tivity to —™™eler—ted motion —nd is this extension ™—ptured ˜y the gener—l ™ov—ri—n™e of its l—wsc st is routinely —llowed th—t the spe™i—l theory of rel—tivity s—tises the prin™iple of rel—tivity of inerti—l motion simply ˜e™—use it is vorentz ™ov—ri—ntX its l—ws rem—in un™h—nged in form General covariance and general relativity 795 under — vorentz tr—nsform—tion of the sp—™e —nd time ™oordin—tesF xow iinstein9s gener—l theory is gener—lly ™ov—ri—ntX its l—ws rem—in un™h—nged under —n —r˜itr—ry tr—nsform—tion of the sp—™etime ™oordin—tesF hoes this form—l property —llow the theory to extend the rel—tivity of motion to —™™eler—ted motionc …ntil re™ent de™—desD the m—jority of expositions of gener—l rel—tivity —nswered yes —nd some still doF es e—rly —s IWIUD urets™hm—nn @IWIUA —rgued th—t gener—l ™ov—ri—n™e h—s no re—l physE i™—l ™ontent —nd no ™onne™tion to —n extension of the prin™iple of rel—tivityF ‚—therD the nding — gener—lly ™ov—ri—nt formul—tion of — theory —mounts essenti—lly to — ™h—llenge to the m—them—ti™—l ingenuity of the theoristF ƒkepti™—l sentiments su™h —s these drove — dissident tr—dition th—t h—s grown from — minority in urets™hm—nn9s time to one of the domin—nt tr—ditions —t presentF st h—d derived further support from the development of more sophisti™—ted m—them—ti™—l te™hniques th—t —re now routinely used to give gener—lly ™ov—ri—nt formul—tions of essenti—lly —ll ™ommonly dis™ussed sp—™etime theoriesD in™luding spe™i—l rel—tivity —nd xewtoni—n sp—™etime theoryF pin—llyD to m—nyD iinstein9s st—tements of his views seemed too simple or —˜˜revi—ted to st—nd without further el—˜or—tion or rep—irY where—s their —t reje™tion ˜y the skepti™s seemed too e—syF „hus mu™h energy h—s ˜een devoted to nding w—ys in whi™h the gener—l ™ov—ri—n™e of iinstein9s theory ™—n ˜e seen to ˜e distin™tive even in ™omp—rison with the gener—lly ™ov—ri—nt formul—tons of spe™i—l rel—tivity —nd xewtoni—n sp—™etime theoryF „he ˜est developed of these —ttempts is due to enderson @IWTUA —nd is ˜—sed on the distin™tion of —˜solute from dyn—mi™—l o˜je™tsF qener—l rel—tivity s—tises enderson9s –prin™iple of gener—l inv—ri—n™e9 whi™h ent—ils th—t the theory ™—n employ no nonEtrivi—l —˜solute o˜je™tsF „his prin™iple is oered —s — ™le—rer st—tement of iinstein9s re—l intentions —nd —s giving — pre™ise interpret—tion of iinstein9s repe—ted dis—vow—l of the —˜solutes of xewton9s sp—™e —nd timeF „he purpose of this —rti™le is to review the development of iinstein9s views on gener—l ™ov—ri—n™eD their rel—tion to the found—tions of gener—l rel—tivity —nd the evolution of the ™ontinuing de˜—te th—t spr—ng up —round these viewsF ƒe™tion P —nd Q will review the development of iinstein9s viewsF ƒe™tion R will outline the w—ys in whi™h —ttempts were m—de to re™eive —nd —ssimil—te iinstein9s views in — f—vour—˜le m—nnerF ƒe™tion S will review urets™hm—nn9s f—mous o˜je™tionD iinstein9s response —nd the diverse w—ys in whi™h ˜oth were re™eived in the liter—tureF st in™ludes dis™ussion of modern geometri™—l methods th—t ensure —utom—ti™ gener—l ™ov—ri—n™eF ƒe™tion T reviews the development of the ™h—r—™teriz—tion of — rel—tivity prin™iple —s — symmetry prin™iple r—ther th—n — ™ov—ri—n™e prin™ipleF ƒe™tion U explores the tr—dition of exposition of gener—l rel—tivity th—t simply ignores the entire de˜—te —nd m—kes no mention of prin™iples of gener—l rel—tivity —nd of gener—l ™ov—ri—n™eF ƒe™tion V develops enderson9s theory of —˜solute —nd dyn—mi™—l o˜je™ts —s it rel—tes to iinstein9s viewsF ƒe™tion W ex—mines po™k9s —nd erzelies propos—ls for —lter—tions to the ™ov—ri—n™e of gener—l rel—tivity —nd gives —n histori™—l expl—n—tion of why so m—ny of iinstein9s pronoun™ements on ™oordin—tes —nd ™ov—ri—n™e —re puzzling to modern re—dersF sn the time period ™overed in this review —rti™leD the m—them—ti™—l methods used in rel—tivity theory evolved from — ™oordin—te ˜—sed ™—l™ulus of tensors to — ™oordin—te freeD geometri™ —ppro—™hF „he m—them—ti™—l l—ngu—ge —nd sensi˜ilities used in v—rious st—ges of the —rti™le will m—t™h those of the p—rti™ul—r su˜je™t under reviewF „he —ltern—tive of tr—nsl—ting everything into — single l—ngu—ge would h—rmfully distort the su˜je™t @see se™tion WFPAF 796 J D Norton 2. The background of special relativity 2.1. Lorentz covariance and the relativity of inertial motion iinstein9s @IWHSA ™ele˜r—ted p—per on spe™i—l rel—tivity ˜rought the notion of the ™ov—ri—n™e of — theory to prominen™e in physi™s —nd introdu™ed — theme th—t would ™ome to domin—te iinstein9s work in rel—tivity theoryF „he proje™t of the p—per w—s to restore the prin™iple of rel—tivity of inerti—l motion to ele™trodyn—mi™sF sn its then ™urrent st—teD the theory distinguished — preferred fr—me of restD —lthough th—t fr—me h—d eluded —ll experiment —nd even f—iled to —ppe—r in the o˜serv—tion—l ™onsequen™es of ele™trodyn—mi™s itselfF iinstein9s renowned solution w—s not to modify ele™trodyn—mi™sD ˜ut the ˜—™kground sp—™e —nd time itselfF re devised — theory in whi™h inerti—l fr—mes of referen™e were rel—ted ˜y the vorentz tr—nsform—tionF sf —n inerti—l fr—me h—s g—rtesi—n sp—ti—l ™oordin—tes @x; y; zA —nd time t —nd — se™ond fr—me moving —t velo™ity v in the x dire™tion h—s sp—ti—l ™oordin—tes @; ; A —nd time ™oordin—te D thenD under the vorentz tr—nsform—tionD  a @x tA  a @t vx=c2A  a y  a z @IA where a @I v2=c2A 1=2 —nd c is the speed of lightF ritherto ™l—ssi™—l theory h—d in ee™t employed wh—t w—s shortly ™—lled @˜yD for ex—mpleD v—ue @IWIID pQAA the q—lileiE tr—nsform—tionF  a x tA  a t  a y  a z ƒele™ting suit—˜le tr—nsform—tion l—ws for the eld —nd other qu—ntitiesD iinstein w—s —˜le to show th—t the l—ws of ele™trodyn—mi™s rem—ined un™h—nged under the vorentz tr—nsE form—tionF „h—t isD they were vorentz ™ov—ri—ntF „hereforeD within the sp—™e —nd time of spe™i—l rel—tivityD ele™trodyn—mi™s ™ould no longer pi™k out —ny inerti—l fr—me of referE en™e —s preferredF i—™h inerti—l fr—me w—s fully equiv—lent within the l—ws of the theoryF enything s—id —˜out one ˜y the l—ws of ele™trodyn—mi™s must —lso ˜e s—id of —ll the restF ile™trodyn—mi™s w—s now ™omp—ti˜le with the rel—tivity of inerti—l motionF ‡ith the ex—mple of ele™trodyn—mi™s —s its p—r—digmD the t—sk of ™onstru™ting — spe™i—l rel—tivisti™ version of — physi™—l theory redu™ed essenti—lly to formul—ting its l—ws in su™h — w—y th—t they rem—ined un™h—nged under vorentz tr—nsform—tionF „hus iinstein9s @IWHSD se™tion IHA origin—l p—per pro™eeded to formul—te — modied me™h—ni™s for slowly —™™elerE —ted ele™trons with this propertyF „hermodyn—mi™s soon —lso re™eived some of its e—rliest rel—tivisti™ reformul—tions in the s—me m—nner @see iinstein IWHUD p—rt s†D for ex—mpleAF „he lesson of iinstein9s IWHS p—per w—s simple —nd ™le—rF „o ™onstru™t — physi™—l theory th—t s—tised the prin™iple of rel—tivity of inerti—l motionD it w—s su™ient to ensure th—t it h—d — p—rti™ul—r form—l propertyX its l—ws must ˜e vorentz ™ov—ri—ntF vorentz ™ov—ri—n™e ˜e™—me synonymous with s—tisf—™tion of the prin™iple of rel—tivity of inerti—l motion —nd the whole theory itselfD —s iinstein @IWRHDpQPWA l—ter de™l—redX „he ™ontent of the restri™ted rel—tivity theory ™—n —™™ordingly ˜e summ—rized in one senten™eX —ll n—tur—l l—ws must ˜e so ™onditioned th—t they —re ™ov—ri—nt with respe™t to vorentz tr—nsform—tionsF 2.2. Minkowski's introduction of geometrical methods sn iinstein9s h—ndsD vorentz ™ov—ri—n™e w—s — purely —lge˜r—i™ propertyF ƒp—™e —nd time ™oordin—tes wereD in ee™tD v—ri—˜les th—t tr—nsformed —™™ording to ™ert—in formul—eF rerE m—nn winkowski @IWHVD IWHWA w—s responsi˜le for introdu™ing geometri™ methods —nd thinking into rel—tivity theoryF re expl—ined the ˜—™kground to his —ppro—™h in his more General covariance and general relativity 797 popul—r @IWHWA le™tureF st —mounted to —n inspired ˜ut essenti—lly str—ightforw—rd —ppliE ™—tion of then ™urrent ide—s in geometryF winkowski9s ™olle—gue —t qottingenD pelix uleinD h—d ˜rought — fertile order to the world of IWth ™entury geometryF „h—t world w—s ˜eginE ning to fr—gment —fter the dis™overy th—t geometry did not h—ve to ˜e iu™lide—nF sn his f—mous Erlangen progr—mD ulein @IVUPA proposed ™—tegorizing the new geometries ˜y their ™h—r—™teristi™ groups of tr—nsform—tionsF iu™lide—n geometryD for ex—mpleD w—s ™h—r—™terE ized ˜y the group of rot—tionsD tr—nsl—tions —nd ree™tionsF „he entities of the geometry were the inv—ri—nts of these tr—nsform—tionsF winkowski pointed out th—t geometers h—d ™on™entr—ted on the ™h—r—™teristi™ tr—nsE form—tions of sp—™eF fut they h—d ignored the groups of tr—nsform—tions —sso™i—ted with me™h—ni™sD those th—t ™onne™ted v—rious inerti—l st—tes of motionF winkowski pro™eeded to tre—t these groups in ex—™tly the s—me w—y —s the geometri™ groupsF sn p—rti™ul—r he ™onE stru™ted the geometry —sso™i—ted with the vorentz tr—nsform—tionF „o ˜eginD it w—s not the geometry of — spaceD ˜ut of — spacetimeD —nd the notion of sp—™etime w—s introdu™ed into physi™s —lmost —s — perfun™tory ˜yEprodu™t of the Erlangen progr—mF woreover he found the sp—™etime h—d the hyper˜oli™ stru™ture now —sso™i—ted with — winkowski sp—™etimeF prom this geometri™ perspe™tiveD the formul—tion of — theory th—t s—tised the prin™iple of rel—tivity ˜e™—me trivi—lF yne merely needed to formul—te the theory in terms of the geoE metri™ entities of the sp—™etime|in ee™t the v—rious types of sp—™etime ve™tors winkowski h—d dened|—nd the theory would ˜e —utom—ti™—lly vorentz ™ov—ri—ntF „hus winkowski @IWHVD —ppendixYIWHVD se™tion †A ™ould write down the prin™iple of rel—tivityD for the theory w—s ™onstru™ted purely geometri™—llyF „husD in his exposition of fourEdimension—l ve™tor —lge˜r— —nd —n—lysisD ƒommerfeld @IWIHDURWA ™ould st—teX e™™ording to winkowskiD —s is well knownD one ™—n formul—te the ™ontent of the prinE ™iple of rel—tivity —sX only spacetime vectors m—y —ppe—r in physi™—l equ—tionsF F F 2.3. Covariance versus invariance in special relativity „he dieren™e ˜etween iinstein —nd winkowski9s —ppro—™h to the s—me theory —nd even the s—me form—lism is — pol—rity th—t will persist in v—rious m—nifest—tions throughout the whole development of rel—tivity theoryD ˜oth spe™i—l —nd gener—lF iinstein9s emph—sis is on the —lge˜r—i™ properties of the theoryD the equ—tions th—t express its l—ws —nd their ˜eh—viour under tr—nsform—tionD its covarianceF „hus the s—tisf—™tion of the prin™iple of rel—tivity is est—˜lished ˜y often —rduous —lge˜r—i™ m—nipul—tionF „he equ—tions of the theory —re tr—nsformed under the vorentz tr—nsform—tion —nd the resulting equ—tions —re shown to h—ve preserved their formF winkowski9s emph—sis is on the geometri™ properties of the theoryD on those geometri™ entities whi™h rem—in un™h—nged ˜ehind the tr—nsform—E tionsD its invarianceF „hus winkowski ensures s—tisf—™tion of the prin™iple of rel—tivity ˜y quite dierent me—nsF „he only stru™tures —llowed in ™onstru™ting — theory —re sp—™etime inv—ri—ntsF „his restri™tion ensures ™omp—ti˜ility with the prin™iple of rel—tivity —nd th—t its s—tisf—™tion ™—n ˜e settled ˜y inspe™tionF 3. Einstein's development of general relativity ‡hile it m—y h—ve ˜een some ye—rs in prep—r—tionD the spe™i—l theory of rel—tivity ™o—les™ed into its n—l form quite suddenly so th—t iinstein9s rst p—per on the theory rem—ins one of its ™l—ssi™ expositionsF „he development of gener—l rel—tivity w—s f—r slower —nd more t—ngledF iight ye—rs el—psed ˜etween the in™eption —nd ™ompletion of the theoryD during whi™h time iinstein pu˜lished repe—ted reports on the intermedi—te ph—sesD f—lse turns —nd unproven expe™t—tionsF iven —fter the ™ompletion of the theory iinstein9s —™™ount of its 798 J D Norton found—tions ™ontinued to evolveF „he modern im—ge of iinstein9s view of the found—tions of gener—l rel—tivity is dr—wn f—irly h—ph—z—rdly from pronoun™ements th—t were m—de —t diering times in this evolutionF es — result they —re not —lw—ys ™omp—ti˜leF sndeed the pronoun™ements were sometimes —s mu™h expressions of results —nti™ip—ted —s demonE str—tedF por this re—sonD it would ˜e misle—ding to ™onstru™t —ny single edi™e —nd pro™l—im it iinstein9s —™™ount of the found—tions of gener—l rel—tivityF ‚—ther we sh—ll h—ve to tr—™e the evolution of iinstein9s views —s they were el—˜or—ted —nd modied in p—™e with the development of the theoryF sn developing gener—l rel—tivityD iinstein sought to s—tisfy m—ny requirementsF rowever we sh—ll see th—t his eorts were domin—ted ˜y — single themeD ™ov—ri—n™eD —nd they reE du™ed essenti—lly to —n enduring t—skD exp—nding the ™ov—ri—n™e of rel—tivity theory ˜eyond vorentz ™ov—ri—n™eF 3.1. The early years 1907{1912: principle of equivalence and the relativity of inertia „wo ye—rs —fter his ™ompletion of the spe™i—l theoryD iinstein ˜eg—n developing ide—s th—t would ultim—tely le—d him to the gener—l theory of rel—tivityF sn — n—l spe™ul—tive se™tion of — IWHU review —rti™le on rel—tivity theoryD he r—ised the question of whether the prin™iple of rel—tivity ™ould ˜e extended to —™™eler—ted motion @iinstein IWHUD p—rt †AF „he question w—s immedi—tely understood —s —sking whether he ™ould exp—nd the ™ov—ri—n™e group of rel—tivity theoryF peeling un—˜le to t—™kle the gener—l questionD iinstein ™onsidered the simple ™—se of — tr—nsform—tion from —n inerti—l referen™e fr—me of spe™i—l rel—tivity to — referen™e fr—me in uniform re™tiline—r —™™eler—tionF sn the —™™eler—ted fr—me of referen™e — homogeneous inerti—l eld —risesF fe™—use of the key empiri™—l f—™t the the equ—lity of inerti—l —nd gr—vit—tion—l m—ssD iinstein w—s —˜le to identify this eld —s — gr—vit—tion—l eldF re then m—de the postul—te th—t would domin—te the e—rly ye—rs of his work on gr—vit—tionF sn the wording of iinstein @IWIID se™tion IA F F F we —ssume th—t the systems K ‘inerti—l system in — homogeneous gr—vit—tion—l eld“ —nd KH ‘uniformly —™™eler—ted system in gr—vit—tion free sp—™e“ —re ex—™tly equiv—lentD th—t isD F F F we —ssume th—t we m—y just —s well reg—rd the system K —s ˜eing in sp—™e free from gr—vit—tion—l eldsD if we then reg—rd K —s uniformly —™™eler—tedF „his —ssumption soon —™quired the n—me –hypothesis of equiv—len™e9 @iinstein IWIP—D pQSSA —nd then –prin™iple of equiv—len™e9 @iinstein IWIP˜D pRRQAF „hrough itD iinstein gener—ted — novel theory of st—ti™ gr—vit—tion—l elds @iinstein IWHUD p—rt †D IWIIDIWIP—D˜AF sn itD the now v—ri—˜le speed of light pl—yed the role of the gr—vit—tion—l potenti—lY light from — he—vy ˜ody su™h —s the sun would ˜e red shiftedY —nd light gr—zing — he—vy ˜ody su™h —s the sun would ˜e dee™tedF por our purposesD the import—nt point is th—t iinstein s—w in the prin™iple —n extension of the prin™iple of rel—tivityF gontinuing the —˜ove p—ss—ge he o˜servedX „his —ssumption of ex—™t physi™—l equiv—len™e m—kes it impossi˜le for us to spe—k of the —˜solute —™™eler—tion of the system of referen™eD just —s the usu—l theory of rel—tivity for˜ids us to t—lk of the —˜solute velo™ity of — systemF F F „he prin™iple of equiv—len™e formed just one p—rt of iinstein9s —ss—ult on the pro˜lem of extending the prin™iple of rel—tivityF re h—d —lso to —nswer the more gener—l worry th—t —™™eler—tion seemed distinguish—˜le from inerti—l motion ˜y o˜serv—˜le ™onsequen™esD where—s no su™h ™onsequen™es en—˜le us to distinguish inerti—l motion from restF xewton h—d driven home the point in the ƒ™holium to the henitions of fook I of his €rin™ipi— @ITVUAF re noted th—t the —˜solute of rot—tion of w—ter in — ˜u™ket w—s reve—led ˜y the o˜serv—˜le ™urv—ture of the w—ter9s surf—™eF „he inerti— of the w—ter w—s responsi˜le for this ee™tD le—ding it to re™ede from the —xis of rot—tionF General covariance and general relativity 799 iinstein found his —nswer to xewton in his re—ding of irnst w—™hF w—™h @IVWQDpPVRA pointed out th—t —ll th—t w—s reve—led in xewton9s ˜u™ket thought experiment w—s ™orE rel—tion ˜etween the ™urv—ture of the w—ter —nd its rot—tion with respe™t to the i—rth —nd other ™elesti—l ˜odiesF „hus iinstein @IWIP™A w—s delighted to report his IWIP theory ent—iled ™ert—in we—k eld ee™ts th—t promised to ™onvert this ™orrel—tion into — physi™—l inter—™tionD with the rot—tion of the st—rs with respe™t to the w—ter dire™tly ™—using the ™urv—ture of its surf—™eF re found th—t the inerti— of — test m—ss is in™re—sed if it is surE rounded ˜y — shell of inerti—l m—sses —nd th—tD if these s—me m—sses —re —™™eler—tedD they tend to dr—g the test m—ss with itF „hese results r—ised the possi˜ility of —n ide— whi™h he —ttri˜uted @pQWA dire™tly to w—™hX F F F the entire inerti— of — point m—ss is —n inter—™tion with the presen™e of —ll the rem—ining m—sses —nd ˜—sed on — kind of inter—™tion with themF iinstein @IWIQDpIPTIA soon ™—lled this ide— the –hypothesis of the rel—tivity of inerti—F9 gle—rly if — theory ™ould ˜e found th—t implemented this hypothesisD iinstein would h—ve su™™eeded in gener—lizing the prin™iple of rel—tivity to —™™eler—tionF porD in su™h — theoryD the preferred set of inerti—l fr—mes would ™e—se to ˜e —n —˜solute fe—ture of the ˜—™kground sp—™e —nd timeY the disposition of inerti—l fr—mes of referen™e would merely ˜e —n —™™ident of the over—ll distri˜ution of m—tter in the universeF roweverD ˜y the middle of IWIPD iinstein w—s still f—r from su™h — theoryF sn ™on™luding his response to — polemi™—l —ss—ult ˜y w—x e˜r—h—mD iinstein @IWIPdD ppIHTQ{RA des™ri˜ed his proje™t in terms of the exp—nsion of the ™ov—ri—n™e of the ™urrent theory of rel—tivity —nd his hope th—t –the equ—tions of theory of rel—tivity th—t —lso em˜r—™ed gr—vit—tion would ˜e inv—ri—nt with respe™t to —™™eler—tion @—nd rot—tionA tr—nsform—tionsF9 however he ™onfessed th—t –it still ™—nnot ˜e foreseen wh—t form the gener—l sp—™etime tr—nsform—tion equ—tions ™ould h—veF9 „he iinstein who wrote these words in tuly IWIP h—d not yet foreseen th—t his n—me would ˜e irrevo™—˜ly —sso™i—ted with — gener—lly ™ov—ri—nt theoryF 3.2. The `Entwurf' theory 1912{1915: general covariance gained and lost ell this ™h—nged with iinstein9s move to uri™h in eugust IWIPF „here he ˜eg—n ™oll—˜E or—ting with the m—them—ti™i—n w—r™el qrossm—nnD — good friend from his student d—ysF qrossm—n dis™overed for iinstein the existen™e of the –—˜solute dierenti—l ™—l™ulus9 1 of ‚i™™i —nd veviEgivit— @IWHIA —nd pointed out th—t this ™—l™ulus would en—˜le iinstein to ™onstru™t — gener—lly ™ov—ri—nt theoryF „he fo™us of this ™—l™ulus w—s the fund—ment—l qu—dr—ti™ dierenti—l form ' a Xn arsdxrdxs @PA r;s=1 whi™h w—s —ssumed to rem—in inv—ri—nt under —r˜itr—ry tr—nsform—tions of the v—ri—˜les x1; : : : ; xnF yf ™ourse the modern re—der immedi—tely —sso™i—tes this form with the inv—riE —nt line element of — nonEiu™lide—n surf—™e of v—ri—˜le ™urv—tureD su™h —s w—s introdu™ed ˜y q—uss —nd developed ˜y ‚iem—nnF roweverD ‚i™™i —nd veviEgivit—9s x1; : : : ; xn were v—ri—˜les —nd not ne™ess—rily geometri™ ™oordin—tesF „hey were —t p—ins to emph—size th—t wh—t w—s then ™—lled innitesim—l geometry w—s just one of m—ny possi˜le —ppli™—tions of their ™—l™ulusF 1„he ‚i™™iEveviEgivit— ™—l™ulus only l—ter —™quired its modern n—me of –tensor ™—l™ulus9 —fter iinstein —nd qrossm—n @IWIQA ren—med —ll of ‚i™™i —nd veviEgivit—9s –™ontr—v—ri—nt —nd ™ov—ri—nt systems9 —s –tensors9 there˜y extending the formerly r—ther restri™ted ™omp—ss of the term –tensorF9 ƒee xorton @IWWPD —ppendixAF 800 J D Norton es l—te —s IWIPD iinstein h—d not —dopted the fourEdimension—l methods of winkowskiD even though these methods h—d —lre—dy found their rst text ˜ook exposition @v—ue IWIIAF iinstein9s IWIP st—ti™ gr—vit—tion—l theory h—d ˜een developed using essenti—lly the s—me m—them—ti™—l te™hniques —s his IWHS spe™i—l rel—tivity p—perF „hus it is —n odd quirk of history th—tD when iinstein did n—lly immerse himself in the fourEdimension—l sp—™etime —ppro—™hD he turned to exploit — ™—l™ulus whose ™re—tors sought to skirt its geometri™ interpret—tion in f—vour of — ˜ro—der interpret—tionF iinstein —nd qrossm—nn pu˜lished the results of their joint rese—r™h e—rly the following ye—r with iinstein writing the –€hysi™—l €—rt9 —nd qrossm—nn the –w—them—ti™—l p—rtF9 „he theory of the resulting p—per @iinstein —nd qrossm—nn IWIQA is ™ommonly known —s the –Entwurf 9 theory for the title of the p—perF –Entwurf einer verallgemeinerten Relativitatstheorie und einer Theorie der Gravitation9 @–outline of — gener—lized theory of rel—E tivity —nd — theory of gr—vit—tion9AF sts ™entr—l ide— involved the introdu™tion of ‚i™™i —nd veviEgivit—9s fund—ment—l form @PAF „hey st—rted with the inv—ri—nt interv—l of winkowski in dierenti—l form ds2 a c2dt2 dx2 dy2 dz2 @QA where @x; y; z; tA —re the sp—™e —nd time ™oordin—tes of —n inerti—l fr—me of referen™e in — winkowski sp—™etimeF „r—nsforming to —r˜itr—ry ™oordin—tes x for  a I; : : : ; R @QA ˜e™omes 2 ds2 a gdxdx: @RA iinstein employed his prin™iple of equiv—len™e to interpret the m—trix of qu—ntities guv th—t h—d —risen with the introdu™tion of —r˜itr—ry ™oordin—tesF sn the spe™i—l ™—se of the prin™ipleD the tr—nsform—tion from @QA to @RA is from —n inerti—l ™oordin—te system to — uniformly —™™eler—ted ™oordin—te systemF sn th—t redu™es to th—t of @QAD ex™ept th—t c now is — fun™tion o™—f stehDeth™oeomrd—itnr—ixtesof@x™oH;eyH;™ziHeAnFts„hg—t isD @RA ˜e™omes ds2 a c2@xH; yH; zHAdt2 dxH2 dyH2 dzH2: @QHA e™™ording to the prin™iple of equiv—len™eD the presen™e of — gr—vit—tion—l eld w—s the only dieren™e ˜etween the sp—™etime of @Q9A —nd th—t of spe™i—l rel—tivity @QAF „herefore iinstein interpreted the ™oordin—te dependent ™ of @Q9A —s representing — gr—vit—tion—l eld —ndD more gener—llyD the g of @RA —s representing — gr—vit—tion—l eldF iinstein —nd qrossm—nn pro™eeded to develop essenti—lly —ll the m—jor ™omponents of the n—l gener—l theory of rel—tivityF tust one eluded themF „he sp—™etimes represented ˜y @QAD @Q9A —nd @RA —re —ll —tF „o tre—t the gener—l ™—se of the gr—vit—tion—l eldD nonE—t metri™s must —lso ˜e —dmitted —ndD in the n—l theoryD the de™ision of whi™h —re —dmitted is m—de ˜y the gr—vit—tion—l eld equ—tionsF iinstein expe™ted these equ—tions to t—ke the now f—mili—r form G a T @SA where T is the stressEenergy tensor —nd G — gr—vit—tion tensor ™onstru™ted solely from the metri™ tensor g —nd its deriv—tivesF iinstein —nd qrossm—nn ™onsidered the ‚i™™i tensor —s the gr—vit—tion tensor|just — h—ir9s ˜re—th —w—y from iinstein9s n—l ™hoi™e of the iinstein tensorF rowever they reported th—t the resulting eld equ—tions f—iled to give 2ren™eforth summ—tion over repe—ted indi™es is impliedF iinstein himself did not introdu™e this summ—tion ™onvention until IWITF General covariance and general relativity 801 the xewtoni—n limit in the ™—se of we—kD st—ti™ gr—vit—tion—l eldsF sn their pl—™eD to the —stonishment of modern re—dersD they oered — set of gr—vit—tion—l eld equ—tions th—t w—s not gener—lly ™ov—ri—ntF iinstein then des™ended into — long struggle with his imperfe™t theory th—t l—sted —lmost three intense ye—rs ˜efore he emerged vi™toriously with the n—l gener—lly ™ov—ri—nt theory in h—ndF3 3.3. The hole argument: general covariance condemned huring these three ye—rsD iinstein formul—ted —n —rgument th—t would de™isively redire™t his underst—nding of gener—l ™ov—ri—n™eF re —nd qrossm—nn h—d ˜een un—˜le to nd —™™ept—˜le gener—lly ™ov—ri—nt eld equ—tionsF „he soE™—lled –hole —rgument9 purported to show th—t this ™ir™umst—n™e need not worry them sin™e —ll gener—lly ™ov—ri—nt eld equ—tions would ˜e physi™—lly uninterestingF iinstein pu˜lished the —rgument four times in IWIRD —ppe—ringD for ex—mpleD —s — l—ter —ppendix to the journ—l printing of iinstein —nd qrossm—nn @IWIQAF sts ™le—rest exposition w—s in — review —rti™le @iinstein IWIRD ppIHTT{UAF4 „he —rgument w—s ˜eguilingly simpleF iinstein —sked us to im—gine — region of sp—™eE time devoid of m—tter|the –hole9|in whi™h the stress energy tensor T v—nishedF re now —ssumed th—t we h—d gener—lly ™ov—ri—nt gr—vit—tion—l eld equ—tions —nd th—t g w—s — dsitoinlwu—itttiehonisnyfsotthreemthhixoslHesDpF w—s™nheitt™ihhme—engiernewe—d™™owoooirtdrhdinixn——tetoeusystyssitsdeteemmthtxheehFmoileeitn˜rsiu™ttewi™no—utmrl—dens˜smfeoorgomH tehd—lyntdoto™—odnnieswetrru™f™ortooemrdE —™™ording to the usu—l tensor tr—nsform—tion l—wF „h—t isD the s—me gr—vit—tion—l eld would ˜e represented ˜y g in ™oordin—te system x D —nd ˜y gH  in ™oordin—te system xH F et this point iinstein ee™ted — su˜tle m—nipul—tion th—t is the key to the hole —rguE mentF yne ™ould ™onsider v—ri—˜le x —nd gH @xH A —s — new set of ten fun™tions —gtHhsee@txsoyfAmFtme„nehtf—uritn™™ismtDio—tn—trskiexoftghthee@txve—nArif—u—˜snl™e—tixsoHentFs of ten fun™tions of the yne ™—n now ™onstru™t of the new m—trix gH  —nd ™onsider ™onstru™tion them gH @x —s fun™tions of the A ™—nnot represent old the ™oordin—tes x F „he origin—l g@x A —nd the s—me gr—vit—tion—l eld in dierent ™oordin—te systemsF „hey —re ˜oth dened on ent ™omponentsD sin™e g —nd gH  the —re s—me ™oordin—te system x D dierent fun™tionsF „h—t isD yet they h—ve dierE g@x A —nd gH @x A represent di erent gr—vit—tion—l elds in the same ™oordin—te systemF xowD ˜y their ™onE stru™tionD the fun™tions g@x A —nd gH @x A will ˜e the s—me outside the holeD ˜ut they will ™ome smoothly to dier within the holeF „hus the two sets of fun™tions represent distin™t gr—vit—tion—l eldsF vet us ™—ll them g —nd gHF „he elds g —nd gH —re the s—me outside the hole ˜ut ™ome smoothly to dier within the holeF 3„his f—s™in—ting episode h—s ˜een disse™ted in some det—il with some help from his priv—te ™—l™ul—tion @see ƒt—™hel IWVH —nd xorton IWVRAF 4por further dis™ussion see xorton @IWVUAF 802 J D Norton iinstein h—s presumed the eld equ—tions gener—l ™ov—ri—ntF „hereforeD if they —re s™oolnvsetdru˜™ytiotnhegHg@x@xAFAD„thh—etn they must ˜e solved isD gener—lly ™ov—ri—nt solutions the two distin™t gr—vit—tion—l elds g —nd ˜y gH @xH A —nd therefore —lso ˜y the gr—vit—tion—l eld equ—tions —llow —s gHF iinstein found this out™ome un—™E ™ept—˜leF por the one m—tter distri˜ution outside the hole now ™le—rly f—ils to determine wh—t the gr—vit—tion—l eld would ˜e within the holeF „h—t isD we ™ould spe™ify the m—tter distri˜ution —nd gr—vit—tion—l eld everywhere in sp—™etime ex™epting some m—tterEfree hole th—t ™ould ˜e —r˜itr—rily sm—ll in ˜oth sp—ti—l —nd tempor—l extentF xonetheless geE ner—lly ™ov—ri—nt eld equ—tions would ˜e un—˜le to determine wh—t the gr—vit—tion—l eld would ˜e within this holeF „his w—s — dr—m—ti™ f—ilure of wh—t he ™—lled the l—w of ™—us—lity —nd we might now ™—ll determinismF iinstein deemed the f—ilure su™iently trou˜lesome to w—rr—nt reje™tion of gener—lly ™ov—ri—nt gr—vit—tion—l eld equ—tions —s physi™—lly interE estingF5 3.4. Einstein's 1916 account of the foundations of general relativity: general covariance regained sn xovem˜er IWISD iinstein9s long struggle with his –intwurf9 theory ™—me to — ™loseF ris resist—n™e to gener—l ™ov—ri—n™e n—lly ˜roke under the —™™umul—ting weight of serious pro˜lems in his –intwurf9 theoryF ris return to gener—l ™ov—ri—n™e —nd the n—l gener—l theory of rel—tivity were reported to the €russi—n e™—demy in — series of h—sty ™ommuni™—E tions th—t ™hroni™le the tense ™onfusions of these l—st desper—te d—ysF6 i—rly the following ye—rD iinstein @IWITA sent Annalen der Physik — review —rti™le on the n—l theoryF „he —rti™le9s —™™ount of the theory9s found—tions w—s written with — freedom un—v—il—˜le to iinstein in the d—rk ye—rs of the –intwurf9 theoryF „houghout those ye—rsD iinstein h—d m—int—ined his —llegi—n™e to the rel—tivity of inerti—F „h—t —llegi—n™e h—d to rest prin™ip—lly on — sin™ere hope of wh—t might ˜e demonstr—˜leF re h—d not demonstr—ted the un™ondition—l rel—tivity of inerti— in his –intwurf9 theoryY he w—s still sure only of we—k eld ee™ts ™omp—ti˜le with the rel—tivity of inerti— @iinstein IWIQD se™tion WA —nd simil—r to those he h—d found in his IWIP theoryF wore vexingD howeverD w—s the very pu˜li™ f—ilure of gener—l ™ov—ri—n™eD whi™h ™ompromised the ™l—im th—t he w—s extending the prin™iple of rel—tivityF iinstein did not report on equ—lly serious pro˜lems th—t h—d ˜ef—llen the prin™iple of equiv—len™eF „he simple IWHUGIWII version of the prin™iple required only equiv—len™e of uniform —™™eler—tion —nd — homogeneous gr—vit—tion—l eldF ‰et in the n—l version of the IWIP theoryD the prin™iple h—d to ˜e restri™ted to innitesim—lly sm—ll regions of sp—™eF iinstein found the need for this restri™tion extremely puzzling sin™e the restri™tion w—s not invoked to homogenize —n inhomogeneous eldF ‡orseD in the –intwurf9 theoryD even this restri™tion f—iled to s—ve this form of the prin™ipleD whi™h h—d to ˜e reported —s — result of his e—rlier IWIP theory @see xorton @IWVSD se™tion RFQA for — dis™ussionAF fy IWITD iinstein9s pro˜lem with gener—l ™ov—ri—n™e h—d ev—por—ted —nd with them the pro˜lems with the prin™iple of equiv—len™eF „hus the IWIT review —rti™le ™ould ™ommen™e with — more ™ondent —™™ount of the theory9s found—tions whi™h rem—ins tod—y one of 5st w—s pointed out mu™h l—ter ˜y ƒt—™hel @IWVHAD using m—them—ti™—l notions not —v—il—˜le to iinstein in IWIQD th—t the new gr—vit—tion—l eld g0 w—s gener—ted from g —s the ™—rry —long g0 a hg under the indeterminism th—t dieomorphism worried iinstein h indu™ed ˜y so profoundly the ™oordin—te tr—nsform—tion is now routinely o˜liter—ted —s —xgD—utogexf0reF ed„ohme —sso™i—ted with —r˜itr—ry dieomorphism so th—tD while g —nd g0 m—y ˜e m—them—ti™—lly distin™tD they —re not judged to represent physi™—lly distin™t gr—vit—tion—l elds @see ‡—ld IWVRD pRQVAF 6iinstein IWISF por diss™ussion of this episodeD see xorton @IWVRD se™tions UDVAF General covariance and general relativity 803 the most widely known of iinstein9s —™™ountsF „he exposition ˜eg—n with — series of now f—mili—r ™onsider—tions —ll of whi™h drove tow—rds gener—l ™ov—ri—n™eF foth spe™i—l rel—tivity —nd ™l—ssi™—l me™h—ni™sD iinstein reportedD suered —n episteE mologi™—l defe™tF st w—s illustr—ted with iinstein9s v—ri—nt of xewton9s ˜u™ketF „wo uid ˜odies hover in sp—™eF „hey —re in —n o˜serv—˜le st—te of ™onst—nt rel—tive rot—tion —˜out — line th—t ™onne™ts themF sn spite of the o˜vious symmetry of this setupD iinstein supE posed th—t one sphere S1 proves to ˜e spheri™—l when surveyed —nd the other S2 proves to ˜e —n ellipsoid of revolutionF gl—ssi™—l me™h—ni™s —nd spe™i—l rel—tivity ™ould expl—in the dieren™e ˜y supposing th—t the rst sphere is —t rest in —n inerti—l fr—me of referen™eD introdu™ed ˜y iinstein into the —rgument —s — –privileged q—lile—n sp—™eD9 —nd th—t the se™ond is notF „his expl—n—tionD iinstein o˜je™tedD viol—tes the –dem—nd of ™—us—lity9D for these privileged fr—mes —re –merely f—™titious ™—uses9 —nd not —n o˜serv—˜le thingF „he true ™—use of the dieren™e must lie outside the systemD iinstein ™ontinuedD immedi—tely identifying the true ™—use in the disposition of dist—nt m—ssesF sn ee™t iinstein used his ex—mple to ™on™lude th—t the only theory th—t ™ould s—tisf—™torily —™™ount for this ex—mple w—s one th—t s—tised the requirement of the rel—tivity of inerti—F eny su™h theoryD iinstein ™ontinuedD ™—nnot single out —ny inerti—l fr—me —s preferredF „hereforeX The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. —long this ro—d we —rrive —t —n extension of the postul—te of rel—tivityF @iinstein9s emph—sisA iinstein then introdu™ed the prin™iple of equiv—len™e in the form given —˜ove in se™tion QFI in whi™h it —sserts the equiv—len™e of uniform —™™eler—tion —nd — homogeneous gr—vit—E tion—l eldF „he prin™iple is used to suggest th—t — theory whi™h implements — gener—lized prin™iple of rel—tivity will —lso ˜e — theory of gr—vit—tionF iinstein then turns to de—l with — ™ompli™—tion th—t —rises from using —™™eler—ted fr—mes of referen™e in spe™i—l rel—tivityF sn —™™eler—ted fr—mesD in p—rti™ul—r in rot—ting fr—mesD geometry ™e—ses to ˜e iu™lide—n —nd ™lo™ks —re slowed in — positionEdependent m—nnerF es — result it turns out th—t one ™—n no longer e—sily dene sp—™e —nd time ™oordin—te systems ˜y the f—mili—r oper—tions of l—ying out rods —nd using st—nd—rd ™lo™ksF „his —pp—rent ™ompli™—tion|—nd not the need for — gener—liz—tion of the prin™iple of rel—tivity|le—ds iinstein to propose gener—l ™ov—ri—n™eX7 „he method hitherto employed for l—ying ™oordin—tes into the sp—™eEtime ™ontinuum in — denite m—nner thus ˜re—ks downD —nd there seems to ˜e no other w—y whi™h would —llow us to —d—pt systems of ™oordin—tes to the fourEdimension—l universe so th—t we might expe™t from their —ppli™—tion — p—rti™ul—rly simple formul—tion of the l—ws of n—tureF ƒo there is nothing for it ˜ut to reg—rd —ll im—gin—˜le systems of ™oordin—tesD on prin™ipleD —s equ—lly suit—˜le for the des™ription of n—tureF „his ™omes to requiring th—tX The general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are co-variant with respect to any substitutions whatever (generally covariant). st is ™le—r th—t — physi™s theory whi™h s—tises this postul—te will —lso ˜e suit—˜le for the gener—l postul—te of rel—tivityF por the sum of all su˜stitutions in —ny ™—se in™ludes the those whi™h ™orrespond to —ll rel—tive motions of threeEdimension—l systems of ™oordin—tes @iinstein9s emph—sisA 7e footnote —t the word –im—gin—˜le9 w—s omitted from the st—nd—rd €errett —nd teery inglish tr—nsl—tionF st s—ysX –rere we do not w—nt to dis™uss ™ert—in restri™tions whi™h ™orrespond to the requirement of unique ™oordin—tion —nd of ™ontinuityF9 „his now essenti—lly unknown footnote shows th—t iinstein did —t le—st on™e —pologize for his f—ilure to spe™ify pre™isely whi™h group of tr—nsform—tions w—s intended ˜y –—ny su˜stitutions wh—teverF9 804 J D Norton ‡hy did iinstein not simply insist th—t the gener—liz—tion of the prin™iple of rel—tivity to —™™eler—ted motion for™es gener—l ™ov—ri—n™ec pollowing the —n—logy with vorentz ™ov—riE —n™eD the gener—lized prin™iple of rel—tivity would require —n extension of the ™ov—ri—n™e of the theory to in™lude tr—nsform—tions ˜etween fr—mes in —r˜itr—ry st—tes of motionF fut gener—l ™ov—ri—n™e extends it even furtherF st in™ludes tr—nsform—tions th—t h—ve nothing to do with ™h—nges of st—tes of motionD su™h —s the tr—nsform—tion ˜etween g—rtesi—n —nd pol—r sp—ti—l ™oordin—tesF futD —s iinstein indi™—tesD he feels ™ompelled to go to this l—rger group sin™e he ™—n see no n—tur—l w—y of restri™ting the sp—™etime ™oordin—te systemF 3.5. The point-coincidence argument smmedi—tely following the —˜ove st—tement of the requirement of gener—l ™ov—ri—n™eD iinE stein g—ve —nother —rgument for gener—l ™ov—ri—n™e whi™h tohn ƒt—™hel h—s ™onveniently l—˜elled the –pointE™oin™iden™e —rgument9F „h—t this requirement of gener—l ™oEv—ri—n™eD whi™h t—kes —w—y from sp—™e —nd time the l—st remn—nt of physi™—l o˜je™tivityD is — n—tur—l oneD will ˜e seen from the following reexionF ell our sp—™eEtime veri™—tions inv—ri—˜ly —mount to — determin—tion of sp—™eEtime ™oin™iden™esF sfD for ex—mpleD events ™onsisted merely in the motion of m—teri—l pointsD then ultim—tely nothing would ˜e o˜serv—˜le ˜ut the meetings of two or more of these pointsF woreoverD the results of our me—surings —re nothing ˜ut veri™—tions of su™h meetings of the m—teri—l points of our me—suring instruments with other m—teri—l pointsD ™oin™iden™es ˜etween the h—nds of — ™lo™k —nd points on the ™lo™k di—lD —nd o˜served pointEevents h—ppening —t the s—me pl—™e —nd the s—me timeF „he introdu™tion of — system of referen™e serves no other purpose th—n to f—™ilit—te the des™ription of the tot—lity of su™h ™oin™iden™esF ‡e —llot to the universe four sp—™eEtime v—ri—˜lesD x1D x2D x3D x4 in su™h — w—y th—t for every point there is — ™orresponding system of v—lues of the v—ri—˜les x1 F F F x4F „o two ™oin™ident point events there ™orresponds one system of v—lues of the v—ri—˜les x1 F F F x4 iFeF ™oin™iden™e is FF ™h—r—™terized ˜y the identity of the ™oEordin—tesF sfD in the pl—™e of F x4D we introdu™e fun™tions of themD xH1D xH2D xH3D xH4D —s — new system the v—ri—˜les x1 of ™oEordin—tesD so th—t the system of v—lues —re m—de to ™orrespond to one —nother without —m˜iguityD the equ—lity of —ll four ™oEordin—tes in the new system will —lso serve —s —n expression for the sp—™eEtime ™oin™iden™e of the two pointEeventsF es —ll our physi™—l experien™e ™—n ˜e ultim—tely redu™ed to su™h ™oin™iden™esD there is no immedi—te re—son for preferring ™ert—in systems of ™oEordin—tes to othersD th—t is to s—yD we —rrive —t the requirement of gener—l ™oEv—ri—n™eF „his pointE™oin™iden™e —rgument is ™ited very frequently in the liter—ture sin™e IWITF rowE ever its re—l purpose w—s essenti—lly ™ompletely forgotten until it w—s redis™overed —nd reve—led ˜y tohn ƒt—™hel @IWVHAF iinstein9s IWIT exposition of gener—l rel—tivity ™ont—ined — very puzzling omissionF sn the ye—rs immedi—tely pre™edingD ˜y me—ns of the hole —rE gumentD iinstein h—d —pp—rently proved th—t —ny gener—lly ™ov—ri—nt theory would ˜e physi™—lly uninterestingF ‰et there w—s iinstein extolling ex—™tly su™h — theory without expl—ining where the hole —rgument went —str—yF „h—t mel—n™holy t—sk of ™orre™ting his p—st error w—s the re—l fun™tion of the pointE ™oin™iden™e —rgumentF „his w—s pre™isely the use to whi™h the —rgument w—s put in iinE stein9s ™orresponden™e of he™em˜er IWIS —nd t—nu—ry IWIT @see xorton IWVUD se™tion RAF e™™ording to iinstein9s —ssumptionD the physi™—l ™ontent of — theory is fully exh—usted ˜y — ™—t—logue of the sp—™etime ™oin™iden™es it s—n™tionsF „herefore —ny tr—nsform—tion th—t preserves these ™oin™iden™es preserves its physi™—l ™ontentF xow the tr—nsform—tion used in the hole —rgument from the eld g to the m—them—ti™—lly distin™t eld gH is more th—n — General covariance and general relativity 805 ms——llemr™eeoit™nro™—oindrsdefnoin™r—mets—eF ts„ioyhnsetoeremff™oForeorrgodwi—nen—vdteergsFHthpreeoprtrrge—sne—nsnftdortgmhHe——tsri—eomnmef—rptohhmeymsgi—™—ttilo™—glelHlydisFdio‡sntiehn—™ttthe—vteerlpdrisnesdineerttveheresE minism is reve—led in the hole —rgument is — purely m—them—ti™—l freedom —kin to — g—uge freedom —nd oers no o˜st—™le to the physi™—l interest of — gener—lly ™ov—ri—nt theoryF iinstein s™—r™ely ever mentioned the de˜—™le of the hole —rgument —g—in in printF rowE ever it ™ontinued to inform his ide—s —˜out ™ov—ri—n™eD sp—™etimeD eld —nd ™oordin—te systemsF por ex—mpleD in exe™uting the hole —rgumentD in order to ee™t the tr—nsition from some g@x A to gH @x AD one h—s to —ssumeD re—l existen™eD independent of the g in ee™tD or gH F th—t porD the ™oordin—te system x D gur—tively spe—kingD one h—s h—s to remove the eld gH F sn — eld letter gD le—ving the of he™em˜er PTD ˜—re ™oordin—te system x D —nd then insert IWISD to €—ul ihrenfestD iinstein expl—ined the th—t new one defe—ts the hole —rgument ˜y —ssuming —mong other things th—t –the referen™e system sigE nies nothing re—l9F8 ‡e he—r these e™hoes of the hole —rgument when iinstein @IWPPDpPIA pro™l—ims in — w—y IWPH —ddress in veidenX „here ™—n ˜e no sp—™e nor —ny p—rt of sp—™e without gr—vit—tion—l potenti—lsY for these ™onfer upon sp—™e its metri™—l qu—litiesD without whi™h it ™—nnot ˜e im—gined —t —llF „hese s—me e™hoes still rever˜er—te in the IWSP —ppendix to iinstein9s popul—r text Rela- tivity: the Special and the General TheoryD when iinstein @IWSPDpISSA insists F F F — pure gr—vit—tion—l eld might h—ve ˜een des™ri˜ed in terms of the gik @—s fun™tions of the ™oEordin—tesAD ˜y solution of the gr—vit—tion—l equ—tionsF sf we im—gine the gr—vit—tion—l eldD iFeF the fun™tions gikD to ˜e removedD there does not rem—in — sp—™e of the type @IA ‘winkowski sp—™etime“D ˜ut —˜solutely nothingD —nd —lso no –topologi™—l sp—™e9 @iinstein9s emph—sisAF wost re™entlyD the hole —rgument h—s enjoyed — reviv—l in the philosophy of sp—™e —nd time liter—ture whereD in v—ri—nt formD it provides — strong —rgument —g—inst the do™trine of sp—™etime su˜st—ntiv—lism @i—rm—n —nd xorton IWVUAF por further dis™ussion of the ˜—™kE ground —n r—mi™—tions of the hole —nd pointE™oin™iden™e —rguments see row—rd @IWWPA —nd ‚y™km—n @IWWPAF 3.6. The Gottingen defense of general covariance „he most prominent leg—™y of the hole —rgument in the liter—ture on gener—l rel—tivity does not —rise from iinstein9s —n—lysisD howeverF sn IWIS —nd IWIU h—vid ril˜ert @IWISD IWIUA pu˜lished — twoEp—rt p—per on gener—l rel—tivity whi™h proved to ˜e enormously inuenti—lF giting the hole —rgumentD ril˜ert @IWIUDppSW{TQA turned to the question of the –prin™iple of ™—us—lity9F re o˜served th—t his formul—tion of gener—l rel—tivity employed fourteen independent v—ri—˜lesD th—t isD ten metri™—l ™omponents for the gr—vit—tion—l eld —nd four potenti—ls for the ele™trom—gneti™ eldF rowever in the joint theory of gr—vit—tion—l —nd ele™trom—gneti™ eldsD four identities redu™ed the fourteen eld equ—tions to only ten independent equ—tionsF „hese four ™onditions ™ouldD howeverD ˜e —˜sor˜ed in four stipul—tions used to spe™ify — ™oordin—te systemF ril˜ert insisted th—t his underdetermin—tion of the eld w—s not physi™—lF i™hoing the geometri™ themes of his qottingen ™olle—gues ulein —nd the l—te winkowskiD he re™—lled @pTIA –F F F —n —ssertion th—t does not rem—in inv—ri—nt under —ny —r˜itr—ry tr—nsform—tion of the ™oordin—te system is m—rked —s physically meaningless9 @ril˜ert9s emph—sisAF re then —rgued th—t the four degrees of freedom did not le—ve the inv—ri—nt ™ontent of the theory underdeterminedF ris ex—mple w—s —n ele™tron —t rest in some ™oordin—te systemF e ™oordin—te tr—nsform—tion le—ves the ele™tron un™h—nged in the p—st of some inst—nt 8es quoted in xorton @IWVUDp ITWAF 806 J D Norton spe™ied ˜y time ™oordin—te x4 a HD ˜ut sets it in motion in the futureF „he two ™oordin—te des™riptions —re the s—me in the p—stD the ele™tron is —t restD ˜ut in the future only one des™ri˜es the ele™tron —s movingF „he one p—st ™—n extend to dierent futuresF „he dieren™esD howeverD h—ve no physi™—l signi™—n™eD sin™e the relev—nt —ssertions —˜out the ele™tron9s motion —re not inv—ri—ntF yne ™ould m—ke the inv—ri—nt ˜y introdu™ing —n inv—ri—nt ™oordin—te system —d—pted to the sp—™etime geometryD su™h —s the q—ussi—n system ril˜ert ™onsideredF goordin—te ˜—sed —ssertions of the ele™tron9s motion would now ˜e inv—ri—ntD ˜ut they would no longer ˜e underdetermined sin™e the introdu™tion of the q—ussi—n system used up the four rem—ining degrees of freedomF ril˜ert9s depi™tion of the indeterminism of — gener—lly ™ov—ri—nt theory w—s in terms of — ™ount of independent eld v—ri—˜les —nd independent eld equ—tionsF st is the version th—t r—pidly ™—me to —ppe—r most often in the liter—ture @eFgF €—uli IWPID se™tion STAF „he four identities —mong the eld equ—tions th—t —llowed the underdeterminism were only l—ter ™onne™ted with the ™ontr—™ted fi—n™hi identities @see wehr— IWURD se™tion UFQAF eg—in ril˜ert9s dis™ussion —nd his ex—mple of the ele™tron w—s the rst tre—tment of the g—u™hy pro˜lem in gener—l rel—tivityD so th—t the liter—ture on the g—u™hy pro˜lem ™—n tr—™e its des™ent ˜—™k to iinstein9s hole —rgument @see ƒt—™hel IWWPAF9 3.7. Einstein's three principles of 1918 sn w—r™h IWIVD iinstein @IWIVA returned to the question of the fund—ment—l prin™iples of gener—l rel—tivityF es he m—de ™le—r in his introdu™tory rem—rksD the p—per w—s provoked ˜y urets™hm—nn9s @IWIUA ™riti™ism @see se™tion SFP ˜elowAF rowever its purpose w—s to l—y out his underst—nding of the found—tions of his theoryF „his exposition diered from the IWIT —™™ount in —t le—st one m—jor —re—F sn IWITD iinstein —ssumed th—t his gener—lly ™ov—ri—nt theory would s—tisfy the rel—tivity of inerti—D —lthough no proof h—d ˜een givenF et ˜est iinstein would h—ve ˜een —˜le to point to we—k eld ee™ts ™omp—ti˜le with the rel—tivity of inerti—F @„hese we—k eld ee™ts —re of the s—me type —s those he reported in the –intwurf9 theory of iinstein @IWIQD se™tion WA —nd —re des™ri˜ed in his text @iinstein IWPP—D pIHHAAF fy IWIUD iinstein h—d found th—t — simple re—ding of the rel—tivity of inerti— w—s in™omp—ti˜le with his theoryF re reported his f—ilure in —n introdu™tory se™tion @se™tion PA to his f—mous p—per on rel—tivisti™ ™osmology @iinstein IWIUAF yn the ˜—sis of the rel—tivity of inerti—D he expe™ted th—t the inerti— of — ˜ody would —ppro—™h zero if it w—s moved su™iently f—r from other m—sses in the universeF „his expe™t—tion would ˜e re—lized in the theory if the sp—™etime metri™ —dopted ™ert—in degener—te v—lues —t — m—ssEfree sp—ti—l innityF rowever iinstein found th—t su™h degener—te ˜eh—viour w—s in—dmissi˜le in his theoryF snste—d he seemed ™ompelled to postul—te some nonEdegener—te ˜ound—ry ™onditions for the metri™ —t — m—ssEfree sp—ti—l innityD su™h —s winkowski—n v—luesF „his winkowski—n ˜ound—ry ™ondition ˜e™—me the em˜odiment for iinstein of the f—ilure of the rel—tivity of inerti—F por this ˜ound—ry ™ondition m—de — denite ™ontri˜ution to the inerti— of — test ˜ody th—t ™ould not ˜e tr—™ed to other m—ssesF „h—t isD with these ˜ound—ry ™onditionD the inerti— of — ˜ody w—s inuen™ed ˜y the presen™e of other m—ssesD in so f—r —s they —e™ted the metri™ eldF rowever its inerti— w—s not fully determined ˜y the other m—ssesF „hereforeD if the rel—tivity of inerti— w—s to ˜e s—tisedD it w—s ne™ess—ry to —˜olish these —r˜itr—rily postul—ted ˜ound—ry ™onditionsF @„he question of whether this w—s —lso su™ient rem—ined un—ddressedFA iinstein su™™eeded in —˜olishing these ˜ound—ry 9row—rd —nd xorton @forth™omingA ™onje™ture th—t there w—s —n en™ounter in IWIS ˜etween the qottingen resolution of the hole —rgument —nd —n unre™eptive iinsteinD still ™onvin™ed of the ™orre™tness of the hole —rgumentF General covariance and general relativity 807 ™onditions —t sp—ti—l innity ˜y — most ingenious ployX he —˜olished sp—ti—l innity itselfF re introdu™ed the rst of the modern rel—tivisti™ ™osmologiesD the one we now ™—ll the –iinstein universe9D whi™h is sp—ti—lly ™losed —nd niteF „he pri™e iinstein h—d to p—y turned out to ˜e highF sn order for his eld equ—tions to —dmit the iinstein universe —s — solutionD he needed to introdu™e the extr— –™osmologi™—l9 term in his eld equ—tionsF sn his not—tion —nd formul—tion of IWIUD with G representing the Ricci tensor —nd  — ™onst—ntD this me—nt th—t the old eld equ—tions G a  @T 1 2 g T A were repl—™ed ˜y G g a  @T 1 2 g T A „he ™osmologi™—l term is g —nd  is the ™osmologi™—l ™onst—ntF „his development w—s essenti—l ˜—™kground to underst—nding the three prin™iples iinE stein listed in @iinstein IWIVDppPRI{PA —s those on whi™h his theory restedF (a) Principle of relativity. „he l—ws of n—ture —re only —ssertions of timesp—™e ™oinE ™iden™esY therefore they nd their uniqueD n—tur—l expression in gener—lly ™ov—ri—nt equ—tionsF (b) Principle of equivalence. snerti— —nd weight —re identi™—l in essen™eF prom this —nd from the results of the spe™i—l theory of rel—tivityD it follows ne™ess—rily th—t the symmetri™ –fund—ment—l tensor9 @gA determines the metri™ properties of sp—™eD the inerti—l rel—tions of ˜odies in itD —s well —s gr—vit—tion—l ee™tsF ‡e will ™—ll the ™ondition of sp—™eD des™ri˜ed ˜y the fund—ment—l tensorD the –qEeldF9 (c) Mach's principle. the qEeld is determined without residue ˜y the m—sses of ˜odiesF ƒin™e m—ss —nd energy —re equiv—lent —™™ording to the results of the spe™i—l theory of rel—tivity —nd sin™e energy is des™ri˜ed form—lly ˜y the symmetri™ energy tensor @TAD this me—ns th—t the qEeld is ™onditioned —nd determined ˜y the energy tensorF „he sep—r—tion of the prin™iple of rel—tivity —nd w—™h9s prin™iple into two distin™t prinE ™iples w—s ™le—rly the produ™t of iinstein9s experien™e with the ™osmologi™—l pro˜lemF sf the iinstein of IWIT h—d —ssumed th—t the rel—tivity of inerti— would ˜e s—tised —utom—tE i™—lly within — gener—lly ™ov—ri—nt theoryD then the iinstein of IWIV no longer h—r˜oured su™h delusionsF „he IWIV version of the prin™iple of rel—tivity seems to —ssert something less th—n — fully gener—lized rel—tivity of the motion of ˜odiesF sn ee™t it merely —sserts the key thesis of the pointE™oin™iden™e —rgumentX the physi™—l ™ontent of — theory is exh—usted ˜y its ™—t—logue of —llowed sp—™etime ™oin™iden™esF qener—l ™ov—ri—n™e follows from this thesis —s — ™onsequen™eF „he prin™iple of rel—tivity @—A is now supplemented ˜y the new w—™h9s prin™iple @™A —nd it is only their ™onjun™tion th—t ˜egins to resem˜le iinstein9s origin—l go—l of — fully gener—lized rel—tivity of motionF sn ee™t w—™h9s prin™iple @™A w—s intended to ™—pture in — eld theoreti™ setting the oldD w—™hEinspired ™onditions for the metri™ eld —t sp—ti—l innityD whi™hD iinstein reported in IWIUD ™ompromised the rel—tivity of inerti—F ell this w—s —lluded to ˜y iinstein in — footnote to the title –w—™h9s prin™iple9D whi™h —lso —nnoun™ed th—t he w—s introdu™ing the n—me for the rst timeX 808 J D Norton …p to now s h—ve not distinguished prin™iples @—A —nd @™A —nd th—t ™—used ™onfusionF s h—ve ™hosen the n—me –w—™h9s prin™iple9 sin™e this prin™iple is — gener—liz—tion of w—™h9s requirement th—t inerti— ˜e redu™i˜le to —n inter—™tion of ˜odiesF iinstein9s wording of the prin™iple of equiv—len™e @˜A w—s —n interesting dep—rture in so f—r —s it now emph—sized th—t the prin™iple depended on the empiri™—l equ—lity of two qu—ntitiesD inerti—l —nd gr—vit—tion—l m—ssD —nd th—t the ee™t of the prin™iple h—d ˜een to unify them ™ompletelyF rowever there w—s little re—l ™h—nge from iinstein9s e—rlier use of the prin™ipleD —s w—s shown ˜y the rem—inder of the p—r—gr—ph th—t des™ri˜ed the prin™ipleF sn ee™t it g—ve — synopsis of the tr—nsition form the line element @QA to @Q9A —nd @RA —nd the resulting interpret—tion of the nonE™onst—nt ™oe™ients of @Q9A —nd @RA —s representing the gr—vit—tion—l eldD —s well —s the inerti—l —nd geometri™ properties of sp—™etimeF 3.8. Mach's principle forsaken por —ll his eortsD iinstein9s portr—y—l of the found—tions of gener—l rel—tivity h—d still not re—™hed its n—l form with the IWIV listF yver the ye—rs followingD the prin™iple of rel—tivity —nd of equiv—len™e ret—ined their IWIV formsF rowever iinstein ™—me to —˜—ndon w—™h9s prin™ipleF „he seeds of iinstein9s disen™h—ntment with w—™h9s prin™iple were ˜e™oming —pp—rent —s e—rly —s IWIWF iinstein @IWIWD se™tion IA des™ri˜ed its ospringD the ™osmologi™—l term —dded to his IWIS eld equ—tionsD —s –gr—vely detriment—l to the form—l ˜e—uty of the theory9F ‡ith the dis™overy of the exp—nsion of the universeD iinstein form—lly disowed the ™osmologi™—l term @iinstein —nd de ƒitter IWQPAF sn —ny ™—seD the —ugment—tion of his eld equ—tions with the ™osmologi™—l term h—d for™ed neither the rel—tivity of inerti— nor w—™h9s prin™iple into his theoryD for it h—d not elimin—ted the possi˜ility of essenti—lly m—tterEfree solutions of the eld equ—tionsF sn su™h solutionsD the inerti— of — test ˜ody ™ould not ˜e —ttri˜uted to other m—ssesF „hese solutions were the su˜je™t of —n extended ex™h—nge in pu˜li™—tion —n in priv—te ˜etween iinstein —nd de ƒitter tow—rds the end of the IWIHs @ƒee uerz˜erg IWVWAF iinstein —lso ˜eg—n to dist—n™e himself from the rel—tivity of inerti—F ‡here—s the ide— w—s urged without reserv—tion up to IWITD he soon ™—me to des™ri˜e it —s — very signi™—nt ide—D ˜ut one of essenti—lly histori™—l interest onlyF por ex—mpleD iinstein @IWPRD pVUA —ttri˜uted to w—™h the ide— th—t inerti— —rose —s —n unmedi—ted inter—™tion ˜etween m—ssesF fut he dismissed it ™—su—lly —s –logi™—lly possi˜leD ˜ut ™—nnot ˜e ™onsidered seriously —ny more tod—y ˜y us sin™e it is —n —™tionE—tE—Edist—n™e theory9F10 iinstein @IWPRD pWHA did still m—int—in th—t the metri™ is fully determined ˜y ponder—˜le m—sses in — sp—ti—lly nite ™osmology —™™ording to his theoryD —lthough the term –w—™h9s prin™iple9 w—s not usedF es time p—ssedD iinstein h—d fewer —nd fewer kind words for this w—™hi—n —ppro—™h to inerti—F re expl—ined in IWRT for ex—mple in his Autobiographical notes @IWRWD pPUA w—™h ™onje™tures th—tD in — truly re—son—˜le theoryD inerti— would h—ve to depend upon the inter—™tion of the m—ssesD pre™isely —s w—s true for xewton9s other for™esD — ™on™eption th—t for — long time s ™onsidered in prin™iple the ™orre™t oneF st presupposes impli™itlyD howeverD th—t the ˜—si™ theory should ˜e of the gener—l type of xewton9s me™h—ni™sX m—sses —nd their inter—™tion —s the origin—l ™on™eptsF ƒu™h —n —ttempt —t — resolution does not t into — ™onsistent eld theoryD —s will ˜e immedi—tely re™ognizedF ris IWIV w—™h9s €rin™iple h—d ˜een —n —ttempt to tr—nsl—te this requirement on m—sses —nd their inter—™tions into eld theoreti™ termsD ˜ut he soon seemed to lose enthusi—sm even for this enterpriseF „he di™ulty w—s th—t the IWIV prin™iple required th—t the metri™ eld g ˜e determined ˜y the m—sses of ˜odies —s represented ˜y the stressEenergy tensor 10„he s—me point is m—de less for™efully in iinstein @IWPPD ppIU{IVA —nd iinstein @IWPP—D pSTA General covariance and general relativity 809 TF rowever this g—ve — prim—ry determining fun™tion to — qu—ntityD TD whi™h iinstein @IWRWD pUIA reported he h—d —lw—ys felt w—s –— form—l ™ondens—tion of —ll things whose ™omprehension in the sense of — eld theory is still pro˜lem—ti™9 —nd one th—t w—s –merely — m—keshift9F iinstein g—ve — n—l synopsis of w—™h9s prin™iple in — letter of pe˜ru—ry PD IWSR to pelix €ir—ni in the ye—r prior to his de—thF giting the —˜ove di™ulty with the stressEenergy tensor —nd the f—™t th—t this tensor presumes the metri™D he l—˜eled his IWIV version of w—™h9s prin™iple –— ti™klish ——ir9 —nd ™on™luded –sn my opinion we ought not to spe—k —˜out w—™h9s prin™iple —ny moreF911 3.9. Einstein's causal objection to absolutes ‡hen iinstein disowed the rel—tivity of inerti— —nd w—™h9s prin™ipleD he —™tu—lly disowed somewh—t less th—n it rst seemedF foth these prin™iples were introdu™ed to solve — pro˜lem in e—rlier theories of sp—™e —nd timeX these theories were defe™tive in the w—y they used inerti—l systems —s ™—usesF iinstein still ™le—rly m—int—ined th—t the pro˜lem w—s serious —nd th—t his gener—l theory of rel—tivity h—d solved itF rowever he h—d origin—lly thought the solution w—s ˜est expressed in terms inspired ˜y his re—ding of w—™hY th—t isD —s — gener—lized rel—tivity of the motion of ˜odiesF es he put it in iinstein @IWIQD pIPTHA „o t—lk of the motion —nd therefore —lso —™™eler—tion of — ˜ody e in itself h—s no me—ningF yne ™—n only spe—k of the motion or —™™eler—tion of — ˜ody rel—tive to other ˜odies fD g etcF ‡h—t holds in kinem—ti™ rel—tion of —™™eler—tion ought —lso to hold for the inerti—l resist—n™eD with whi™h ˜odies oppose —™™eler—tion F F F re w—s led —w—y from this w—™hi—n ™h—r—™teriz—tion of the solution ˜y his work on w—™h9s prin™iple —nd the ™osmologi™—l pro˜lemD —s well —s his preferen™e for eld r—ther th—n ˜ody —s — primitive notionF ‡e sh—ll see th—t his m—ture ™h—r—™teriz—tion of the solution w—s th—t gener—l rel—tivity —llowed sp—™e —nd time to ˜e mut—˜leF „hey no longer just —™ted ™—us—llyD they ™ould —lso ˜e —™ted upon —ndD in this senseD h—d lost their —˜solute ™h—r—™terF sn iinstein9s m—ture viewD it is this spe™i—l ™—us—l property th—t distinguishes gener—l rel—tivity from e—rlier theories —nd possi˜ly even justies the n—me –gener—l rel—tivity9D in so f—r —s it is the eld theoreti™ tr—nsl—tion of iinstein9s origin—l notion of the gener—lized rel—tivity of the motion of ˜odiesF sn the e—rly ye—rs of iinstein9s theoryD the ™—us—l defe™t w—s lo™—ted most prominently in the mere f—™t of the older theories9 use of —n inerti—l referen™e system —s — ™—useF „hus in iinstein9s IWIT review —rti™leD he sought to —™™ount for the ™entrifug—l ˜ulges in — rot—ting uid ˜ody @see se™tion QFR —˜oveAF „o s—y th—t the ˜ody ˜ulges ˜e™—use it rot—ted with respe™t to —n inerti—l fr—me of referen™e is to introdu™e — –merely factitious ™—useD —nd not — thing th—t ™—n ˜e o˜served9 @IWITD pIIQAF „his s—me ex—mple is tre—ted simil—rly in iinstein @IWIR—DppQRR{TAD iinstein @IWIU—A m—kes ™le—r the sort of ™—use th—t he would nd —™™ept—˜le in his popul—r exposition of rel—tivityF sn ™h ˆˆs he —sks for the re—son for the preferred st—tus of inerti—l systemsF re dr—ws —n —n—logy with two p—ns of w—ter on — g—s r—ngeF yne is ˜oilingD one is notF „he dieren™eD iinstein insistsD only ˜e™omes s—tisf—™torily expl—ined when we noti™e the ˜luish —me under the ˜oiling p—n —nd none under the otherF iinstein soon ™—me to stress — dierent —spe™t of these e—rlier theories —s ™—us—lly deE fe™tiveF re identied this —spe™t with their —˜solute ™h—r—™terF sn his Meaning of relativity @IWPP—D pSSA he wrote in p—rody of xewton9s v—tin 11„r—nsl—tion from „orretti @IWVQD pPHPA with –dem Mach'schen 9 Prinzip rendered —s –w—™h9s prin™iple9F 810 J D Norton F F F from the st—ndpoint of the spe™i—l theory of rel—tivity we must s—yD continuum spatii et temporis est absolutum. sn this l—tter st—tement —˜solutum me—ns not only –physi™—lly re—l9D ˜ut —lso –independent in its physi™—l propertiesD h—ving — physi™—l ee™tD ˜ut not itself inuen™ed ˜y physi™—l ™onditions9F end he ™ontinued to expl—in th—t su™h —˜solutes —re o˜je™tion—˜le sin™e @ppSS{TA F F F it is ™ontr—ry to the mode of thinking in s™ien™e to ™on™eive of — thing @the sp—™etime ™ontinuumA whi™h —™ts itselfD ˜ut whi™h is not —™ted uponF „he text immedi—tely turned to w—™h9s ide—s —ndD l—ter @ppWW{IHVA to the we—k eld ee™ts ™omp—ti˜le with the rel—tivity of inerti— —nd his IWIU eld formul—tion of this ide— in — sp—ti—lly ™losed ™osmologyF eround the s—me timeD iinstein9s ˜riefer summ—ries —dvertised gener—l rel—tivity —s elimin—ting the —˜soluteness of sp—™e —nd time @iinstein IWUPD pPTHAX12 ƒp—™e —nd time were there˜y divested not of their re—lity ˜ut of their ™—us—l —˜soluteness|iFeF—e™ting ˜ut not —e™tedF sn these ˜riefer summ—riesD iinstein w—s no longer insisting th—t the sp—™etime metri™ w—s to ˜e fully determined ˜y the distri˜ution of m—ssesF ƒp—™e —nd time h—d lost their —˜soluteness simply ˜e™—use they were no longer immut—˜leF fy the IWSHsD —s iinstein expl—ined to €ir—niD he no longer endorsed his IWIV w—™h9s prin™ipleF rowever he did ret—in the ide— th—t the e—rlier theories were ™—us—lly defe™tive in —dmitting su™h —˜solutes @eFgF iinstein IWSHD pQRVA —ndD —s he expl—ined in the –™ompletely revised9 @pHA IWSR —ppendix to his Meaning of relativity @IWPPD pIQW{RHAD gener—l rel—tivity h—d resolved the pro˜lem —s its essenti—l —™hievementX st is the essenti—l —™hievement of the gener—l theory of rel—tivity th—t it h—s freed physi™s from the ne™essity of introdu™ingD the –inerti—l system9 @or inerti—l systemsAF F F F „here˜y ‘in e—rlier theories“D sp—™e —s su™h is —ssigned — role in the system of physi™s th—t distinguishes it from —ll other elements of physi™—l des™riptionF st pl—ys — determining role in —ll pro™essesD without in its turn ˜eing inuen™ed ˜y themF „his view of the de™ien™y of e—rlier theories —nd gener—l rel—tivity9s —™hievement is not one th—t grew in the w—ke of iinstein9s disen™h—ntment with w—™h9s prin™ipleF ‚—therD it w—s present even in his e—rliest writings ˜ene—th the ™on™erns for the rel—tive motion of ˜odies —nd the o˜serv—˜ility of ™—usesF iinstein @IWIQD ppIPTH{IA m—kes the essenti—l pointX F F F in ‘theories ™urrent tod—y“D the inerti—l system is introdu™edY its st—te of motionD on the one h—ndD is not ™onditioned ˜y the st—tus of o˜serv—˜le o˜je™ts @—nd therefore ™—used ˜y nothing —™™essi˜le to per™eptionA ˜utD on the other h—ndD it is supposed to determine the rel—tions of m—teri—l pointsF e footnote e—rlier in the p—r—gr—ph —lso tried to identify wh—t w—s so uns—tisf—™tory —˜out inerti—l systemsF ‡h—t is uns—tisf—™tory —˜out this is th—t it rem—ins unexpl—ined how the inerti—l system ™—n ˜e singled from other systemsF „hus we h—ve here the enduring ™ore of the ™luster of ide—s th—t led iinstein to the rel—tivity of inerti— —nd w—™h9s prin™ipleX his ™on™ern th—tD through their introdu™tion of inerti—l systemsD e—rlier theories —llowed —˜solutes th—t —™ted ˜ut ™ould not ˜e —™ted uponF pin—llyD we m—y —sk whether the –essenti—l —™hievement9 of gener—l rel—tivityD the elimE in—tion of the —˜solute inerti—l systemsD follows —utom—ti™—lly from gener—l ™ov—ri—n™e in iinstein9s viewD so th—t gener—l ™ov—ri—n™e would then truly —mount to — gener—lized prinE ™iple of rel—tivity in — form —d—pted to — eld theoryF st is h—rd to nd — ™le—r —nswer in iinstein9s writingsF ris IWIV ™—t—logue of three prin™iples suggested th—t the requirement of gener—l ™ov—ri—n™e @–@—A prin™iple of rel—tivity9A needed to ˜e supplemented ˜y someE thing —ddition—l @–@™A w—™h9s prin™iple9A to re—lize fully the gener—l rel—tivity of motionF 12ƒee —lso iinstein @IWPPD pIVD IWPRD pVVAF General covariance and general relativity 811 iinstein9s text suggests this without ™le—rly st—ting itD for iinstein @IWIVD pPRIA introdu™es the three prin™iples with the rem—rk th—t they —re –in —ny ™—se in no w—y independent from e—™h other9F „his it is not ™le—r whether these p—rti™ul—r two of the three prin™iples re—lly —re independent orD if they —re notD whether gener—l ™ov—ri—n™e somehow le—ds to w—™h9s prin™ipleF €erh—ps the ˜est —nswer we will nd is iinstein9s repe—ted insisten™e th—t gener—l ™ov—ri—n™eD in ™onjun™tion with the requirement of simpli™ityD le—ds us dire™tly to gener—l rel—tivity @seeD for ex—mpleD iinstein @IWSPD pp ISP{QD IWRWD ppUI{QD IWQQD pPURAAF end it is this theory th—t elimin—tes the —˜soluteness of the inerti—l systemF 4. The favourable text-book assimilation of Einstein's view: fragmentation and mutation elthough iinstein h—d to struggle to g—in —™™ept—n™e of this theory in its e—rliest ye—rs @espe™i—lly prior to IWITAD ˜y IWPH iinstein9s new theory w—s widely ™ele˜r—tedF „he extr—v—g—nt pu˜li™ity surrounding the su™™ess of iddington9s IWIW e™lipse expedition h—d even l—un™hed iinstein into the popul—r press —nd pu˜li™ eyeF huring this periodD the v—st m—jority of —™™ounts of iinstein9s theory merely sought to re™—pitul—te iinstein9s own —™™ountF „hus ˜eg—n the tr—dition of writing in wh—t s ™—ll the f—vour—˜le —ssimil—tion of iinstein9s view —nd whi™h is to ˜e reviewed in this se™tionF s sh—ll ™onsider —n —™™ount of the found—tions of gener—l rel—tivity f—vour—˜le to iinstein9s view if it n—mes some or —ll of iinstein9s three prin™iples of IWIV —s found—tions of the theoryX prin™iple of rel—tivityG™ov—ri—n™eD prin™iple of equiv—len™e —nd w—™h9s prin™ipleY it must in™lude —t le—st the rst prin™ipleF „wo things will ˜e™ome ™le—r —˜out the f—vour—˜le re™eption of iinstein9s —™™ount of the found—tions of gener—l rel—tivityF pirstD it is very widespre—d —nd still — m—jor tr—dition tod—yF ƒe™ondD wh—t is often oered —s — re™—pitul—tion of iinstein9s —™™ount|even if only t—™itly|™—n dier in very signi™—nt w—ys from wh—t iinstein re—lly s—idF wost prominentlyD the rel—tivity of inerti— —nd w—™h9s prin™iple is only infrequently reported —s p—rt of the found—tions of gener—l rel—tivity in more te™hni™—l expositionsF „his disf—vour is not — response to iinstein9s own l—ter disillusionment with w—™h9s prin™ipleF prom the e—rliest momentsD the prin™iple f—iled to nd — pl—™e in the m—jority of —™™ounts within more te™hni™—l expositionsF ‚—ther the f—vour—˜le —™™ounts r—pidly st—˜ilizedD most ™ommonlyD into lo™—ting the found—tions of gener—l rel—tivity in the prin™iple of equiv—len™e —nd the prin™iple of rel—tivityF iven hereD these —™™ounts h—ve f—iled to rem—in f—ithful to iinstein9s viewpointF „hey —lmost ex™lusively employ —n innitesim—l prin™iple of equiv—len™eD — v—ri—nt form th—t iinstein never endorsed —nd w—s quite dierent in outlook from iinstein9s own formF sn order to g—uge the m—gnitude —nd ™h—r—™ter of the f—vour—˜le re™eptionD this se™tion will review the f—vour—˜le —™™ounts of the found—tions of gener—l rel—tivity —s they h—ve —ppe—red in the text˜ooks on gener—l rel—tivityF „he review is —lso limited prin™ip—lly to expositions th—t either proved — selfE™ont—ined exposition of tensor ™—l™ulus or su™ient dierenti—l geometry for gener—l rel—tivity or presume su™h knowledge in the re—der —nd th—t pro™eed —t le—st —s f—r —s — formul—tion of the gr—vit—tion—l eld equ—tionsF ‡e should note —lso th—t the f—vor—˜le re™eption extends ˜eyond the re—lm of rel—tivity theoryF eguirre —nd ur—use @IWWID pSHVA —re prep—red to l—˜el — me™h—ni™s —s –gener—l rel—tivisti™9 merely ˜e™—use it is gener—lly ™ov—ri—ntF te—n iisenst—edt @IWVTD IWVWA h—s des™ri˜ed the rising —nd f—lling fortunes of gener—l rel—tivityF efter —n initi—l period of gre—t interest —nd —™tivity in the l—te IWIHs —nd e—rly IWPHsD the theory fell into de™—des of negle™t ˜e™—use of m—ny f—™torsX — sense th—t the theory h—d only slender ™onrm—tionD th—t its pr—™ti™—l utility to physi™ists w—s sm—ll —nd 812 J D Norton th—t the theory h—d ˜een e™lipsed ˜y the developments in qu—ntum theoryF „he IWTHs s—w — new vigour in work on the theoryD in p—rt due to — renewed interest in empiri™—l tests of the theory —nd to the exploit—tion of newD more sophisti™—ted m—them—ti™—l toolsF sn the followingD the f—vour—˜le re™eptions is divided into periods ree™ting these shifts in intensity of workF pirstD howeverD s will review the spe™i—l pro˜lem of the prin™iple of equiv—len™eF 4.1 Einstein's principle of equivalence as a covariance principle and its later misrepresentation „here —re m—ny inst—n™es of l—ter —™™ounts misrepresenting iinstein9s ide—sF xone is —s univers—l —nd ™omplete —s the l—ter tre—tments of iinstein9s prin™iple of equiv—len™eF sn his MK——™se™iaiefsnl—ei—rng—ngrt—iionvnfgietRr—wtetiiil—toalhntis—vryleitssytpeeDelmid™tiwnitnseotreseKipnpeFr™geriis—v—elenvsrtien——lg—snttndi—votKitetemyHdue—tnnnht—d—™ot™fKeftrlehHeree——mtpsery—dinss9Dts™eeiipmsilniensottfyKepi™nHio™—o@—rIrledWoPi—nfP™———™tlDeellpsehSriu—Usnt{weVifdrAoirtt–jmihunelsgnytF writesX F F F there is nothing to prevent our ™on™eiving this gr—vit—tion—l eld —s re—lD th—t isD the ™on™eption th—t KH is –—t rest9 —nd — gr—vit—tion—l eld is present we ™—n ™onsider —s equiv—lent to the ™on™eption th—t only K is —n –—llow—˜le9 system of ™oEordin—tes —nd no gr—vit—tion—l eld is presentF „he —ssumption of the ™omplete physi™—l equiv—len™e of the systems of ™oordin—tesD K —nd KHD we ™—ll the –prin™iple of equiv—len™e9Y F F F ‘it“ signies —n extension of the prin™iple of rel—tivity to ™oEordin—te systems whi™h —re in nonEuniform motion rel—tive to e—™h otherF sn f—™tD through this ™on™eption we —rrive —t the unity of n—ture of inerti— —nd gr—vit—tionF iinstein howeverD is ne—rly univers—lly understood —s urging — r—ther dierent prin™ipleD whi™h s sh—ll ™—ll the –innitesim—l prin™iple of equiv—len™e9F e ™—noni™—l formul—tion is given in €—uli @IWPID pIRSAX por every innitely sm—ll world region @iFeF — world region whi™h is so sm—ll th—t the sp—™eE —nd timeEv—ri—tion of gr—vity ™—n ˜e negle™ted in itA there —lw—ys exists — ™oordin—te system K0@X1; X2; X3; X4A in whi™h gr—vit—tion h—s no inuen™e either on the motion of p—rti™les or —ny other physi™—l pro™essF „he key ide— here is th—t in —dopting — su™iently sm—ll region of sp—™etimeD —n —r˜itr—ry gr—vit—tion—l eld ˜e™omes homogenous —nd ™—n ˜e tr—nsformed —w—y ˜y — suit—˜le ™hoi™e of ™oordin—te systemF „his prin™iple exists in m—ny v—ri—nt formsF ƒometimes it is strengthE ened to require th—t when the gr—vit—tion—l eld is tr—nsformed —w—y we re™over spe™i—l rel—tivity lo™—lly @for ex—mpleD wisner et al. IWUQD pQVTA ‡ith somewh—t dierent qu—liE ™—tionsD €—uli9s innitesim—l prin™iple ™orresponds to hi™ke9s –strong equiv—len™e prin™iple9 @‚oll et al.D IWTRD pRRRAF hi™ke9s –we—k equiv—len™e prin™iple9D howeverD requires only the uniqueness of gr—vit—tion—l —™™eler—tionD whi™h —mounts to requiring th—t the tr—je™tories of free f—ll of suit—˜ly ide—lized ˜odies —re independent of their ™onstitutionsF …nlike most other writesD €—uli @IWPID pIRSA —™knowledged th—t his innitesim—l verE sion of the prin™iple of equiv—len™e diered from iinstein9sD suggesting th—tD where iinE stein9s prin™iple —pplied only to homogeneous gr—vit—tion—l eldsD €—uli9s version w—s for the –gener—l ™—se9F rowever the dieren™es r—n f—r deeper th—n €—uli —llowed —nd pert—in to quite fund—ment—l questions of the role of the prin™iple of equiv—len™e in gener—l relE —tivityF „hese dieren™es ™—n ˜e summ—rized in three essenti—l —spe™ts of the prin™iple whi™h rem—ined xed throughout iinstein9s writings on gener—l rel—tivityD from the e—rliest moments in IWHUD to his n—l ye—rs in the IWSHsX13 13„he ™—se for these dieren™es ˜etwee iinstein9s version —nd the ™ommon innitesim—l version General covariance and general relativity 813  iinstein9s prin™iple of equiv—len™e w—s — ™ov—ri—n™e prin™ipleF ƒpe™i—l rel—tivity required the ™omplete physi™—l equiv—len™e of —ll inerti—l ™oordin—te sysE temsY for iinsteinD gener—l rel—tivity required the ™omplete equiv—len™e of —ll ™oordin—te systemsF iinstein9s prin™iple of equiv—len™e required the ™omplete equiv—len™e of — set of ™oordin—te systems of intermedi—te sizeX inerti—l systems plus uniformly —™™eler—ted ™oorE din—te systemsF „h—t isD the prin™iple s—n™tioned the extension of the ™ov—ri—n™e of spe™i—l rel—tivity ˜eyond vorentz ™ov—ri—n™e ˜ut not —s f—r —s gener—l ™ov—ri—n™eF „husD for iinE steinD the prin™iple of equiv—len™e w—s — rel—tivity prin™iple intermedi—te in r—nge ˜etween the prin™iple of rel—tivity of spe™i—l rel—tivity —nd of gener—l rel—tivityF „he point is so import—nt for our ™on™erns here th—t it is helpful to h—ve it in iinstein9s own words @IWSHD pQURAX „his is the gist of the prin™iple of equiv—len™eX sn order to —™™ount for the equ—lity of inert —nd gr—vit—tion—l m—ss within the theory it is ne™ess—ry to —dmit nonEline—r tr—nsform—tions of the four ™oordin—tesF „h—t isD the group of vorentz tr—nsform—tions —nd hen™e the set of –permissi˜le9 ™oordin—te systems h—s to ˜e extendedF yrD more su™™in™tlyD in —n —rti™le devoted to expli™—ting pre™isely wh—t he intended with his prin™iple of equiv—len™eD iinstein @IWIT—D pTRIA wrote in emph—sized textX „he requirement of gener—l ™ov—ri—n™e of equ—tions em˜r—™es th—t of the prin™iple of equiv—len™e —s — quite spe™i—l ™—seF „he fun™tion of the —ltern—tiveD innitesim—l prin™iple of equiv—len™e is to stipul—te th—t — sp—™etime of gener—l rel—tivity with —n —r˜itr—ry gr—vit—tion—l eld is in some sense lo™—lly|th—t isD in innitesim—l regions|like the sp—™etime of spe™i—l rel—tivityF @iinstein o˜je™ted in ™orresponden™e with ƒ™hli™k to the l—tter9s use of this ide—D pointing out to ƒ™hli™k th—t the sense in whi™h spe™i—l rel—tivity holds lo™—lly must ˜e so we—k th—t —™™elE er—ted —nd un—™™eler—ted p—rti™les ™—nnot ˜e distinguishedF por det—ilsD see xorton @IWVSD se™tion WAFA es — ™ov—ri—n™e prin™ipleD iinstein9s version of the prin™iple served no su™h fun™tionF „herefore it w—s inv—ri—˜ly restri™ted in the following rel—ted w—ysX  iinstein9s prin™iple of equiv—len™e w—s —pplied only in spe™i—l rel—tivity to wh—t we now would ™—ll winkowski sp—™eEtimesF „h—t isD the inerti—l ™oordin—te system K of iinstein9s formul—tion of the prin™iple is not some kind of free f—ll ™oordin—te system of gener—l rel—tivityF st is simply —n inerti—l ™oordin—te system of spe™i—l rel—tivityF „hus the ™oordin—te systems K —nd KH —re ˜oth ™oordin—te systems of — winkowski sp—™etimeF fe™—use of thisD we would now ˜e in™lined to pi™ture the entire prin™iple —s oper—ting within spe™i—l rel—tivityF „his seems not to h—ve ˜een iinstein9s viewF re seems to h—ve reg—rded spe™i—l rel—tivity supplemented with the prin™iple of equiv—len™e —s h—ving more physi™—l ™ontent th—n spe™i—l rel—tivity —loneF „he supplemented theory h—d — wider ™ov—ri—n™e —nd it de—lt with — new phenomenonD homogeneous gr—vit—tion—l eldsF  iinstein9s prin™iple of equiv—len™e w—s not — pres™ription for tr—nsforming —w—y arbitrary gr—vit—tion—l eldsY it w—s just — re™ipe for ™re—ting — special type of gr—vit—E tion—l eldF iinstein9s prin™iple of equiv—len™e g—ve — re™ipe for ™re—ting — homogeneous gr—vit—tion—l eld ˜y tr—nsforming to — uniformly —™™eler—ted ™oordin—te systemF „he innitesim—l prin™iple gives — re™ipe for tr—nsforming —w—y —n —r˜itr—ry gr—vit—tion—l eldX one rst homogenizes it ˜y ™onsidering —n innitesim—l region of sp—™etime —nd then tr—nsforms it —w—y ˜y the reverse tr—nsform—tion of iinstein9s prin™ipleF iinstein repe—tedly insisted th—t his prin™iple of equiv—len™e did not —llow one to tr—nsform —w—y —n —r˜itr—ry gr—vit—tion—l eldD ˜ut only gr—vit—tion—l els of — quite spe™i—l typeD those produ™ed ˜y —™™eler—tion of the prin™iple is l—id out is some det—il in xorton @IWVSA 814 J D Norton of the ™oordin—te systemF @iinstein devotes — p—r—gr—ph of ne—r p—ge length to this point @IWIT—D ppTRH{IAF ƒee xorton @IWVSD se™tion PAFA14 4.2. The early years: 1916{1930 iinstein h—d n—med w—™h9s prin™iple —s one of the three fund—ment—l prin™iples of gener—l rel—tivityF roweverD the prin™iple or its pre™ursorD the rel—tivity of inerti—D h—s pl—yed the le—st role in —™™ounts of the found—tions of gener—l rel—tivityF „ypi™—lly the prin™iple does not —ppe—r in the dis™ussion of the found—tions of the theoryF sf it —ppe—rs in —n expositionD it —rises most ™ommonly l—ter in the ™ontext of the ™osmologi™—l pro˜lem —nd not —lw—ys in — f—vour—˜le lightD even in expositions otherwise well disposed to iinstein9s viewpointF „his p—ttern w—s set —n the e—rliest momentsF sn IWIT —nd IWIU the hut™h —stronomer de ƒitter took up the t—sk of —llowing the qerm—ns —nd fritish to ex™h—nge more th—n —rtill—ry shellsF re presented — three p—rt report to the fritis™h roy—l estronomi™—l ƒo™iety on iinstein9s new theory of gr—vit—tion @ de ƒitter IWITAF ‡hilst otherwise f—vour—˜le to iinsteinD its se™ond p—rt ™on™luded with ™riti™ism of iinstein9s notion of the rel—tivity of inerti—F hevelopment of this ™riti™ism ™ontinued in the third p—rtF iinstein9s IWIU work on the ™osmologi™—l pro˜lem —nd his IWIV formul—tion of w—™h9s prin™iple did not improve the re™eption of his ide—s on the origin of inerti—F v—ue9s @IWPIDppIUW{VHA e—rly gener—l rel—tivity text mentions them only in p—ssing —s in™omp—ti˜le with winkowski—n ˜ound—ry ™onditions —t sp—ti—l innityF re nds the whole question physi™—lly too un™l—ried to w—rr—nt further dis™ussionF €—uli @IWPIA does give the question more ™over—geD ˜ut only in — l—terD ™losing se™tion @se™tion TPA iinstein9s ide—s on the rel—tivity of inerti— gured more prominently in more popul—r expositions of gener—l rel—tivityF por ex—mple preundli™h @IWIWD se™tion RAD „hirring @IWPPD se™tion ˆ†AD forn @IWPRD ™h †ssD se™tion IA —nd uop @IWPQD pp P{SD IWI{SA tre—t the rel—tivity of inerti—F sndeedD the more popul—r the textD the more likely we —re to nd these ide—s used to expl—in the found—tions of gener—l rel—tivityF „he liter—ture on w—™h9s prin™iple h—s ˜e™ome enormous —nd is ourishing tod—yF rowE ever its ™on™erns h—ve ™ome to diverge from the ™on™erns of this —rti™leD gener—l ™ov—ri—n™e —nd the found—tions of gener—l rel—tivityF „he interested re—der is referred to ‚einh—rt @IWUQA —nd „orretti @IWVQD ppIWR{PHPA for further dis™ussionF ‡h—t is most import—nt for our ™on™erns is th—t the m—jority of expositions of relE —tivity theory from this period emph—size the gener—l ™ov—ri—n™e of gener—l rel—tivity —s espe™i—lly import—ntF yf ™ourse this emph—sis w—s justied if only for the novelty of gener—l ™ov—ri—n™eF rowever the —™hievement of gener—l ™ov—ri—n™e w—s —lso routinely —ssumed to ensure —utom—ti™ s—tisf—™tion of — gener—lized prin™iple of rel—tivityF sn some expositions this —ssumption w—s dis™ussed in det—ilD in others it w—s merely suggested ˜y l—˜elling the requirement of gener—l ™ov—ri—n™eD — prin™iple of rel—tivityF e™™ounts th—t emph—size gener—l ™ov—ri—n™e —nd presume —n —utom—ti™ ™onne™tion to — gener—lized prin™iple of relE —tivity in™ludeX de ƒitter @IWITD ppUHH{HPAD preundli™h @IWIWD pPVAD g—rmi™he—l @IWPHD ™h †ssAD €—ge @IWPHD pQVUAD ƒ™hli™k @IWPHD ppSP{QAD gunningh—m @IWPID ™h †ssAD he honder @IWPID ppIH{IRAD v—ue @IWPID pPIAD €—uli @IWPID se™tion SPAD ‡eyl @IWPID se™tion PUAD fe™E querel @IWPPAD uottler @IWPPD ppIVV{WAD „hirring @IWPPD pISIAD uop @IWPQAD forn @IWPRD ™h †ssAD ‚ei™hen˜—™h @IWPRD pIRIAD veviEgivit— @IWPTD pPWRAD vevinson —nd eisler @IWPWD pUHAF ƒome of these —™™ounts expli™itly invoke iinstein9s pointE™oin™iden™e —rgument to 14iinstein himself never employed the tri™k of homogenizing —n —r˜itr—ry gr—vit—tion—l eld ˜y ™onsidering innitesim—l regions of sp—™etimeF sn IWIPD when his prin™iple still de—lt only with homogenous gr—vit—tion—l eldsD he w—s for™ed to restri™t it to innitesim—l regions of sp—™e to over™ome ™ert—in te™hni™—l di™ulties with this theory of st—ti™ gr—vit—tion—l eldsF ‡hen they were over™omeD the restri™tion dis—ppe—redF ƒee xorton @IWVSD se™tion RFQAF General covariance and general relativity 815 est—˜lish gener—l ™ov—ri—n™eF „hey in™lude de ƒitter @IWITD ppUHH{HPAD g—rmi™he—l @IWPHD ™h †ssAD ƒ™hli™k @IWPHD ppSP{QAF w—ny of the expositions —lso pl—™e gre—t emph—sis on the prin™iple of equiv—len™eF e few from the very e—rliest ye—rs st—te the prin™iple in ex—™tly iinstein9s f—shionX „hirring @IWPPD pIHWAD uop @IWPQD pIIHA @—lso g—rmi™h—el @IWPHD pVHAD —lthough ™riti™—llyAF ythers employ the now f—mili—r innitesim—l prin™iple of equiv—len™eD other v—ri—nt formul—tions of the prin™iples or give v—gue ™h—r—™teriz—tions of the prin™iple th—t defy ™le—r ™l—ssi™—tionF „he following —t le—st n—me — prin™iple of equiv—len™e in the found—tions of gener—l rel—tivityX preundli™h @IWPPD se™tion SSAD uottler @IWPPD p IWPAD forn @IWPR ™h †ssAD ‚ei™hen˜—™h @IWPRD pp IRI{PAF 4.3. The lean years: 1930{1960 huring these three le—n de™—des for gener—l rel—tivityD the volume of pu˜li™—tion fell to the merest tri™kleF ‡ithin th—t tri™kleD iinstein9s view of gener—l ™ov—ri—n™e rem—ined — domin—nt themeF e™™ounts of gener—l rel—tivity whi™h emph—sized the gener—l ™ov—ri—n™e of the theory —nd either expli™itly or t—™itly took this gener—l ™ov—ri—n™e to extend the prin™iple of rel—tivity in™ludeX fergm—nn @IWRPD ™h ˆAD ƒ™hrodinger @IWSHD pPAD woller @IWSPD ™h †ssAD tord—n @IWSSD se™tion IRAD ur—tzer @IWSTD se™tion ISAD f—rgm—nn @IWSUD pITPAD „onnel—t @IWSUD ™h ˆsA ell ˜ut ƒ™hrodinger —nd tord—n introdu™e — prin™iple of equiv—len™e ˜y n—meF woller @IWSPD ppPIW{PHA introdu™es gener—l rel—tivity with — dis™ussion of the rel—tivity of inerti—F „olm—n @IWQRD pQ —nd ™h †sA is ex™eption—l in oering iinstein9s three prin™iples of IWIV|the prin™iple of ™ov—ri—n™eD the prin™iple of equiv—len™e —nd w—™h9s prin™iple|—s the found—tions of gener—l rel—tivityF rowever his version of the prin™iple of equiv—len™e is the innitesim—l version never endorsed ˜y iinstein —nd he —™™epts urets™hm—nn9s view of the physi™—l v—™uity of the prin™iple of ™ov—ri—n™eD while insisting with iinstein on its heuristi™ v—lueF 4.4. Rebirth 1960{1980 „he ren—iss—n™e of gener—l rel—tivity in the IWTHs ˜rought ™le—rer divisions in the liE ter—ture on the found—tions of gener—l rel—tivityF es we sh—ll see ˜elowD one in™re—singly import—nt str—nd either simply ignored iinstein9s view of the found—tions of the theory or ˜e™—me quite strident in its denun™i—tion of iinstein9s viewF enother sought to rep—ir iinstein9s —™™ount in the f—™e of su™h —ss—ultsF e m—jor p—rt of the liter—tureD howeverD ™ontinued in simple —ssent with iinstein9s viewD only m—king sm—ller —djustment —™™ording to t—steF wost ™ommonlyD —™™ounts in this l—st ™—tegory found ˜oth —n innitesim—l prin™iple of equiv—len™e —nd the prin™iple of gener—l ™ov—ri—n™e in the found—tions of gener—l rel—tivityF ƒu™h —™™ounts in™ludeX ‡e˜er @IWTID se™tions IFQD PFRAD fergm—nn @IWTID IWTPAD v—wden @IWTPD ™h TAD ‚osser @IWTRD se™tions IPFID IPFPAD w™†ittie @IWTSD ™h RAD ‰ilm—z @IWTSD ™h ISD ITAD ƒkinner @IWTWD ™h QAD h—vis @IWUHD SFsFPAD €r—s—nn— @IWUID pref—™eD ™h IAD w—vrides @IWUQD se™tions sssFRD ssFSAD €—p—petrou @IWURD sntrodu™tionD se™tion IVAD €—thri— @IWURD ™h TDUAD fowler @IWUTD ™h WAD edlerD f—zin —nd ƒ™hier @IWUUD pTH —nd se™tion SFIAD ƒteph—ni @IWUUD se™tion VFIAD „reder et al. @IWVHD sntrodu™tionAF wost of these —™™ounts expli™itly ™onne™ted gener—l ™ov—ri—n™e with — gener—lized prin™iple of rel—tivityD either in n—me or ˜y expli™it dis™ussionF „hese in™ludeX fergm—n @IWTID IWTPAD v—wden @IWTPD ™h TAD ‚osser @IWTRD se™tions IPFID IPFPAD ‰ilm—z @IWTSD ™h ISD ITAD €r—s—nn— @IWUIAD w—vrides @IWUQD se™tions sssFRDAD €—p—petrou @IWURD sntrodu™tionAD €—thri— @IWURD ™h TDAD fowler @IWUTD ™h WAD edlerD f—zin —nd ƒ™hier @IWUUD pTH —nd se™tion SFIAD ƒteph—ni @IWUUD se™tion VFIAD 816 J D Norton „reder et al. @IWVHD sntrodu™tionAF ƒkinner @IWTWD se™tion QFQFIA reported th—t the prin™iple of gener—l rel—tivity required something ˜eyond the prin™iple of ™ov—ri—n™eX –the l—ws of physi™s must determine the geometry of sp—™etime —ppropri—te for — p—rti™ul—r physi™—l ™ir™umst—n™e9F „wo —™™ounts portr—yed gener—l ™ov—ri—n™e —s — gener—lized prin™iple of rel—tivity ˜ut did not pl—™e the prin™iple of equiv—len™e ˜y n—me in the found—tions of gener—l rel—tivityX gh—ron @IWTQD le™$on VAD etw—ter @IWURAF w—™h9s prin™iple is mentioned ˜y v—wden @IWTPD pIQQAF ‡ork on gener—l rel—tivity in this period —lso g—ve rise to — v—ri—nt form of the prin™iple of gener—l ™ov—ri—n™eF ‡ein˜erg @IWUUD ppWI{PA dened his prin™iple of gener—l ™ov—ri—n™e —sX st st—tes th—t — physi™—l equ—tion holds in — gener—l gr—vit—tion—l eldD if two ™onditions —re metX IF „he equ—tion holds in the —˜sen™e of gr—vit—tionX th—t isD it —grees with the l—ws of spe™i—l rel—tivity when the —ne ™onne™tion the metri™ eld v—nishesF g equ—ls the winkowski tensor  —nd when PF „he equ—tion is gener—lly ™ov—ri—ntY th—t isD it preserves its form under — gener—l ™oordin—te tr—nsform—tion x 3 xHF „he noveltyD of ™ourseD is th—t the se™ond ™ondition —lone is usu—lly t—ken —s the prin™iple of gener—l ™ov—ri—n™eD where—s the rst looks like — form of the innitesim—l prin™iple of equiv—len™eF sndeed ‡ein˜erg presents the prin™iple —s —n —ltern—te form of the innitesiE m—l prin™iple of equiv—len™e —nd shows how it follows from the prin™iple of equiv—len™eF re insists th—t it is not — rel—tivity prin™iple like the vorents inv—ri—n™e of spe™i—l rel—tivityF fose @IWVHD ™h IA lo™—tes the found—tions of gener—l rel—tivity in — lo™—l prin™iple of equivE —len™e —nd its reEexpression in — two ™ondition prin™iple of gener—l ™ov—ri—n™e equiv—lent to ‡ein˜erg9sF ƒimil—rly poster —nd xighting—le @IWUWD ppxiExiiiA lo™—te the found—tions of gener—l rel—tivity in —n innitesim—l prin™iple of equiv—len™e —nd — version of the prin™iE ple of gener—l ™ov—ri—n™e essenti—lly the s—me —s ‡ein˜erg9sF „hey strengthen ‡ein˜erg9s ™ondition PF to re—d ‘P9F“ the equ—tion is — tensor equ—tion @iFeF it preserves its form under gener—l ™oordin—te tr—nsform—tionAF „he strengthening lies in the f—™t th—t not only tensor equ—tions —re ™ov—ri—nt under —r˜iE tr—ry ™oordin—te tr—nsform—tionsF ƒee —lso „reder et al. @IWVHA 4.5. Recent years since 1980 „he ye—rs sin™e IWVH h—ve seen no resolution of the dis—greements over the found—tions of gener—l rel—tivityF es we sh—ll see l—terD the liter—tures th—t reje™t iinstein9s —™™ount or seek m—jor rep—irs ™ontinue to ourishF et the s—me timeD — signi™—nt liter—ture ret—ins — viewpoint —lmost —s ™lose to iinstein9s —s the f—vour—˜le re™eption in the IWPHsF fro—dlyD in this l—tter liter—tureD the found—tions of gener—l rel—tivity —re still lo™—ted within —n innitesim—l prin™iple of equiv—len™e —nd — prin™iple of gener—l ™ov—ri—n™eF „wo —™™ounts oer essenti—lly ‡ein˜erg9s viewF foth ƒtr—um—nn @IWVRD ™h PA —nd uenyon @IWWHD ™h IA ˜—se gener—l rel—tivity on —n innitesim—l prin™iple of equiv—len™eF @uenyon dis™usses ˜oth hi™ke9s we—k —nd strong versionD with the l—tter —mounting to —n innitesim—l prin™ipleFA uenyon @IWWHD se™tion TFRA gives — formul—tion of the prin™iple of gener—l ™ov—ri—n™e whi™h is essenti—lly ‡ein˜erg9s —s strengthened ˜y poster —nd xightinE g—le @see —˜oveAF ‡ithout expli™itly introdu™ing the n—meD prin™iple of gener—l ™ov—ri—n™eD ƒtr—um—nn @IWVRD se™tion IFQA provides two requirements whi™h —re –— m—them—ti™—l forE mul—tion of the prin™iple of equiv—len™e9F „he rst is —™tu—lly the prin™iple of minim—l ™ouplingD — version of the prin™iple of equiv—len™e @„r—utm—n IWTSD enderson IWTUD pQQUD General covariance and general relativity 817 enderson —nd q—ute—u IWTWAF „he se™ond requirement is essenti—lly ‡ein˜erg9s version of the prin™iple of gener—l ™ov—ri—n™eF he peli™e —nd gl—rke @IWWHD pU{IQA lo™—te the found—tions of gener—l rel—tivity in the f—mili—r innitesim—l prin™iple of equiv—len™e —nd prin™iple of gener—l ™ov—ri—n™eF g—meli @IWVPD se™tion IFRD IFSA lo™—tes the found—tion of the theory in these s—me two prin™iplesF re doesD howeverD deline—te three versions of the prin™iple of gener—l ™ov—ri—n™e whi™hD he notesD —re –not quite equiv—lent9F IF ell ™oordin—te systems —re equ—lly good for st—ting the l—ws of physi™sF ren™e —ll ™oordin—te systems should ˜e tre—ted on the s—me footingD tooF PF „he equ—tions th—t des™ri˜e the l—ws of physi™s should h—ve tensori—l forms —nd ˜e expressed in — fourEdimension—l ‚iem—nni—n sp—™etimeF QF „he equ—tions des™ri˜ing the l—ws of physi™s should h—ve the s—me form in —ll ™oordin—te systemsF illis —nd ‡illi—ms @IWVVD se™tion SFPA lo™—te the found—tions of the theory in —n innitesim—l prin™iple of equiv—len™e —nd wh—t they ™—ll —n extension of the prin™iple of rel—tivityX –the l—ws of physi™s —re the s—me for —ll o˜serversD no m—tter wh—t their st—te of motion9F „he term prin™iple of gener—l ™ov—ri—n™e is not mentionedF ƒexl —nd …r˜—ntke @IWVQA tre—t —ll three of iinstein9s prin™iples of IWIVF „he prin™iple of equiv—len™e @se™tion IFPA is given most emph—sisD —lthough in it innitesim—l formF w—™h9s prin™iple —nd the prin™iple of gener—l ™ov—ri—n™e —re mentioned only —pp—rently for histori™—l interest @se™tion RFSAD with the l—tter oered —s iinstein9s —ttempt to s—tisfy the formerF pin—llyD d9svorno @IWWPD ™h WA in — ™h—pter entitled –„he €rin™iples of qener—l ‚el—tivity9D —™knowledges th—t these prin™iples h—ve ˜een — sour™e of mu™h ™ontroversyF roweverD —s prin™iples fund—ment—l to gener—l rel—tivity or —t le—st serious ™—ndid—tes for themD he presents iinstein9s three prin™iples of IWIVD the enderson —nd q—utre—u prin™iple of minim—l ™oupling —nd — prin™iple of ™orresponden™e @with xewtoni—n gr—vit—tion theory —nd spe™i—l rel—tivity in the limiting ™—sesAF „he innitesim—l prin™iple of eqiv—len™e is presented —s the –key prin™iple9F w—™h9s prin™iple is given three formul—tionsD —ll ™losely ™onne™ted with iinstein9s ™osmologi™—l ide—s of IWIU —nd IWIVF d9svorno nds the –full import9 of the prin™iple of gener—l rel—tivity @9—ll o˜servers —re equiv—lent9A ™ont—ined in the prin™iple of gener—l ™ov—ri—n™e @–the equ—tions of physi™s should h—ve tensori—l form9AF endD the hole —rgumentD whi™h gured so prominently in iinstein9s e—rly thinking —˜out gener—l ™ov—ri—n™eD is dis™ussed in se™tion IQFTF „o my knowledgeD this is the rst time the hole —rgument h—s ˜een dis™ussed in — gener—l rel—tivity text in over h—lf — ™enturyF „he hole —rgument h—s —lso re™ently re—ppe—red in the physi™s journ—l liter—tureF ƒeeD for ex—mpleD ‚ovelli @IWWIAF 5. Is general covariance physically vacuous? 5.1 Kretschmann's objection: the point-coincidence argument turned against Einstein sn the tr—dition th—t is skepti™—l of iinstein9s —™™ount of the found—tions of gener—l rel—E tivityD the ˜est known of —ll o˜je™tions is due to urets™hm—nn @IWIUD ppSUS{TAF re ˜eg—n his p—per with the rem—rksF15 15s h—ve suppressed urets™hm—nn9s footnotes in this p—ss—ge to other liter—tureF por further dis™ussion see xorton @IWVPD se™tion VAF ƒee —lso row—rd —nd xorton @forth™omingA for spe™ul—tion th—t these footnotes dire™t re—ders to iinstein9s un—™knowledged sour™e for his pointE™oin™iden™e —rgumentD urets™hm—nn @IWISA3 818 J D Norton „he forms in whi™h dierent —uthors h—ve expressed the postul—te of the vorentzE iinstein theory of rel—tivity|—nd espe™i—lly the forms in whi™h iinstein h—s re™ently expressed his postul—te of gener—l rel—tivity|—dmit the following interpret—tion @in the ™—se of iinsteinD it is required expli™itlyAX e system of physi™—l l—ws s—tises — rel—tivity postul—te if the equ—tions ˜y me—ns of whi™h it is represented —re ™ov—ri—nt with respe™t to the group of sp—tioEtempor—l ™oordin—te tr—nsform—tions —sso™i—ted with th—t postul—teF sf one —™™epts this interpret—tion —nd re™—lls th—tD in the n—l —n—lysisD —ll physi™—l o˜serv—tions ™onsist in the determin—tion of purely topologi™—l rel—tions @–™oin™iden™es9A ˜etween o˜je™ts of sp—tioEtempor—l per™eptionD from whi™h it follows th—t no ™oordin—te system is privileged ˜y these o˜serv—tionsD then one is for™ed to the following ™on™lusionX fy me—ns of — purely m—them—ti™—l reformul—tion of the equ—tions representing the theoryD —nd withD —t mostD m—them—ti™—l ™ompli™—tions ™onne™ted with th—t reformul—tionD —ny physi™—l theory ™—n ˜e ˜rought into —greement with —nyD —r˜itr—ry rel—tivity postul—teD even the most gener—l oneD —nd this without modifying —ny of its ™ontent th—t ™—n ˜e tested ˜y o˜serv—tionF urets™hm—nn9s point is th—t there must ˜e something more to — rel—tivity prin™iple th—n ™ov—ri—n™eF por he —rgues th—t we ™—n t—ke —ny theory —nd reformul—te it so th—t it is ™ov—ri—nt under —ny group of tr—nsform—tions we pi™kY the pro˜lem is not physi™—lD it is merely — ™h—llenge to our m—them—ti™—l ingenuityF sn ˜riefD gener—l ™ov—ri—n™e is physi™—lly v—™uousF „his —t le—stD is how urets™hm—nn9s point h—s ˜een understood —lmost univers—lly —nd it is —lmost wh—t he —™tu—lly —rguedF ris re—l o˜je™tion w—s — little more su˜tleF st depended on — nonEtrivi—l —ssumption th—t virtu—lly —ll l—ter ™omment—tors f—il to report16 ell physi™—l o˜serv—tions ™onsist in the determin—tion of purely topologi™—l rel—tions @–™oin™iden™es9A ˜etween o˜je™ts —nd sp—tioEtempor—l per™eptionF „his —ssumption is ™le—rly re™ogniz—˜le to us —s the ˜—si™ premise of iinstein9s own pointE ™oin™iden™e —rgument @see se™tion QFS —˜oveAF „here ™—n ˜e no question of the import—n™e of this —ssumption to urets™hm—nn9s point even though it is ˜uried in the gr—mm—r of his st—tementF e little l—terD he repe—ts it @pSUWAF F F F —™™ording to the investig—tions of ‚i™™i —nd veviEgivit— @IWHIA it m—y s™—r™ely ˜e dou˜ted th—t one ™—n ˜ring —ny physi™—l system of equ—tions into — gener—lly ™ov—ri—nt form without —lter—tion of its o˜serv—tion—lly test—˜le ™ontentF „his is o˜vious from the ˜eginningD if one on™e —g—in re™—lls th—t stri™tly only purely topologi™—l f—™ts of n—tur—l phenomen— orD —™™ording to iinsteinD ™oin™iden™es —re o˜serv—˜leF „husD —llowing th—t urets™hm—nn9s mention of –topologi™—l f—™ts9 —lludes to his own version of the pointE™oin™iden™e —rgument @see row—rd —nd xortonD forth™omingAD we nd th—t urets™hm—nn9s re—l o˜je™tion is thisX if we accept the point-coincidence argument, then any theory can be given a formulation of arbitrary covariance. „his is — most striking revers—l of fortunesF „he pointE™oin™iden™e —rgument h—d ˜een iinstein9s s—lv—tion from the hole —rgument —nd permitted his return to gener—l ™ov—ri—n™eF roweverD in —dvo™—ting the pointE™oin™iden™e —rgumentD iinstein h—d in ee™t —lre—dy —greed to virtu—lly everything in urets™hm—nn9s o˜je™tionF „o est—˜lish the —dmissi˜ility of gener—l ™ov—ri—n™e for his own theoryD iinstein h—d —llowed th—t the physi™—l ™ontent of — theory resides solely in the o˜serv—˜le ™oin™iden™es it s—n™tionsF ƒin™e these ™oin™iden™es —re preserved under —r˜itr—ry ™oordin—te tr—nsform—tionD the physi™—l ™ontent of — theory is un—e™ted ˜y the —doption of — gener—lly ™ov—ri—nt formul—tionF ‡h—t urets™hm—nn noti™ed w—s th—t this —rgument depended on nothing pe™uli—r to gener—l rel—tivityD so it 16s ™—nnot resist spe™ul—ting th—t this misre—ding is —t le—st in p—rt due to the ˜ewildering ™omplexity of his qerm—n proseD whi™h h—s ˜een disent—ngled ™onsider—˜ly in the —˜ove tr—nsl—tionF „his tr—nsl—tion —lso slightly ™orre™ts the tr—nsl—tion of xorton @IWWPD se™tion VFIAF General covariance and general relativity 819 ™ould equ—lly ˜e used to est—˜lish the —dmissi˜ility of — gener—lly ™ov—ri—nt formul—tion of —ny theoryF eg—in it did not depend on the f—™t th—t the ™ov—ri—n™e group w—s the gener—l groupD so the s—me —rgument est—˜lished the —dmissi˜ility of formul—tions of —ny theory of —r˜itr—ry ™ov—ri—n™eF 5.2. Einstein's reply iinstein @IWIVA responded to urets™hm—nn9s o˜je™tionF r—ving l—id out the three prin™iE ples upon whi™h he ˜elieved gener—l rel—tivity to ˜e ˜—sedD he turned to urets™hm—nn9s o˜je™tionD whi™h he rest—ted ™orre™tly with its now lost premise @p PRPAX gon™erning @—A ‘prin™iple of rel—tivity“D rerr urets™hm—nn o˜serves th—t — prin™iple of rel—tivityD formul—ted in this w—yD m—kes no —ssertions over physi™—l re—lityD iFeF over the content of the l—ws of n—tureY r—therD it is only — requirement on their m—them—ti™—l formulationF „h—t isD sin™e —ll physi™—l experien™e rel—tes only to ™oin™iden™esD it must —lw—ys ˜e possi˜le to represent experien™es of the l—wful ™onne™tion of these ™oin™iden™es ˜y gener—lly ™ov—ri—nt equ—tionsF „herefore he ˜elieves it ne™ess—ry to ™onne™t —nother requirement with the requirement of rel—tivityF iinstein h—d little ™hoi™e ˜ut to —™™ept urets™hm—nn9s pointF „he —ltern—tive w—s to renoun™e the pointE™oin™iden™e —rgument th—t he h—d —dvertised so widelyF rowever he tried to s—lv—ge something of the spe™i—l ™onne™tion ˜etween gener—l ™ov—ri—n™e —nd gener—l rel—tivity in the heuristi™s of theory ™hoi™eF re ™ontinuedX s ˜elieve rerr urets™hm—nn9s —rgument to ˜e ™orre™tD ˜ut the innov—tion proposed ˜y him not to ˜e ™ommend—˜leF „h—t isD if it is ™orre™t th—t one ™—n ˜ring —ny empiri™—l l—w into gener—lly ™ov—ri—nt formD the prin™iple @—A still possesses — signi™—nt heuristi™ for™eD whi™h h—s —lre—dy proved itself ˜rilli—ntly in the pro˜lem of gr—vit—tion —nd rests on the followingF yf two theoreti™—l systems ™omp—ti˜le with experien™eD the one is to ˜e preferred th—t is the simpler —nd more tr—nsp—rent from the st—ndpoint of the —˜solute dierenti—l ™—l™ulusF vet one ˜ring xewtoni—n gr—vit—tion—l me™h—ni™s into the form of —˜solutely ™ov—ri—nt equ—tions @four dimension—lA —nd one will ™ert—inly ˜e ™onvin™ed th—t prin™iple @—A ex™ludes this theoryD not theoreti™—llyD ˜ut pr—™ti™—lly3 „hus iinstein seems to —™™ept urets™hm—nn9s o˜je™tionD ˜egrudginglyD with — qu—li™—tion on the role of gener—l ™ov—ri—n™e in theory ™hoi™e —nd with the reserv—tion th—t gener—l ™ov—ri—n™e in —ll theories would ˜e impr—™ti™—lF sndeed it is ironi™ th—t the version of the prin™iple of rel—tivity given in this s—me p—per ˜y iinstein @quoted in se™tion QFU —˜oveA essenti—lly just rest—tes urets™hm—nn9s pointF17 ‡h—tever ™on™ession iinstein m—de to urets™hm—nn seems to h—ve h—d — lesser ee™t on iinstein9s l—ter writingsF re does o™™—sion—lly —llow th—t gener—l ™ov—ri—n™e is –more ™h—r—™teristi™ of the m—them—ti™—l form of this theory ‘of gener—l rel—tivity“ th—n its physi™—l ™ontent9 @IWPRD pWH{IAF yr th—t the –requirement ‘of gener—l ™ov—ri—n™e“ @™om˜ined with th—t of the gre—test possi˜le logi™—l simpli™ity of the l—wsA limits the n—tur—l l—ws ™on™erned in™omp—r—˜ly more strongly th—n the spe™i—l prin™iple of rel—tivity9 @IWSPD p ISQAF „he heuristi™ role of simpli™ity in ™onne™tion with gener—l ™ov—ri—n™e w—s emph—sized in his Autobiographical Notes @IWRWD pTSAF fut this emph—sis seemed to ˜e forgotten ˜y pUQD where he re™—lledX –‡e h—ve —lre—dy given physical re—sons for the f—™t th—t in physi™s inv—ri—n™e under the wider ‘gener—l“ group h—s to ˜e required9F @iinstein9s emph—sisA wore 17„he only dieren™e is th—t urets™hm—nn —llows the pointE™oin™iden™e —rgument to justify — formul—tion of —ny ™ov—ri—n™eD where—s iinstein sees it for™ing — gener—lly ™ov—ri—nt formul—tion —s the –uniqueD n—tur—l expression9 of the theoryF €resum—˜ly this is ˜e™—use — gener—lly ™ov—ri—nt formul—tion —dds the le—st to the ™—t—log of ™oin™iden™esF ƒee iinstein to fessoD t—nu—ry QD IWITD —s quoted in xorton @IWWPD pPWVA 820 J D Norton ™ommonlyD howeverD the qu—li™—tion over simpli™ity is simply not mentionedF st does not —ppe—r —t the relev—nt point in his textD iinstein @IWPP—D pTIAF eg—inD iinstein @IWSHD pQSPA insistsD without expli™it mention of simpli™ity ™onsider—tions th—t F F F the prin™iple of gener—l rel—tivity imposes ex™eedingly strong restri™tions on the theoreti™—l possi˜ilitiesF ‡ithout this restri™tive prin™iple it would ˜e pr—™ti™—lly imE possi˜le for —ny˜ody to hit on the gr—vit—tion—l equ—tions F F F row ™—n we re™on™ile iinstein9s ™on™ession to urets™hm—nn —nd his ™ontinuing emph—sis on the import—n™e of gener—l ™ov—ri—n™ec „he —nswer m—y well lie in iinstein9s f—mous pro™l—m—tion of his IWQQ rer˜ert ƒpen™er le™tureD whi™h reve—led — met—physi™s not present expli™itly in iinstein9s writings of IWIVX yur experien™e hitherto justies us in ˜elieving th—t n—ture is the re—liz—tion of the simplest ™on™eiv—˜le m—them—ti™—l ide—sF s —m ™onvin™ed th—t we ™—n dis™over ˜y me—ns of purely m—them—ti™—l ™onstru™tions the ™on™epts —nd l—ws ™onne™ting them with e—™h otherD whi™h furnish the key to underst—nding the of n—tur—l phenomen— F F F the ™re—tive prin™iple resides in m—them—ti™sF ‡hen iinstein replied to urets™hm—nn th—t one ought to pi™k of two empiri™—lly vi—˜le systems the simpler —nd more tr—nsp—rent within the —˜solute dierenti—l ™—l™ulusD he m—y h—ve ˜een urging something more th—n merely — m—tter of pr—™ti™—l ™onvenien™eF st is not just th—t the simpler is more ™onvenientD so th—t gener—lly ™ov—ri—nt formul—tions of xewtoni—n gr—vit—tion—l —re @he ˜elievedA pr—™ti™—l impossi˜ilitiesF ‡e ™—n re™ognize the truth of — theory in its m—them—ti™—l simpli™ityD end inste—d of ˜eing physi™—lly v—™uousD gener—l ™ov—ri—n™e is the right l—ngu—ge in whi™h to seek this simpli™ityF v—ter writers who endorsed iinstein9s IWIV reply to urets™hm—nn m—y well h—ve —rmed — more extreme met—physi™s th—n they re—lized3 5.3. Generally covariant formulations of Newtonian mechanics sn IWIV iinstein sought to prote™t the spe™i—l ™onne™tion ˜etween gener—l ™ov—ri—n™e —nd his gener—l theory of rel—tivity ˜y issuing — ™h—llengeX nd — gener—lly ™ov—ri—nt formul—tion of xewtoni—n gr—vit—tion—l me™h—ni™sF re h—d ™ondently predi™ted th—t should —nyone try the result would ˜e unwork—˜le pr—™ti™—llyF iinstein w—s shortly proved wrongF g—rt—n @IWPQA —nd priedri™hs @IWPUA found serE vi™e—˜leD gener—lly ™ov—ri—nt formul—tions of xewtoni—n gr—vit—tion theoryF iinstein w—s right in so f—r —s these gener—lly ™ov—ri—nt formul—tions were more ™omplex th—n gener—l rel—tivityF rowever iinstein w—s quite wrong in predi™ting th—t su™h formul—tions would not ˜e us—˜le pr—™ti™—llyF elthough they —re not —s —ttr—™tive — host for routine ™—l™ul—tion —s the f—r simpler q—lile—n ™ov—ri—nt formul—tionD they —re of the s—me order of ™omplexity —s other theories routinely ex—mined in physi™sF rowever there —re ™ert—in ™ir™umst—n™es in whi™h their use is prefer—˜le if not m—nd—toryF sn —n —rti™le ™omp—ring xewtoni—n —nd rel—tivisti™ theories of gr—vit—tionD „r—utm—n @IWTTD pRIQA pointed out su™h ™omp—rison ™—n re—lly only ˜e ee™ted reli—˜ly if the two theories under ™omp—rison —re formul—ted in the s—me m—them—ti™—l l—ngu—geF ytherwise it is h—rd to —s™ert—in whi™h dieren™es —re physi™—l —nd whi™h —re —™™idents of the dieren™es in formul—tionF ƒin™e gener—l rel—tivity is known only in — gener—lly ™ov—ri—nt formul—tionD this me—ns we ought to ™omp—re it only with the gener—lly ™ov—ri—nt formul—tion of xewtoni—n theoryF @por simil—r sentimentsD see —lso r—v—s @IWTRD pWQWA —nd w—l—ment @IWVTD p IVIAFA por this re—sonD — few expositions of rel—tivity in™lude — tre—tment of xewtoni—n sp—™etie theory in — gener—lly ™ov—ri—nt formul—tionD —lthough the pr—™ti™e is not ™ommonF ƒee for ex—mple „r—utm—n @IWTRD ™h SAD —nd wisner et al @IWUQD ™h IPAF sn the philosophy of sp—™e —nd time liter—tureD howeverD the use of the gener—l ™ov—ri—nt formul—tion of xewtoni—n General covariance and general relativity 821 theory is ˜e™oming st—nd—rdD even —t the introdu™tory levelD see i—rm—n —nd priedm—n @IWUQAD i—rm—n @IWURD ppPUT{UAD priedm—n @IWVQD ™h sssAD w—l—ment @IWVTA —nd xorton @IWWP—AF elthough ˜oth g—rt—n —nd priedri™hs were very mu™h ™on™erned with the rel—tionship ˜etween their work —nd iinstein9s gener—l theory of rel—tivityD it is striking th—t neither m—de the o˜vious point th—t their work h—d seriously we—kened iinstein9s IWIV reply to urets™hm—nn —nd r—ised very serious dou˜ts over iinstein9s ™l—im to h—ve gener—lized the prin™iple of rel—tivity to —™™eler—tionF18 st is only l—ter th—t this o˜vious point —˜out gener—lly ™ov—ri—nt formul—tions of xewtoni—n theory is m—deX they provide —n inst—nt—tion of urets™hm—nn9s ™l—im th—t —ny theory ™—n ˜e m—de gener—lly ™ov—ri—ntF ƒee r—v—s @IWTRD p WQWA —nd wisner et al @IWUQD pQHPAF 5.4. Automatic general covariance: coordinate free geometric formulation st did not need the l—˜ours of g—rt—n —nd priedri™hs to show th—t theories other th—n gener—l rel—tivity —dmitted gener—lly ™ov—ri—nt formul—tionsF sn — sense this possi˜ility h—d ˜een known for — long timeF es €—inleve pointed out —s e—rly —s IWPI in his dis™ussion of gener—l rel—tivity @IWPID pVUUAD v—gr—ngi—n me™h—ni™s h—s —lw—ys ˜een inv—ri—nt under —r˜iE tr—ry spatial tr—nsform—tionF eg—inD the moment iinstein —pplied the —˜solute dierenti—l ™—l™ulus of ‚i™™i —nd veviEgivit— to rel—tivity theory in IWIQD it w—s o˜vious th—t spe™i—l rel—tivity ™ould ˜e given gener—lly ™ov—ri—nt formul—tionF sn this formD spe™i—l rel—tivity is simply the theory of sp—™etime with line element @RAD where g is symmetri™ with vorentz sign—ture —nd whose ‚iem—nnEghristoel ™urv—ture tensor v—nishesF „h—t iinstein never em˜r—™ed this o˜vious possi˜ility suggests th—t his underst—nding of gener—l ™ov—ri—n™e w—s — little more ™omplex th—n the simple one supposed in urets™hm—nn9s o˜je™tionF19 €erE h—ps for this re—son or perh—ps just for its simpli™ityD the vorentz ™ov—ri—nt formul—tion of spe™i—l rel—tivity rem—ins popul—r tod—yF „he possi˜ility of formul—ting spe™i—l rel—tivity in —r˜itr—ry ™oordin—tesD howeverD w—s expli™itly re™ognized in the liter—ture quite e—rly @see for ex—mple urets™hm—nn @IWIUD pSUWAD he honder @IWPSD ™h IAD po™k @IWSWD ™h s†D pQSHAF e num˜er of ™omment—tors h—ve o˜served th—t ‚i™™i —nd veviEgivit—9s ™—l™ulus vinE di™—tes urets™hm—nn9s o˜je™tion in the sense th—t it provides the ne™ess—ry m—them—ti™—l —pp—r—tus for nding gener—lly ™ov—ri—n™e formul—tion of –pr—™ti™—lly any —ssumed l—w9 @‡hitt—ker IWSID †ol ssD p ISWA or –—lmost —ny l—w9 @xorth IWTSD pSVAF „his possi˜ility h—s not re—lly ˜een exploited widely in the rel—tivity liter—ture until the IWTHs —nd IWUHs with the introdu™tion of wh—t wisnerD „horne —nd ‡heeler @IWUQA l—˜el —s the –geometri™9 or – ™oordin—te free9 —ppro—™hF „his —ppro—™h is ˜—sed on ‚i™™i —nd veviEgivit—9s ™—l™ulusF roweverD —s w—s pointed out in se™tion QFP —˜oveD the ™—l™ulus w—s ™re—ted expli™itly —s —n —˜str—™t ™—l™ulusD —s independent —s possi˜le from geometri™ notionsF „he ™—l™ulus w—s signi™—ntly —ltered to —rrive —t its modern geometri™ in™—rn—tionF st is now —ugmented with geometri™ ide—s from topologyF „he most signi™—nt —ugment—tions —re the modern ide—s of — dierenti—l m—nifold —nd of — geometri™ o˜je™t of †e˜len —nd ‡hitehe—d @IWQPAD —s well —s —n —˜str—™tD —lge˜r—i™ —ppro—™h to ve™torsD tensors —nd the likeD —ttri˜uted to 18„hus rom—n @IWQPD pIUUA m—kes no mention of g—rt—n9s —nd priedri™h9s work when he rem—rks th—t the gener—l prin™iple of rel—tivity –holds in ex—™tly the s—me words for the xewtoni—n theory ‘—s for gener—l rel—tivity“F ‚—ther the rem—rk is supported merely ˜y o˜serving th—t the prin™iple requires only th—t the m—them—ti™—l expression of — theory ˜e independent of the ™oordin—tes system —nd does not restri™t the theory9s ™ontentF 19sndeedD —s he m—de ™le—r through his prin™iple of equiv—len™eD he held th—t —n extension of the ™ov—ri—n™e of spe™i—l rel—tivity ˜eyond vorentz ™ov—ri—n™e w—s — physi™—l extension of the theoryY his prin™iple of equiv—len™e tells us th—t extending the ™ov—ri—n™e to uniformly —™™eler—ted ™oordin—tes now —llows the theory to em˜r—™e the phenomenon of gr—vit—tion in — spe™i—l ™—seF 822 J D Norton g—rt—n @wisner et al IWUQD ™h V —nd WAF „hese methods ˜e™—me st—nd—rd in the IWTHs —nd IWUHs through su™h expositions of rel—tivity theory —s „r—utm—n @IWTSAD r—wking —nd illis @IWUQAD wisnerD „horne —nd ‡heeler @IWUQAD ƒ—™hs —nd ‡u @IWUUAF pollowing their methodsD we would ™h—r—™terize spe™i—l rel—tivity —s — theory of winkowski sp—™etimesF „h—t isD the theory h—s models hM; gabi ‡here M is — ™onne™tedD fourEdimension—lD dierenti—˜le m—nifold —nd gab is — symmetri™ se™ond r—nk tensor of vorentz sign—ture whi™h is —tD so th—t it s—tises the equ—tion Rabcd a H ‡here Rabcd is the ‚iem—nnEghristoel ™urv—ture tensorF „here —re o˜vious extensions if one wishes to in™lude further eldsD su™h —s — w—xwell eld —nd ™h—rge uxF ƒimil—rlyD gener—l rel—tivityD is the theory with models hM; gab; Tabi where now gab need not ˜e —tF Tab is the se™ond r—nkD symmetri™ stressEenergy tensorD whi™h m—y ˜e required to s—tisfy further –energy ™onditions9 @r—wking —nd illisD IWUQD se™tion RFQAF „he metri™ tensor gab —nd Tab —re rel—ted ˜y the gr—vit—tion—l eld equ—tion Gab a Tab where Gab is the iinstein tensor —nd  — ™onst—nt e typi™—l geometri™ formul—tion of xewtoni—n sp—™etime theory without —˜solute rest @—fter w—l—ment IWVTA h—s models D M; ta; hab; raE „mheterti™heisorsye™9sontedmrp—nork—D lsmymetmrie™trisi™tDasDmisoo—tshmnooontEhvD—nnoisnhEivn—gn™isohnitnrg—v™—orEiv—en™ttoternseolrdDFh„abhD ewshpi—™hti—isl degener—te through its sign—ture @HDIDIDIAD ra is — smooth deriv—tive oper—torD ™onferring —ne stru™ture on the sp—™etimeF „hese stru™tures s—tisfy orthogon—lity —nd ™omp—ti˜ility ™onditions habta a H ratb a rahbc a H w—ny —ltern—tiveD further ™onditions ™—n ˜e imposed upon this ˜—si™ sp—™etime stru™tureD for ex—mpleD —™™ording to whether we wish to —dd gr—vit—tion —s distin™t s™—l—r eld —nd le—ve the ˜—™kground sp—™etime —t or whether we wish to in™orpor—te gr—vit—tion into the sp—™etime —s ™urv—ture —fter the model of gener—l rel—tivity @see priedm—n IWVQD ™h sssAF „hese —re —ll inst—n™es of — gener—lD geometri™ formul—tion of sp—™etime theoriesF ell su™h theories h—ve models General covariance and general relativity 823 hM; O1; O2; : : : ; Oni @TA where O1; O2; : : : ; On —re now just n geometri™ o˜je™t elds su˜je™t to ™ert—in ™onstr—ining equ—tionsF †irtu—lly —ll theories of sp—™e —nd time now given serious ™onsider—tion ™—n ˜e formul—ted in this w—yF 20 ƒu™h theories —re —utom—ti™—lly gener—lly ™ov—ri—nt in — sense th—t —™tu—lly follows from the de nitions of the m—them—ti™—l stru™tures used in the formul—tionF pollowing ƒt—nd—rd denitions @eFgF fishop —nd qold˜erg IWTVD ™h ID r—wking —nd illis IWUQD ™h PD „orretti IWVQD —ppendixAD —n nEdimension—l dierenti—˜le m—nifold is — ™onne™tedD topologi™—l sp—™e with — set of ™oordin—te ™h—rtsD su™h th—t every point of the topologi™—l sp—™e lies in the dom—in of — ™oordin—te ™h—rtD whi™h is — homeomorphism of —n open set of the sp—™e with RnF „he set of ™oordin—te ™h—rts form — m—xim—l or ™omplete —tl—s in so f—r —s the —tl—s ™ont—ins every ™oordin—te ™h—rt th—t ™—n ˜e ™onstru™ted in the usu—l w—y from its ™oordin—te ™h—rts ˜y CkEtr—nsform—tions on RnF k is some positive integer orD most ™ommonlyD innityF „he next step is ™ompli™—ted ˜y the v—gueness of the denition of –geometri™ o˜je™t9F st is given ˜y †e˜len —nd ‡hitehe—d @IWQPD pRTA —s –—n inv—ri—nt whi™h is rel—ted to the sp—™e ‘under ™onsider—tion“9 where —n inv—ri—nt is –—nything whi™h is un—ltered ˜y tr—nsform—tions of ™oordin—tes9F21 „hus for our purposesD it is prudent to —ssume th—t our geometri™ o˜je™t eldsD —re like enderson9s @IWTUD pISA –lo™—l geometri™—l o˜je™ts9F „hey —re represented ˜y — nite set of num˜ers for e—™h point in the m—nifold in e—™h ™oordin—te ™h—rts —nd whi™h tr—nsform under ™oodin—te tr—nsform—tion in — w—y th—t respe™t tr—nsitivityD identity —nd inversionF „hese num˜ers —re the geometri™ o˜je™t9s ™omponents in the ™oordin—te ™h—rtsF vet us s—y th—t — geometri™ o˜je™t eld O h—s ™omponents Oik::: where the integer v—lued i; k; : : : represents — suit—˜le set of index l—˜elsF gom˜iningD we now —rrive —t the sense in whi™h —ny theory with models @TA is gener—lly ™ov—ri—ntF sf N is —ny –lo™—l ™oordin—te neigh˜ourhood9 of MD —n open set of N th—t is the dom—in of some ™oordin—te ™h—rt xiD then the restri™tion of the model @TA to x will ˜e represented ˜y hA; @O1Aik:::; : : : ; @OnAik:::i @UA ‡here A is the r—nge of xi —nd the rem—ining stru™tures —re the ™omponents of the o˜je™ts O1; : : : On in the ™oordin—te ™h—rt xiF „he theory is gener—lly ™ov—ri—nt in the sense th—t if @UA is — ™oordin—te represent—tion of the model @TAD then so is —ny represent—tion deriv—˜le from @UA ˜y —r˜itr—ry Ck tr—nsform—tionF „his is sometimes known —s –passive general covariance9 €ut more ˜rieyD on™e we h—ve formul—ted — theory —s h—ving models of the form @TAD thenD ˜uilt into the denitions of the stru™tures used is the possi˜ility of representing the models in ™oordin—te systems th—t —re rel—ted ˜y the —r˜itr—ry tr—nsform—tions of iinstein9s gener—l ™ov—ri—n™eF @wore pre™iselyD they —re rel—ted ˜y Ck tr—nsform—tions if the m—nifold h—s — Ck m—xim—l —tl—s of ™oordin—te ™h—rtsFA „hese ™oordin—te represent—tions ˜eh—ve 20„h—t is not to s—y th—t —ll intelligi˜le theories of sp—™e —nd time must —dmit su™h — formul—tionF ‡ith — pre™ise denition of geometri™ o˜je™t in h—ndD it is just — m—tter of m—them—ti™—l p—tien™e to ™onstru™t — sp—™etime theory without su™h — formul—tionF ye ™ould ˜eginD for ex—mpleD ˜y ™onsidering sp—™etimes whose event sets —re very l—rge ˜ut nite —nd do not —dmit smooth ™oordin—te ™h—rtsF 21„he still v—gue –rel—ted to sp—™e9 ™l—use is —n —ttempt to —void the pro˜lem th—t –F F F stri™tly spe—kingD —nythingD su™h —s — pl—nt or —n —nim—lD whi™h is unrel—ted to the sp—™e whi™h we —re t—lking —˜outD is —n inv—ri—nt9F 824 J D Norton ex—™tly like the ™omponents of the gener—lly ™ov—ri—n™e formul—tion of theories used ˜y iinstein —nd others in the e—rly ye—rs of gener—l rel—tivityF st is to this —utom—ti™ gener—l ™ov—ri—n™e th—t „hirring @IWUWD p ITTA referred when he wrote et the time of the ˜irth of gr—vit—tion theoryD the requirement of gener—l ™ov—ri—n™e provided some relief from the l—˜or p—insD ˜ut l—ter on it w—s more often — sour™e of ™onfusionF „he ™on™ept of — m—nifold in™orpor—tes it —utom—ti™—lly when the denition used equiv—len™e ™l—sses of —tl—sesD —nd hen™e only ™h—rt independent st—tements —re reg—rded —s me—ningfulF „his progr—m is ˜y no me—ns unique to gr—vit—tion theory| we h—ve —lso followed in in ™l—ssi™—l me™h—ni™s —nd ele™trodyn—mi™sF „he ˜ig dieren™e ‘in gener—l rel—tivity“ is th—t the metri™ g on M is now not determined — prioriF ‡hile the use of these geometri™ methods h—s ˜e™ome st—nd—rd in modern work on gener—l rel—tivityD it should ˜e noted th—t their domin—n™e is not viewed univers—lly with unmixed joyF ‡ein˜erg @IWUPD pref—™eA notes th—t —n emph—sis on these methods tends to o˜s™ure the import—n™e of the prin™iple of equiv—len™e within the theory —nd the n—tur—l ™onne™tions to qu—ntum theoryF pin—llyD there is — notion th—t is loosely du—l to the notion of p—ssive gener—l ™ov—ri—n™e des™ri˜ed —˜oveF st is the notion of `active general covariance'F „he m—in m—them—ti™—l dieren™e is th—t the —™tive version employs m—ps on the m—nifold w of the models @TA r—ther th—n tr—nsform—tions ˜etween ™oordin—te ™h—rtsF st ™—n ˜e dened —s followsF vet h ˜e —n —r˜itr—ry dieomorphism 22 from M to MF „hen — theory with models of the form @TA is gener—lly ™ov—ri—nt in the —™tive sense if every stru™ture hhM; h£O1; h£O2; : : : ; h£Oni @THA is — model whenever hM; O1; O2; : : : ; Oni @TA is — modelF sn —dditionD it is routinely —ssumed th—t the stru™ture @TA —nd @T9A represent the s—me physi™—l ™ir™umst—n™e @eFgFD in the ™—se of gener—l rel—tivityD see r—wking —nd illis IWUQD pSTAF „his —ssumption h—s ˜een ™—lled –vei˜niz equiv—len™e9 @i—rm—n —nd xorton IWVUAF w—ny theories —re gener—lly ™ov—ri—nt in the —™tive senseF e su™ient ™ondition for —™tive gener—l ™ov—ri—n™e is th—t the o˜je™t elds O1; O2; : : : ; On th—t ™—n ˜e in™luded in the models @TA —re determined solely ˜y tensor equ—tionsF „hus gener—l rel—tivity is ™ov—ri—nt in this sense —s —re versions of spe™i—l rel—tivity —nd xewtoni—n sp—™etime theoryF €—ssive gener—l ™ov—ri—n™e involves no physi™—lly ™ontingent prin™iplesF yn™e models of the form of @TA —re sele™tedD p—ssive gener—l ™ov—ri—n™e follows —s — m—tter of m—them—ti™—l denitionD no m—tter wh—t the physi™—l ™ontent of the theoryF €—ssive gener—l ™ov—ri—n™e involves no physi™—lly ™ontingent prin™iplesD yn™e models of the form of @TA —re sele™tedD p—ssive gener—l ™ov—ri—n™e follow —s — m—tter of m—them—tE i™—l denitionD no m—tter wh—t the physi™—l ™ontent of the theoryF „his is not the ™—se with —™tive gener—l ™ov—ri—n™eGvei˜niz equiv—len™eF ƒtru™tures @TA —nd @T9A —re m—them—tE i™—lly independent stru™turesD th—t they represent the s—me physi™—l ™ir™umst—n™e is —n —ssumption dependent on the properties of the physi™—l ™ir™umst—n™e —nd our methods of ™oordin—ting the stru™tures to itF „he dieren™es ˜etween su™h p—irs of stru™tures —s @TA —nd @T9A —re gener—lly of — n—ture th—t m—ke it uninteresting to suppose —nything other th—n 22por ex—mpleD if M is — Ch m—nifoldD then h might ˜e —ny Ch dieomorphism in the sense of r—wking —nd illis @IWUQD p PQAF General covariance and general relativity 825 vei˜niz equiv—len™eF roweverD it h—s ˜een —rgued @i—rm—n —nd xorton IWVUD xorton IWVVA th—t —t le—st one do™trineD sp—™etime su˜st—ntiv—lismD must deny vei˜niz equiv—len™eF23 ƒin™e the —ssumption of —™tive gener—l ™ov—ri—n™eGvei˜niz equiv—len™e is — physi™—l —ssumption —l˜eit we—kD it does require physi™—l —rguments to support itF st turns out th—t iinstein9s two ™ele˜r—ted —rguments|the pointE™oin™iden™e —rgument —nd the hole —rgument|™—n ˜e put in to modern forms th—t support —™tive gener—l ™ov—ri—n™eGvei˜niz equiv—len™eF e™™ording to the modernized pointE™oin™iden™e —rgumentD the two dieoE morphi™ models @TA —nd @T9A would —gree on —ll o˜serv—˜lesD for —ll th—t is o˜serv—˜le —re ™oin™iden™es th—t —re preserved ˜y the dieomorphismF „hereforeD if we deny vei˜niz equiv—len™eD we would h—ve to insist th—t the two dieomorphi™ models represent distin™t physi™—l ™ir™umst—n™esD even though no possi˜le o˜serv—tion ™ould pi™k ˜etween themF „o ™onstru™t the modernized hole —rgumentD we ™onsider some neigh˜ourhood H of the m—nifold M in models @TA —nd @T9A —nd pi™k — dieomorphism h th—t is the identity outside H ˜ut ™omes smoothly to dier from it within HF „he the two dieomorphi™ models will ˜e the s—me outside H ˜ut will ™ome smoothly do dier within HF ‡e now h—ve — m—them—ti™—l indeterminismD in the sense th—t the fullest spe™i™—tion of the model outside H will f—il to determine how it is to ˜e extended into H —™™ording to the theoryF „his indeterminism is usu—lly dismissed —s — purely m—them—ti™—l g—uge freedom —sso™i—ted with —™tive gener—l ™ov—ri—n™eF sf we deny vei˜niz equiv—len™e —nd insist th—t the two models represent distin™t physi™—l ™ir™umst—n™esD then we ™onvert this g—uge freedom into — physi™—l indeterminismF „he dieren™es ˜etween the models within H must now represent — dieren™e of physi™—l ™ir™umst—n™esF ‡hi™h will o˜t—in within H ™—nnot ˜e determined ˜y the fullest spe™i™—tion of the physi™—l ™ir™umst—n™es outside HD no m—tter how sm—ll H is in sp—ti—l —nd tempor—l extensionF por further dis™ussion of the dieren™es ˜etween —™tive —nd p—ssive gener—l ™ov—ri—n™eD see xorton @IWVWD se™tion PFQAF 5.5. Later responses to Kretschmann's objection urets™hm—nn9s o˜je™tion is pro˜—˜ly the single most frequently mentioned of —ll o˜je™tions to iinstein9s views on the found—tions of gener—l rel—tivityF es s h—ve —lre—dy indi™—ted —˜oveD howeverD the o˜je™tion whi™h —ppe—rs univers—lly under urets™hm—nn9s n—me in the liter—ture is —™tu—lly — ™onsider—˜ly redu™ed version of wh—t urets™hm—nn re—lly s—idF st is ™ommonly reported —s the —ssertion th—t gener—l ™ov—ri—n™e is physi™—lly v—™uousD sin™e it is merely — ™h—llenge to our m—them—ti™—l ingenuity to ˜ring —ny theory into gener—lly ™ov—ri—nt formF por the purposes of this se™tionD whi™h reviews l—ter responses to the o˜je™tionD s will t—ke –urets™hm—nn9s o˜je™tion9 to ˜e this redu™ed versionD for th—t is the one th—t w—s responded toF issenti—lly no one other th—n iinstein seemed to re—lize th—t urets™hm—nn h—d ˜—sed his o˜je™tion on — ™ontingent —ssumptionD the premise of the pointE ™oin™iden™e —rgumentF „h—t —ssumption|th—t –the l—ws of n—ture —re only —ssertions of timesp—™e ™oin™iden™es9|is so nonEtrivi—l th—t iinstein —™tu—lly m—de it the st—tement of his IWIV version of the prin™iple of rel—tivityF sn l—ter liter—tureD urets™hm—nn9s o˜je™tion is ™ommonly —™™eptedF snst—n™es in whi™h urets™hm—nn is ™ited ˜y n—me in™lude r—v—s @IWTRD pWQWAD ‚indler @IWTWD pIWTAD i—rm—n 23et presentD howeverD there is no ™onsensus in the philosophy of sp—™e —nd time liter—ture over the ™onne™tion ˜etween sp—™etime su˜st—ntiv—lismD vei˜niz equiv—len™e —nd the hole —rgumentD with virtu—lly every ™on™eiv—˜le position ˜eing defendedF ƒee f—rtels @IWWQAD futtereld @IWVUD IWVVD IWVWAD i—rm—n @IWVWD ™h WAD xorton @IWWP—D se™tion SFIPAD g—rtwright —nd roefer @forth™omingAD w—udlin @IWVVD IWWHAD ‚yn—siewi™z @forth™oming @—AD @˜AAD ƒt—™hel @forth™omingAD „eller @forth™omE ingAD wundy @IWWPAF 826 J D Norton @IWURD pPUIAD priedm—n @IWUQD pSSAD ‚—y @IWVUD pUHAF eg—in urets™hm—nn9s —ssertion of the physi™—l v—™uity of gener—l ™ov—ri—n™e m—y ˜e m—de without n—ming urets™hm—nnF snst—n™es in™lude ƒil˜erstein @IWWPD ppPP{QAD ƒzekeres @IWSSD pPIPAD po™k @IWSWD pQUHD ˜ut see p xviAD „hirring @IWUWD pITTAF iinstein9s IWIV response to urets™hm—nn —lso ™omm—nds ™onsider—˜le —ssentF iinE stein9s response is en™—psul—ted in the simple rem—rk th—t gener—l ™ov—ri—n™e is physi™—lly v—™uous —loneY however it —™hieves physi™—l ™ontent —nd signi™—nt heuristi™ for™e when it is supplemented ˜y the requirement th—t the l—ws of n—ture t—ke simple formsF „his viewpoint is —dvo™—ted ˜yX €—inleve @IWPID pVUUAD „olm—n @IWQRD ppQQD ITT{TUA24D fridgE m—n @IWRWD ppQQW{RHD QRSAD ‡hitt—ker @IWSID vol ssD pISWAD ‡e˜er @IWTID pIS{ITAD ƒkinner @IWTWD p QPRAD edlerD f—zin —nd ƒ™hier @IWUUD pIRSAF yh—ni—n @IWUTD ppPSQ{RA st—tes urets™hm—nn9s o˜je™tion —nd quotes iinstein9s IWIV reply —t lengthD ˜ut he pro™eeds to elu™id—te iinstein9s response in terms of the requirement of gener—l inv—ri—n™e of the —˜E solute o˜je™t tr—dition @see se™tion V ˜elowAF sn his IWIV reply to urets™hm—nnD iinstein urged the heuristi™ power of gener—l ™ov—ri—n™e —nd the ˜—sis of his ˜rilli—nt su™™ess with gener—l rel—tivityF d9svorno @IWWPD pIQIA ™omes ™losest to this viewpoint when he suggests th—t we ™—nnot ignore gener—l ™ov—ri—n™eD even if it is v—™uousD pre™isely ˜e™—use it was of su™h import—n™e to iinsteinD r—ther th—n ˜e™—use of some —s yet unre—lized heuristi™ powerF fut perh—ps wisner et al @IWUQD se™tion IPFSA ™—pture iinstein9s met—physi™s most ™le—rly when they re™—pitul—te urets™hm—nn9s o˜je™tion —nd retort fut —nother viewpoint is ™ogentF st ™onstru™ts — powerful sieve in the form of — slightly —ltered —nd slightly more ne˜ulous prin™ipleX –x—ture likes theories th—t —re simple when st—ted in ™oordin—te freeD geometri™ l—ngu—ge9 F F F e™™ording to this prin™ipleD x—ture must love gener—l rel—tivityD —nd it must h—te xewtoni—n theoryF yf —ll theories ever ™on™eived ˜y physi™istsD gener—l rel—tivity h—s the simplestD most eleg—nt geometri™ found—tionsF F F fy ™ontr—stD wh—t di—˜oli™—lly ™lever physi™ist would ever foist on m—n — theory with su™h — ™ompli™—ted geometri™ found—tion —s xewtoni—n theoryc „here —re o˜vious pro˜lems with this viewF „o ˜eginD it would seem th—t the view is pl—inly f—lseF „he very simplest l—wsD whi™h n—ture ought to love the mostD —re just in™omE p—ti˜le with experien™eF por ex—mpleD it would ˜e very simple of —ll of sp—™eD time —nd the distri˜ution of m—tter were homogeneousY ˜ut they —re not homogeneousF ƒo x—ture9s preferen™es ™—n only ˜e exer™ised —mong the more ™ompli™—ted dregs th—t rem—in —fter experien™e h—s dr—ined of the truly simple|x—ture9s preferen™e here is — r—ther ™ontrived oneF xextD it is not ™le—r ˜y wh—t rules we —re to judge whi™h of two theories is the simplerF st ™—nnot just ˜e — m—tter of intuitive impressionsD sin™e then we h—ve no w—y of —djudiE ™—ting dis—greementsF fut even — ˜—si™ ™ount of the num˜er of m—them—ti™—l stru™tures in — theory is h—rd to do un—m˜iguouslyF25 fondi @IWSWD pIHVAD howeverD endorses the view th—t gener—l ™ov—ri—n™e is physi™—lly v—™uous —nd points out th—t ™onserv—tion l—ws expli™E itly involving gr—vit—tion—l energyEmomentum in gener—l rel—tivity —re not tensori—lD ˜ut pseudoEtensori—lF pin—llyD it is not o˜vious why n—ture should ˜e so kind —s to prefer l—ws th—t we hum—ns deem simpleF „hus xorth @IWTSD pSVA muses th—t the virtue of simpli™ity for ™ov—ri—nt l—ws might merely ˜e th—t they —re more likely to ˜e —™™epted ˜y othersF wy own view is th—t one should not look on simpli™ity —s resulting from the emotion—l —tt—™hments of x—tureF ‚—ther it —rises from the l—˜ours of theorists who h—ve ™onstru™ted l—ngu—ges in whi™h x—ture9s ™hoi™es —ppe—r simpleF ‡hether x—ture9s further ™hoi™es will ™ontinue to —ppe—r simple in some l—ngu—ge seems to me —n entirely ™ontingent m—tter 24„olm—n gives urets™hm—nn9s o˜je™tions in its full form insof—r —s the possi˜ility of gener—lly ™ov—ri—nt formul—tion is t—ken to follow ne™ess—rily from the pointE™oin™iden™e —rgumentF 25ss the stressEenergy tensor of pressureless dustD T ab a UaUbD ™ounted —s one stru™ture T ab or —s twoD the m—tter density  —nd the fourEvelo™ity eld Uac General covariance and general relativity 827 —nd one t—kes — gre—t risk elev—ting —ny l—ngu—ge to the st—tus of x—ture9s ownF es we explore new dom—ins of physi™—l l—wD the one thing th—t is most ™le—r is x—ture9s surprising vers—tility in frustr—ting our n—tur—l expe™t—tionsF rowever this does not me—n th—t there is no v—lue in simpli™ityF ep—rt from its pr—gm—ti™ v—lueD it h—s —n epistemi™ v—lueF „he more ™ompli™—ted — theoryD the more likely we —re to h—ve introdu™ed stru™tures with no ™orrel—tions in re—lityY —nd the more ™ompli™—ted — theoryD the h—rder it will ˜e to test for these physi™—lly irrelev—nt stru™turesF ‡e should prefer the simpler theory —nd seek l—ngu—ges th—t m—ke our theories simpleD ˜ut not ˜e™—use x—ture is simpleF ‚—therD if we restri™t ourselves to simpler theoriesD we —re more likely to know the truth when we nd itF „here is — v—ri—tion of iinstein9s response to urets™hm—nn th—t —voids the di™ult questions over simpli™ityF sts over—ll ee™t is to dire™t us tow—rds simpler theories ˜y restri™ting the stru™tures we ™—n employ in our formul—tionsF st fo™uses on the pro™ess of nding gener—lly ™ov—ri—nt formul—tions of —r˜itr—ry l—wsF sf we restri™t the num˜er of —ddition—l m—them—ti™—l stru™tures th—t ™—n ˜e introdu™ed in this pro™essD it m—y no longer ˜e possi˜le to ™onstru™t — gener—lly ™ov—ri—nt formul—tion for some l—wsD so th—t we on™e —g—in h—ve —n interesting division ˜etween gener—lly ™ov—ri—nt —nd other theoriesF po™k @IWSWD p xviA des™ri˜es the ide— in its most gener—l form F F F the requirement of ™ov—ri—n™e of equ—tions h—s gre—t heuristi™ v—lue ˜e™—use it limits the v—riety of possi˜le forms of equ—tions —nd there˜y m—kes it e—sier to ™hoose the ™orre™t onesF roweverD one should stress th—t the equ—tions ™—n so ˜e limited only under the ne™ess—ry ™ondition th—t the num˜er of fun™tions introdu™ed is —lso limitedY if one permits the introdu™tion of —n —r˜itr—ry num˜er of new —uxili—ry fun™tionsD pr—™ti™—lly —ny equ—tion ™—n ˜e given ™ov—ri—nt formF „r—utm—n @IWTRD ppIPP{QA illustr—tes how unrestri™ted —dmission of new stru™tures —llows ™onstru™tion of — gener—lly ™ov—ri—nt formul—tion of equ—tions th—t ™le—rly —re ™oordin—te dependentF re ™onsiders the equ—tion A1 a H the v—nishing of the rst ™omponent of — ™ove™tor Aa in some ™oordin—te systemF sf ua is the ™oordin—te ˜—sis ve™tor eld —sso™i—ted with the x1 ™oordin—teD then this l—w —dmits gener—lly ™ov—ri—nt formul—tion —s uaAa a H „he vill—in is the ve™tor eld uaD sin™e @pIPQA one should not introdu™e su™h —ddition—l stru™tures in —ddition to those —lre—dy present in the —xioms of the theory @eFgF the metri™ tensorD —ne ™onne™tionA —nd to those th—t —re ne™ess—ry to des™ri˜e the physi™—l systemF sf we now —pply this thinking to gener—l rel—tivityD we —rrive —t — popul—r me—ns of inje™ting ™ontent into the gener—l ™ov—ri—n™e of gener—l rel—tivityF sn — vorentz ™ov—riE —nt version of spe™i—l rel—tivityD the metri™—l properties of sp—™etime —re not represented expli™itlyF sn the tr—nsition to the gener—lly ™ov—ri—ntD gener—l theory of rel—tivityD these properties ˜e™ome expli™it —s — new stru™tureD the metri™ tensor gabF st is required th—t this new stru™ture represent some denite physi™—l element of re—lity —nd nut just ˜e — m—them—ti™—l ™ontriv—n™e introdu™ed to for™e through gener—l ™ov—ri—n™eF „he metri™ tenE sor s—tises this requirement in so f—r —s it represents the gr—vit—tion—l eld —s well —s the metri™—l properties of sp—™etimeF €—uli @IWPID pISHA des™ri˜es this out™ome 828 J D Norton F F F urets™hm—nn F F F took the view th—t the postul—te of gener—l ™ov—ri—n™e does not m—ke —ny —ssertions —˜out the physi™—l content of the physi™—l l—wsD ˜ut only —˜out their m—them—ti™—l formulationD —nd iinstein : : : entirely ™on™urred with this viewF „he gener—lly ™ov—ri—nt formul—tion of the physi™—l l—ws —™quires — physi™—l ™ontent only through the prin™iple of equiv—len™eD in ™onsequen™e of whi™h gr—vit—tion is deE s™ri˜ed solely ˜y the gik —nd these l—tter —re not given independently from m—tterD ˜ut —re themselves determined ˜y eld equ—tionsF ‡e nd — simil—r view in forel @IWPTD pIUP{QAD ‡eyl @IWPID ppPPT{UA ‚ei™hen˜—™h @IWPRD pIRIAD enderson @IWTUD IWUI|see se™tion VFI ˜elowAD qr—ves @IWUID pIQVA —nd even —s re™ently —s ‡—ld @IWVRD pSUA who formul—tes the prin™iple of gener—l ™ov—ri—n™e —s „he prin™iple of gener—l ™ov—ri—n™e in this ™ontext ‘preErel—tivisti™ —nd rel—tivisti™ physi™s“ st—tes th—t the metri™ of sp—™e is the only qu—ntity pert—ining to sp—™e th—t ™—n —ppe—r in the l—ws of physi™sF ƒpe™i™—lly there —re no preferred ve™tor elds or preferred ˜—ses of ve™tor elds pert—ining only to the stru™ture of sp—™e whi™h —ppe—r in —ny l—w of physi™sF @re ™—utions th—t the –the phr—se ’pert—ining to sp—™e4 does not h—ve — pre™ise me—ning9FA foth €—uli —nd ‡eyl stress — spe™i—l —spe™t of the physi™—l ™h—r—™ter of the metri™ in their dis™ussionsX the metri™ is not given a priori ˜ut it is inuen™ed or determined ˜y the m—tter distri˜ution vi— inv—ri—nt eld equ—tionsF „his wouldD of ™ourseD rule out gener—lly ™ov—ri—nt formul—tions of spe™i—l rel—tivityF ‡eylD in p—rti™ul—rD sees this —s the de™isive property of gener—l rel—tivityF –ynly this f—™t justies us in —ssigning the n—me ’gener—l theory of rel—tivity4 to our re—soning F F F 9 he wrote @pPPTAF purtherD he emph—sized the result th—t –gr—vit—tion is — mode of expression of the metri™—l eld9 —nd th—t –this —ssumptionD r—ther th—n the postul—te of gener—l inv—ri—n™eD seems to the —uthor to ˜e the re—l pivot of the gener—l theory of rel—tivity9 @ppPPT{UAF ‡e sh—ll see th—t this theme will ˜e in™orpor—ted into the —˜solute o˜je™t —ppro—™h @see se™tion V ˜elowAF e pr—™ti™—l di™ulty still rem—insF et the most fund—ment—l levelD the gener—l prin™iple is ™le—rly ™orre™tX we should deny —dmission to theories or stru™tures th—t do not represent elements of re—lityF „he hope is th—t this restri™tion will preserve — unique —sso™i—tion ˜etween gener—l ™ov—ri—n™e —nd the gener—l theory of rel—tivityF rowever the prin™iple m—y well not ˜e su™iently pre™isely formul—ted to h—ve —ny for™e in re—listi™ ex—mplesF gonsider the stru™tures dtaD hab —nd raD introdu™ed in ™onstru™ting — gener—lly ™ov—ri—nt formul—tion of xewtoni—n theoryF ere they —dmissi˜le or notc xoti™e th—t €—uli —nd ‡eyl9s emph—sis on the dyn—mi™ ™h—r—™ter of the metri™ m—y not help us hereF sn versions of xewtoni—n gr—vit—tion theoryD the gr—vit—tion—l eld is in™orpor—ted into the —ne stru™ture ra whi™h then h—s simil—r dyn—mi™—l properties to the metri™ of gener—l rel—tivityF „he str—tegy so f—r h—s ˜een to —ugment the requirement of gener—l ™ov—ri—n™e with —ddition—l requirements th—t m—ke it nonEtrivi—lF st turns out th—t there is —n extremely simple w—y of —ugmenting the prin™iple of gener—l ™ov—ri—n™e so th—t we ™—nnot render gener—lly ™ov—ri—nt su™h theories —s spe™i—l rel—tivity —nd versions of xewtoni—n theory th—t do not in™orpor—te the gr—vit—tion—l eld into —ne stru™tureF sn ˜oth these ™—sesD the —sso™i—ted gener—lly ™ov—ri—nt formul—tions h—ve the property th—t they ™—n ˜e simplied ˜y reintrodu™ing restri™ted ™oordin—te systemsF „his is not so in the ™—se of gener—l rel—tivityD so we ™—n pi™k ˜etween these ™—ses ˜y insisting th—t the gener—lly ™ov—ri—nt formul—tion not —dmit simpli™—tionF fergm—nn @IWRPD pISWA expli™itly in™orpor—tes this requirement into the st—tement of the prin™iple of gener—l ™ov—ri—n™eX „he hypothesis th—t the geometry of physi™—l sp—™e is represented ˜est ˜y — form—lism whi™h is ™ov—ri—nt with respe™t to gener—l ™oordin—te tr—nsform—tionsD —nd th—t — restri™tion to — less gener—l group of tr—nsform—tions would not simplify th—t form—lism is ™—lled the principle of general covariance General covariance and general relativity 829 et rst this seems like — purely ad hoc ™ontriv—n™eF rowever fergm—nn9s propos—l ™onne™ts dire™tly with the view th—t rel—tivity prin™iples —re geometri™ symmetry prin™iplesD —s we sh—ll see in se™tion TFP ˜elowF eltern—tivelyD fondi @IWSWD pIHVA ™—lls the propos—l into question ˜y re™—lling po™k9s use of h—rmoni™ ™oordin—tes to redu™e the ™ov—ri—n™e of gener—l rel—tivity @see se™tion W ˜elowAF „here h—ve ˜een other studies of the rel—tionship ˜etween — theory —nd its geneE r—lly ™ov—ri—nt reformul—tion —nd these studies —rrive —t ™on™lusions un™omfort—˜le for urets™hm—nn9s o˜je™tionF ƒ™hei˜e @IWWID IWVIA h—s ™onsidered the rel—tionship within — more pre™ise form—l settingF re ™on™ludes th—t it is simply not o˜vious th—t —ny geomeE try of restri™ted ™ov—ri—n™e ™—n —lw—ys ˜e re™—st in — gener—lly ™ov—ri—nt formul—tionF €ost @IWTUA ™on™ludes th—t the pro™ess of rendering theories gener—lly ™ov—ri—nt is f—r from —utoE m—ti™ trivi—lity —nd must ˜e tre—ted with some ™—reF sn the ™—se of ele™trom—gneti™ theoryD he shows how dierent w—ys of rendering the theory gener—lly ™ov—ri—nt —™tu—lly le—d to distin™t theoriesF w—shoon @IWVTA simil—rly emph—sizes th—tD while —ny theory ™—n ˜e renE dered gener—lly ™ov—ri—ntD the m—nner in whi™h it is done ™—n h—ve physi™—l ™onsequen™esD in p—rti™ul—rD in the me—surements of —™™eler—ted o˜serversF w—ny —uthors —re prep—red to —™™ept urets™hm—nn9s o˜je™tion ˜ut feel th—t it h—s to ˜e qu—lied in signi™—nt w—ys if the true signi™—n™e of gener—l ™ov—ri—n™e is to ˜e —ppre™i—tedF ‡hile —™™epting urets™hm—nn9s o˜je™tions —nd th—t — requirement of gener—l ™ov—ri—n™e is not — rel—tivity prin™iple like th—t of spe™i—l rel—tivityD ‡ein˜erg @IWUPD ppWPD III{QA ™h—r—™terizes gener—l ™ov—ri—n™e —s —kin to the g—uge inv—ri—n™e of ele™trom—gneti™ eldsF e™™epting urets™hm—nn9s o˜je™tionD funge @IWTUD se™tion QFIFQA o˜serves th—t if gener—l ™ov—ri—n™e is understood —s simply requiring form inv—ri—n™e of l—wsD then it does ˜e™ome — purely m—them—ti™—l requirementF „herefore he ™on™ludes th—t gener—l ™ov—ri—n™e is to ˜e understood —s — regul—tive r—ther th—n ™onstitutive prin™ipleF w—vrides @IWUQD pTTA —lso —™™epts urets™hm—nn9s o˜je™tion ˜ut sees the signi™—n™e of the prin™iple in —˜sorption of —™™eler—tion into the nonEiu™lide—n stru™ture of sp—™etimeF —h—r @IWVWD se™tion VFIA —ppro—™hes the pro˜lem with — distin™tion introdu™ed ˜y logi™i—ns ˜etween —n o˜je™t l—ngu—ge —nd its met—l—ngu—geF sn this ™ontextD the o˜je™t l—ngu—ge ™ont—ins the —ssertions —˜out physi™s systems —nd the met—l—ngu—ge ™ont—ins —ssertions —˜out the o˜je™t l—ngu—geF ‡hether — ˜ody of o˜je™t l—ngu—ge —ssertionsD su™h —s xewtoni—n theoryD is gener—lly ™ov—ri—nt is not itself —n o˜je™t l—ngu—ge —ssertionF st ˜elongs to the met—l—ngu—geF ‡e m—y ˜e —˜le to nd — gener—lly ™ov—ri—nt formul—tion of xewtoni—n theory whi™h is logi™—lly equiv—lent to the origin—l q—lile—n ™ov—ri—nt versionF rowever the met—Elevel property of gener—l ™ov—ri—n™e is not inherited ˜y the origin—l formul—tionD for met—Elevel properties —re not tr—nsmitted ˜y logi™—l equiv—len™eF „herefore we ™—nnot s—y th—t xewtoni—n theory is itself gener—lly ™ov—ri—ntF ƒever—l other —uthors h—ve —ppro—™hed gener—l ™ov—ri—n™e —s — prin™iple of oper—ting — met—Elevel of l—ngu—geF ƒee qr—ves @IWUID ppIRQ{UAF sn p—rti™ul—rD „orne˜ohm @IWSPD se™tion RIA ™h—r—™terizes the prin™iple of gener—l ™ov—ri—n™e —s — ™losure rule oper—ting on — met—Elevel in whi™h one qu—nties over ™oordin—te systems employed in st—tements of physi™—l l—wsF pin—llyD see uu™h—r @IWVVA for — rein™—rn—tions of the issues r—ised ˜y the de˜—te of urets™hm—nn9s o˜je™tion in r—miltoni—n dyn—mi™s —nd ™—noni™—l qu—ntiz—tion of gener—lly ™ov—ri—nt systemsF 6. Is the requirement of general covariance a relativity principle? 6.1. Disanalogies with the principle of relativity of special relativity sn —ddition to —™™us—tions th—t his prin™iple of gener—l ™ov—ri—n™e is physi™—lly v—™uousD iinstein9s tre—tment of gener—l ™ov—ri—n™e h—s ˜een ˜esieged ˜y ™ontinuing ™ompl—ints 830 J D Norton th—t the —™hievement of gener—l ™ov—ri—n™e does not —mount to — gener—liz—tion of the prin™iple of rel—tivity to —™™eler—tionF „hese ™ompl—ints h—ve ™ome in m—ny dierent formsF ƒome of the e—rliest m—ke the o˜vious point th—t su™h —n extension of the prin™iple of rel—tivity to —™™eler—ted motion seems to ˜e —tly ™ontr—di™ted ˜y the simplest o˜serv—tionsF „he prin™iple of rel—tivity of inerti—l motion ts the experien™es of — tr—veller in — tr—in moving uniformly on smooth tr—™ksY nothing within the ™—rri—ge reve—ls the tr—in9s motionF roweverD the s—me is not so if the tr—in —™™eler—tesD —s w—s pointed out —™er˜i™—lly ˜y ven—rd @IWPID pISAD whose involvement in the perse™ution of iinstein in qerm—ny in the IWPHs is well knownX vet the tr—in in ™onsider—tion undert—ke — distin™tD nonEuniform motion F F F sfD —s — resultD everything in the tr—in is wre™ked through the ee™ts of inerti—D while outside everything rem—ins und—m—gedD thenD s ˜elieveD no sound mind would w—nt to dr—w —ny other ™on™lusion th—n th—t the tr—in h—d —ltered its motion with — jolt —nd not the surroundingsF por iinstein9s reply to this ex—™t p—ss—geD see iinstein @IWIV—AF st w—s only in the IWSHs —nd IWTHs th—t su™h longEst—nding worries took — prominent though still disputed pl—™e in the m—instre—m liter—tureF „his dissident view drew strength from su™h eminent rel—tivists —s po™k —nd ƒyngeD who d—red to pro™l—im wh—t few would —dmitX they just ™ould not see how iinstein9s theory gener—lizes the prin™iple of rel—tivity| —nd they even suspe™ted th—t iinstein ™ould not see it eitherF ƒo ƒynge @IWTTD pUA wroteX F F F the gener—l theory of rel—tivityF „he n—me is repellentF ‚el—tivityc s h—ve never ˜een —˜le to underst—nd wh—t th—t word me—ns in this ™onne™tionF s used to think th—t this w—s my f—ultD some —w in my intelligen™eD ˜ut it is now —pp—rent th—t no˜ody ever understood itD pro˜—˜ly not even iinstein himselfF ƒo let it goF ‡h—t is ˜efore us is iinstein9s theory of gr—vit—tionF ƒee —lso ƒynge @IWTRD pQA —nd @IWTHD p ixA where he wrote F F F the geometri™ w—y of looking —t sp—™eEtime ™omes dire™tly from winkowskiF re protested —g—inst the use of the word –rel—tivity9 to des™ri˜e — theory ˜—sed on —n –—˜solute9 @sp—™etimeAD —ndD h—d he lived to see the gener—l theory of rel—tivityD s ˜elieve he would h—ve repe—ted his protest in even stronger termsF sn simil—r veinD po™k @IWSWD pp xvi{xviiiD QTU{VD QUS{TA tre—ted — rel—tivity prin™iple —s st—ting — uniformity of sp—™etimeF „hus spe™i—l rel—tivity —dmits — rel—tivity prin™iple ˜e™—use of the uniformity of — winkowski sp—™etimeF „he sp—™etimes of gener—l rel—tivityD howeverD m—nifest this uniformity only in the innitesim—lD so th—t the n—ming of the theory –gener—l rel—tivity9 or –gener—l theory of rel—tivity9 is simply in™orre™tD ˜etr—ying iinstein9s f—ilure to underst—nd his own theoryF po™k ™ontinued @pQTVA „he f—™t th—t the theory of gr—vit—tionD — theory of su™h —m—zing depthD ˜e—uty —nd ™ogen™yD w—s not ™orre™tly understood ˜y its —uthorD should not surprise usF ‡e should —lso not ˜e surprised —t the g—ps in logi™D —nd even errorsD whi™h the —uthor permitted himself when he derived the ˜—si™ equ—tions of the theoryF sn the history of physi™s we h—ve m—ny ex—mples in whi™h the underlying signi™—n™e of — fund—ment—lly new physi™—l theory w—s re—lized not ˜y its —uthor ˜ut ˜y some˜ody else —n in whi™h he deriv—tion of the ˜—si™ equ—tions proposed ˜y the —uthor proved to ˜e logi™—lly in™onsistentF st is su™ientD to point to w—xwell9s theory of the ele™trom—gneti™ eld FFF ellowing in —ddition th—t the only —dmissi˜le sense of –gener—l rel—tivity9 is —s purely form—l property of gener—l ™ov—ri—n™eD po™k @IWURD pSA ™on™luded „hus we ™—n sum upX gener—l rel—tivity ™—nnot ˜e physi™—lD —nd physi™—l rel—tivity ™—nnot ˜e gener—lF „hese ™onfessions were eng—gingly ™—ndid —nd their i™ono™l—sti™ sentiments found re™eptive General covariance and general relativity 831 —udien™esF „he heresy of dis˜elief in iinstein ˜e™—me respe™t—˜leF po™k —nd ƒynge —reD of ™ourseD not —lone in divor™ing gener—l ™ov—ri—n™e from — generE —liz—tion of the prin™iple of rel—tivity —nd —nnoun™ing the f—ilure of iinstein9s eort in this reg—rdF ƒee for ex—mple v—nd—u —nd vifshitz @IWSID pPPWAD h—vis @IWUHD pPIWA D ‚—ine —nd reller @IWVID pIQSA —nd fondi @IWUWD pIPWAF 6.2 Relativity principles as symmetry principles sf ™ov—ri—n™e prin™iples —re not rel—tivity prin™iplesD then wh—t —re rel—tivity prin™iplesc xew —nswers to this question h—ve ™ome repe—tedly within the tr—dition th—t proposes the divor™e of gener—l ™ov—ri—n™e from — gener—liz—tion of prin™iple of rel—tivityF ‡e sh—ll see th—t they eventu—lly st—˜ilize on the view th—t — rel—tivity prin™iple expresses — symmetry of the sp—™etime stru™tureF yne of the e—rliest propos—ls ™omes from urets™hm—nnF ris f—mous o˜je™tion to gener—l ™ov—ri—n™e —™tu—lly o™™upies — sm—ll p—rt of his lengthy p—per @IWIUAF „he ˜ulk of it is devoted to developing —n —ltern—te interpret—tion of rel—tivity prin™iplesF ris propos—ls —re em˜edded within extended ™—l™ul—tions —nd ™ir™uitous ver˜i—geF „hey —ppe—r to redu™e to the followingF „he key ide— in identifying the rel—tivity prin™iple of some given theory lies not in extending its ™ov—ri—n™eD ˜ut in redu™ing it to the minimum group possi˜leF „his redu™tion must ˜e done in — w—y th—t identies — group —sso™i—ted with the theory9s physi™—l ™ontent r—ther th—n some p—rti™ul—r formul—tion of itF sn the ™—se of spe™i—l rel—tivityD his gener—l propos—l le—ds to the expe™ted resultX the vorentz group expresses the theory9s rel—tivity prin™ipleF gonsider the ˜undle of —ll lightE like worldlines in the theoryF sn the vorentz ™ov—ri—nt formul—tionD this ˜undle is des™ri˜ed ˜y the equ—tion @x1 x01A2 C : : : C @x4 x04A2 a H @VA ‡here x a x1; : : : ; x4 a ict —re the usu—l sp—™etime ™oordin—tes —nd x01; : : : ; x04 some —rE ˜itr—ry origin eventF „his ˜undle is m—pped ˜—™k into itself ˜y —ny vorentz tr—nsform—tion th—t preserves the originF urets™hm—nn —llowed th—t we ™ould extend the usu—l vorentz ™ov—ri—nt formul—tion of the theory even —s f—r —s gener—lly ™ov—ri—nt formul—tionD using the methods of ‚i™™i —nd veviEgivit—F roweverD in — formul—tion of extended ™ov—ri—n™eD —n —llowed tr—nsform—tion willD in gener—lD not m—p this ˜undle ˜—™k into itselfF ‚—therD su™h — tr—nsform—tion will —lter the ™oordin—te im—ge of the ˜undleF eg—inD one ™ould ™onsider — formul—tion whose ™ov—ri—n™e is restri™ted to — group sm—ller th—n the vorentz groupF rowever this formul—tion ™ould only ˜e ™onstru™ted —t the expense of —ltering the physi™—l ™ontent of the theory26F „he vorentz tr—nsform—tion is the formul—tion of minim—l ™ov—ri—n™e f—ithful to the theory9s physi™—l ™ontentF „herefore the vorentz tr—nsform—tion is the group —sso™i—ted with the theory9s rel—tivity prin™ipleF e simil—r —n—lysis in the ™—se of gener—l rel—tivity le—ds to — quite dierent resultF sn ee™t urets™hm—nn nds th—t the single mem˜ered identity group pl—ys the s—me role in gener—l rel—tivity —s does the vorentz group in spe™i—l rel—tivityF es — result he ™—n —rrive —t — ™on™lusion th—t dire™tly ™ontr—di™ts einstein9s @pTIHA 26row urets™hm—nn —rrived —t this ™ru™i—l ™on™lusion is — little un™le—r to meF ƒu™h — formul—E tion would need to repl—™e @VA ˜y —nother formul— or formul—e of more restri™ted ™ov—ri—n™e —nd presum—˜ly urets™hm—nn held th—t —ny su™h formul—e would h—ve to —lter the physi™—l ™ontent of @VAF por ex—mpleD to viol—te vorentz ™ov—ri—n™eD the new formul— might pi™k out one or other sp—ti—l dire™tion —s preferredD where—s equ—tion @VA des™ri˜ing the ˜undle —dmits no su™h preferred dire™tionsF 832 J D Norton „herefore iinstein9s theory s—tises no rel—tivity prin™iple —t —ll in the sense developed ‘e—rlier in the p—per“Y on the ˜—sis of its ™ontentD it is — ™ompletely —˜solute theoryF „o —rrive —t this resultD urets™hm—nn ™onsidered the ˜undle of lightElike worldlines —nd of free m—teri—l p—rti™les within the theoryF re found the former xed the ™omponents of the metri™ tensor g up to — multipli™—tive f—™tor  —nd the l—tter for™ed  to ˜e — ™onst—ntF @xoti™e th—t these —re now f—mili—r resultsF sn modern l—ngu—geX ™onform—l stru™ture xes the metri™ up to — ™onform—l f—™tor —nd spe™ifying —ne stru™ture for™es the f—™tor to ˜e ™onst—ntFA pin—lly ™onsider—tion of sp—™etime ™urv—ture rules out —ny v—lue of  other th—n unityF „hus the physi™—l ™ontent of the theory xes the metri™—l ™omponentsF fut on™e these ™omponents —re xedD the ™oordin—te system is xed —nd no ™ov—ri—n™e tr—nsform—tion rem—insY in ee™t the ™ov—ri—n™e group h—s ˜e™ome the identity group —nd one h—s no rel—tivity prin™ipleF urets™hm—nn —lso showed th—t the s—me result ™ould ˜e —rrived —t in —nother w—yF es long —s the sp—™etime metri™ is su™iently nonEuniformD it is possi˜le to dene — unique sp—™etime ™oordin—te system for e—™h metri™ ˜y setting the four ™oordin—tes equ—l to unique ™urv—ture inv—ri—ntsF „his on™e —g—in redu™es the ™ov—ri—n™e group to the identityF pin—lly urets™hm—nn ™ould extr—™t one n—l ˜low from his ™—l™ul—tionsF sn ee™t he ™ould ™on™lude th—t the vorentz group w—s the l—rgest group possi˜le for —ny rel—tivity prin™iple in — sp—™etime theory of the type of spe™i—l —nd gener—l rel—tivity @pTIHAX e physi™—l theoryD whi™h —™™ords —n o˜serv—tion—lly —™™essi˜le me—ning to the extern—l prin™iple Z   ds a H where ds2 a gdxdx of — sp—™eEtime m—nifold of winkowski norm—l form of the line element or posits th—t the inv—ri—nt metri™—l ™h—r—™ter of the m—nifold is in some other w—y in prin™iple o˜serv—˜le to the s—me extentD ™—n s—tisfy no ˜ro—der rel—tivity postul—te in the sense ‘developed e—rlier in the p—per“ th—n th—t of the origin—l iinsteini—n theory of rel—tivityF urets™hm—nn9s propos—l h—s ˜een ™riti™ized —t length ˜y enderson @IWTTAF re —rgues th—t the propos—l f—ils sin™e one ™—n too re—dily redu™e the ™ov—ri—n™e of — theory to the identityF ris ex—mples in™lude ele™trodyn—mi™s —nd spe™i—l rel—tivityD provided th—t we —dd some other stru™tureD su™h —s — s™—l—r eldD to the winkowski sp—™etimeF g—rt—n @IWPUA g—ve — less ˜elli™ose —nd m—them—ti™—lly more perspi™—™ious ™h—r—™teriE z—tion of the dieren™e ˜etween the gener—l ™ov—ri—n™e of gener—l rel—tivity —nd the vorentz ™ov—ri—n™e of spe™i—l rel—tivityF qener—l rel—tivity threw into physi™s —nd philosophy the —nt—gonism th—t existed ˜eE tween the two prin™iple dire™tors of geometryD ‚iem—nn —nd uleinF „he sp—™eEtimes of ™l—ssi™—l me™h—ni™s —nd or spe™i—l rel—tivity —re of the type of uleinD those of gener—l rel—tivity —re of the type of ‚iem—nnF …nder ulein9s irl—ngen progr—m — wide r—nge of geometries were —ll ™h—r—™terized ˜y their —sso™i—ted groups —nd the geometri™ entities they studied were the inv—ri—nts of those groupsF „he key —spe™t of these irl—ngen progr—m geometries|the iu™lide—nD the proje™tiveD the —neD the ™onform—l —nd others|w—s th—t —ll the sp—™es were homogeneousF sn the ‚iem—nn tr—ditionD one ™onsidered — sp—™e —nd — group of tr—nsform—tionsF fut the geometri™ entities investig—ted —re no longer the inv—ri—nts of the tr—nsform—tionsD for in this ™—se there —re essenti—lly noneF snste—d one is interested in the inv—ri—nts of — qu—dr—ti™ dierenti—l formD the fund—ment—l or metri™—l formD th—t is —djoined to the sp—™eF es resultD the groups —sso™i—ted with geometries in the two tr—ditions h—ve very dierent signi™—n™eF „he sp—™etime geometry of spe™i—l rel—tivityD —s introdu™ed ˜y winkowskiD is in the tr—dition of uleinF es — result its ™h—r—™teristi™ groupD the vorentz groupD is General covariance and general relativity 833 —sso™i—ted with the homogeneity of the sp—™etimeF qener—l rel—tivity lies in the ‚iem—nn tr—dition —ndD —s — resultD its gener—l group of tr—nsform—tion is —sso™i—ted with no su™h homogeneityF ƒesm—t @IWQUD ppQVP{QA g—ve — more —lge˜r—i™ ™h—r—™teriz—tion of why the felt the gener—l ™ov—ri—n™e of gener—l rel—tivity h—d f—iled to implement — gener—liz—tion of the prin™iple of rel—tivityF ‡h—t w—s needed w—s — theory whose l—ws would rem—in un™h—nged in form under tr—nsform—tions ˜etween —ll fr—mes of referen™e in™luding —™™eler—ted onesD in the s—me w—y th—t the l—ws of spe™i—l rel—tivity rem—ined inv—ri—nt under vorentz tr—nsE form—tionF „he gener—l ™ov—ri—n™e of gener—l rel—tivity just did not do thisF …nder the tr—nsform—tions of gener—l ™ov—ri—n™eD su™h —s tr—nsform—tion ˜etween g—rtesi—n —nd pol—r ™oordin—tesD the expression for ˜—si™ tensors do ™h—ngeF ‡h—t gener—l ™ov—ri—n™e does —llowD howeverD is th—t — tensorD su™h —s the iinstein tensorD ™—n ret—in its zero v—lue in empty sp—™e under these tr—nsform—tionsD even though its expression ™h—ngesF ƒesm—t9s point seems to ˜e pre™isely the point th—t ‡ein˜erg @IWUPD pWPA is m—king when he expl—ins the dieren™e ˜etween the vorentz inv—ri—n™e of spe™i—l rel—tivity —nd gener—l ™ov—ri—n™eF yne ™ouldD he notesD exp—nd the ™ov—ri—n™e of xewton9s se™ond l—w ˜y tr—nsforming it under vorentz tr—nsform—tionF roweverD — new qu—ntityD the velo™ity of the ™oordin—te fr—me would —ppe—r in the tr—nsformed equ—tionF „he requirement th—t this velo™ity does not —ppe—r in the tr—nsformed equ—tion is wh—t we ™—ll the €rin™iple of ƒpe™i—l ‚el—tivityD or –vorentz inv—ri—n™e9 for shortD —nd this requirement pl—™es very powerful restri™tions on the origin—l equ—tionF ƒimil—rlyD when we m—ke —n equ—tion gener—lly ™ov—ri—ntD new ingredients will enterD th—t isD the metri™ tensor g —nd the —ne ™onne™tion F „he dieren™e is th—t we do not require th—t these qu—ntities drop out —t the endD —nd hen™e we do not o˜t—in —ny restri™tion on the equ—tions to st—rt withY r—ther we exploit the presen™e of g —nd  to represent gr—vit—tion—l elds po™k @IWSUA @see —lso po™k IWSWD p xiii{xivD ITTA g—ve — synthesis of —ll these ide—sX the homogeneity of sp—™es in the ulein tr—ditionD the m—pping ˜—™k into themselves of urets™hm—nn9s ˜undle of lightElike —nd inerti—l worldlines —nd he g—ve it in —n —lge˜r—i™ form indi™—ted ˜y ƒesm—t —nd ‡ein˜ergF sn ™onsidering the uniform or homogeneous sp—™etime of spe™i—l rel—tivityD he expl—ined @pQPSAX „he property of sp—™etime ˜eing homogeneous me—ns th—t @—A there —re no privileged points in sp—™e —nd in timeY @˜A there —re no privileged dire™tionsD —nd @™A there —re no privileged inerti—l fr—mes @th—t —ll fr—mes —re moving uniformly —nd in — str—ight line with respe™t to one —nother —re on the s—me footingAF „he uniformity of sp—™e —nd time m—nifests itself in the existen™e of the vorentz groupF sn p—rti™ul—rD the equ—lity of points in sp—™e —nd time ™orresponds to the possi˜ility of — displ—™ementD the equ—lity of dire™tions ™orresponds to th—t of sp—ti—l rot—tionsD —nd the equ—lity of inerti—l fr—mes ™orresponds to — spe™i—l vorentz tr—nsform—tionF po™k then g—ve this ™ondition m—them—ti™—l expressionF „he vorentz tr—nsform—tion le—ves un™h—nged the winkowski line element ds2 a dx20 dx21 dx22 dx23 a gdxdx @WA where the x0; : : : ; x3 —re the usu—l sp—™etime ™oordin—tes of the vorentz ™ov—ri—nt formul—E tionF „his s—me ™ondition ™—n ˜e st—ted in —r˜itr—ry ™oordin—tes in whi™h the line element @WA ˜e™omes 834 J D Norton ds2 a gdxdx „he m—them—ti™—l expression of the homogeneity of the winkowski sp—™etime is now st—ted —s the preserv—tion of the fun™tion—l form of the ™omponents of the metri™ in some ™l—ss of s™thyoseotrendmeinw—xt™eoDmstyhpseotnenmeintstFws„iolhlf —˜ttheiespDmoifsesttirh˜ie™lemgtHeotrt—ir™r—ehn—tshfsoe™romsm—mtpoeon—fuennnet™swtigo™nosoirondfsixnoHm—te—es—srty˜hsiettergm—ryx—H™roeionrtdhwienhx—i™theF „h—t isD gH @xHA a g@xA @IHA where the equ—lity must ˜e re—d —s holding for equ—l numeri™—l v—lues of the qu—druples x™eoxmp—rpenosdsnexesnH—tFsh„gohmisot™groe—nnndesiifttoyiormnofiistnh™teoonsgpsHi—d™eerut—inm˜dleyersmitnho™reeeu˜rseoust—thrli™™toteionvrsedoitrnh—t—rtn—enmssyfesortreemlmy—srteixoqnui—rrnuindlegFxtHhe—rntedlt—htieet in indistinguish—˜le f—shion to the metri™ tensorF „he set of ™oordin—te systems with this property —re rel—ted ˜y — ten p—r—meter group whi™h ™orresponds to the vorentz groupF xoti™e th—t the —lge˜r—i™ expression for the tr—nsform—tions from x to xH in the vorentz group ™—n no longer ˜e the f—mili—r formul—e @IA of iinstein9s origin—l IWHS p—perF por ex—mpleD in gener—lizing the ™oordin—tesD the ™oordin—te system of @WA m—y rem—in inerti—l ˜ut with the g—rtesi—n sp—ti—l ™oordin—tes repl—™ed ˜y pol—r ™oordin—tesD in whi™h ™—se the expression for the vorentz tr—nsform—tion would h—ve to ˜e —ltered ™orrespondinglyF roweverD wh—tever m—y ˜e their —ltered formD the tr—nsform—tion equ—tions must le—ve un™h—nged the fund—ment—l form of the ™omponents of the metri™ tensorF ytherwise the sp—™etime would distinguish ˜etween two inerti—l ™oordin—te systemsD in viol—tion of this uniformityF „h—t is the ™ondition expressed in @IHAF „he distin™tion ˜etween simple ™ov—ri—n™e —nd tr—nsform—tion of form @IA seems to ˜e distin™tion ˜etween fu™hd—hl9s @IWVID pPWA –improper9 —nd –proper form inv—ri—n™e9F sn his ex—mpleD the equ—tion gijSiSj a H @where ƒ is — s™—l—r eld —nd ™omm—s denote dierenti—tionA is improperly form inv—ri—nt if the tr—nsformed equ—tion just ret—ins this form —sD s—yD gi0j0Si0Sj0 a HF st is properly form inv—ri—nt if the gi0j0 of the tr—nsformed equ—tion —lso rem—in the s—me fun™tions of the new ™oordin—tes —s the untr—nsformed gij were of the oldF po™k9s propos—l now rel—tes dire™tly to fergm—nn9s @IWRPD pISWA st—tement of the prinE ™iple of gener—l ™ov—ri—n™e —s given in se™tion SFSF —˜oveF e™™ording to @IHAD — geneE r—lly ™ov—ri—nt formul—tion of spe™i—l rel—tivity will —dmit — ten p—r—meter su˜group of tr—nsform—tion|the vorentz tr—nsform—tion|th—t preserves the fun™tion—l form of the ™omponents of the metri™ tensor gF st ™—n do so in m—ny dierent w—ysF yne merely sele™ts some —r˜itr—ry ™oordin—te system in whi™h the winkowski metri™ h—s ™omponents g —nd —llows ™ondition @IHA to gener—te the su˜groupF sf one ˜egins with the usu—l di—gE on—l form of the metri™D D one —rrives —t the usu—l form of the vorentz tr—nsform—tion @IAF i—™h of these su˜groups is —sso™i—ted with — formul—tion of spe™i—l rel—tivity of reE du™ed ™ov—ri—n™e of the p—rti™ul—r fun™tion—l form of the metri™—l ™omponents th—t rem—in un—ltered —™™ording to @IHA will ˜e ˜uilt into its l—wsF „herefore fergm—nn9s st—tement of the prin™iple of gener—l ™ov—ri—n™e will judge the gener—lly ™ov—ri—nt formul—tion of spe™i—l rel—tivity to ˜e in—dmissi˜le —nd thus preserves — distin™tion ˜etween the ™ov—ri—n™e of gener—l rel—tivity —nd of spe™i—l rel—tivityF xoti™e —lso th—t the formul—tions of spe™i—l rel—tivity of redu™ed ™ov—ri—n™e —re now of — form ™omp—ti˜le with ulein9s irl—ngen progr—mD sin™e the ‚iem—nni—n qu—dr—ti™ dierenti—l form —re no longer tr—nsformed merely ™ov—ri—ntly within the theoryF „husD in General covariance and general relativity 835 —™™ord with g—rt—n9s o˜serv—tionsD the tr—nsform—tion groups of the formul—tions —re now —sso™i—ted with the homogeneity of the sp—™etimeF po™k9s ™ondition @IHA h—s —n immedi—te expression in the geometri™ —ppro—™h to sp—™eE time theoriesF vet h ˜e the du—l m—nifold dieomorphism of the ™oordin—te tr—nsform—tion dened on — winkowski hM; gabiF „hen po™k9s ™ondition @IHA ˜e™omes h £ gab a gab @IIA —nd the group of tr—nsform—tions s—tisfying this ™ondition is the vorentz groupF27 „h—t isD the vorentz group is the group of dieomorphisms th—t —re the symmetry tr—nsform—tions or isometries of the winkowski metri™F @‡—ldD IWVRD pp RVD THD RQVAF „he existen™e of this group expresses the uniformity of the winkowski sp—™etimeF ‡ith this terminologyD we ™—n summ—rize why po™k —nd others ˜elieve th—t the tr—nE sition from spe™i—l to gener—l rel—tivity h—s f—iled to gener—lize the prin™iple of rel—tivityF „wo groups —re —sso™i—ted with the formul—tions of — theoryX its ™ov—ri—n™e group ™h—r—™E terizes purely form—l —spe™ts of its formul—tionY its symmetry group ™h—r—™terizes — physi™—l f—™tD the degree of uniformity of the sp—™etime —nd this uniformity group —llows the theory to s—tisfy — rel—tivity prin™ipleF sn the tr—nsition from — vorentz ™ov—ri—nt formul—tion of spe™i—l rel—tivity to — gener—lly ™ov—ri—nt formul—tion of gener—l rel—tivityD the ™ov—ri—n™e group is exp—ndedF „his isD howeverD merely —n —™™ident of formul—tionF „he symmetry group is —™tu—lly redu™ed from the vorentz group to the identity groupD for the gener—l ™—seF „he identity group is —sso™i—ted with no rel—tivity prin™iple —t —llF „herefore the tr—nsition from spe™i—l to gener—l rel—tivity does not gener—lize the rel—tivity prin™ipleF st er—di™—tes itF 6.3. Coordinate systems versus frames of reference po™k took it —s immedi—te th—t his ™ondition @IHA —utom—ti™—lly re—lized the equiv—len™e of inerti—l fr—mes of referen™e where—s gener—l rel—tivity em˜odies no su™h equiv—len™eF „h—t this is ™orre™t m—y not ˜e immedi—tely ™le—r given th—t su™h formul—tions of the prin™iple of gener—l ™ov—ri—n™e su™h —s fergm—nn9s do preserve — sense in whi™h the n—tur—l ™ov—ri—n™eF „o give — pre™ise st—tement of this result we require — ™le—rer st—tement of wh—t is — fr—me of referen™eF sn tr—dition—l developments of spe™i—l —nd gener—l rel—tivity it h—s ˜een ™ustom—ry not to distinguish ˜etween two quite distin™t ide—sF „he rst is the notion of — ™oordin—te systemD understood simply —s the smoothD inverti˜le —ssignment of four num˜ers to events 27„o see the tr—nsitionD let the metri™ gab h—ve ™omponents g in some ™oordin—te system —nd ldfuerunot—dmtlehrdqei@uIt—erHd—oArnDmutshpofeorlerpdsmhioi—sfetmoiroemn—holDsfrrwpxoheminstmoo™woqhourj—umddsi—ntrp—u™stpoel—nensssyidoesvefterermnet—thslepsxxfwu0intt@h™oxtitxoAh0Fne—ssfnlo—rutoeirnslf—e™ytooioo™frnotdnh˜idenei—t™ttiwooeoensrexdn@iInxH—tAotFe——„nsnydosetvxgeee0mnnt—esrsh——p—ltleowmwti—hetpdhe h™oporfrdoimn—tpesuxn0d@exr hAF in sf the s—me ™oordin—tes the metri™ —t p h—s systemF gonsider the ™omponents gD then tmheetr™i—™rrhie£dg—albon™—grrmieedtr—i™lo—ntg to hp will h—ve the s—me ™omponents g in the carried along ™oordin—te system —nd the ™—rried —long ™oordin—te system will —ssign ™oordin—tes x to hpF ‡e now see th—t this ™—rried —long metri™ is the s—me —s the origin—l metri™ —t hpD —s @IIA dem—ndsD ˜y ™omp—ring their ™omponents in the origin—l ™oordin—te systemF ‡e tr—nsform the ™—rried —long metri™ ˜—™k from the ™—rried —long ™oordin—te system to the origin—l ˜y me—ns of the ™oordin—te tr—nsform—tion of @IHA —nd nd th—t the ™—rried —long metri™ metri™ —grees hw—isth™otmhepoorniegnints—lgm0 et—rit™ hpD whi™h sin™e the further dis™ussion of the du—lity of ™oordin—te h—s ™oordin—tes xF „herefore the ™—rried —long fun™tion—l forms of tr—nsform—tion —nd gm—n—infodldg0 di—ereomthoerpsh—imsmeFD por see xorton @IWVWD se™tion PFQA 836 J D Norton in sp—™etime neigh˜ourhoodsF „he se™ondD the fr—me of referen™eD refers to —n ide—lized physi™—l system used to —ssign su™h num˜ersF wore pre™iselyD sin™e the physi™—l systems tend to ˜e sp—™eEllingD one is ™on™erned with how su™h hypotheti™—l system would ˜eh—ve were they to ˜e ™onstru™tedF w—ny su™h systems —re possi˜leF por ex—mple one ™—n im—gine sp—™e full of simil—rly ™onstru™ted ™lo™ks —nd —ll of them —tt—™hed to — rigid fr—me of sm—ll rodsF „he ™lo™k re—dings give us the time ™oordin—tes —nd the ™ounting of rods gives us sp—ti—l ™oordin—tesF „o —void unne™ess—ry restri™tionsD we ™—n divor™e this —rr—ngement from metri™—l notionsF pollowing uop™zynski —nd „r—utm—n @IWWPD ppPR{SAD we ™ould require only th—t the sp—™eElling f—mily of ™lo™ks ˜e—r three smoothly —ssigned indi™es @whi™h ™ould fun™tion —s sp—ti—l ™oordin—tesAD th—t the ™lo™ks ti™k smoothlyD —lthough not ne™ess—rily in proper timeD —nd th—t time re—dings v—ry smoothly from ™lo™k to ™lo™kF yf spe™i—l import—n™e for our purposes is th—t e—™h fr—me of referen™e h—s — denite st—te of motion —t e—™h event of sp—™etimeF ‡ithin the ™ontext of spe™i—l rel—tivity —nd —s long —s we restri™t ourselves to fr—mes of referen™e in inerti—l motionD then little of import—n™e depends on the dieren™e ˜etween —n inerti—l fr—me of referen™e —nd the inerti—l ™oordin—te system it indu™esF „his ™omfort—˜le ™ir™umst—n™e ™e—ses immedi—tely on™e we ˜egin to ™onsider fr—mes of referen™e in nonE uniform motion even within spe™i—l rel—tivityF „his ˜e™—me — m—jor pro˜lem for iinstein to negoti—te —s e—rly —s IWHUD when he ˜eg—n to ™onsider uniformly —™™eler—ted fr—mes of referen™e in his new gr—vit—tion theoryF re found @IWHUD se™tion IVA the need to introdu™e ™oordin—te times whi™h ™ould not ˜e re—d dire™tly from ™lo™k me—surementsF ƒimil—rlyD due to the vorentz ™ontr—™tion of rods oriented in the dire™tion of motionD the geometry —sso™i—ted with — uniformly rot—ting fr—me of referen™e ™e—sed to ˜e iu™lide—nF es — resultD sp—ti—l ™oordin—tes ™—n no longer ˜e —ssigned ˜y the usu—l methods with me—suring rodsF „he point of iinstein9s rot—ting disk thought experiment @rst pu˜lished in iinstein @IWIPD se™tion IA —nd ˜est known from iinstein @IWITD se™tion QAA is th—t sp—™etime ™oordin—tes will lose this dire™t metri™—l signi™—n™e on™e we str—y from the f—mili—r inerti—l ™oordin—te systems of spe™i—l rel—tivityF28 ‡ith the —dvent of gener—l rel—tivityD iinstein wished to ™onsider fr—mes of referen™e with —r˜itr—ry st—tes of motionF rowever he deemed it impr—™ti™—l to ret—in even — vestige of the ide—lized physi™—l system of the fr—me of referen™eF sn their pl—™e he simply used —r˜itr—ry ™oordin—te systemsF „he —sso™i—tion of —n —r˜itr—ry ™oordin—te system with —n —r˜itr—ry fr—me of referen™e ˜e™—me st—nd—rd in the liter—ture for m—ny de™—desF „husD for ex—mple fergm—nn @IWTPD pPHUA expl—ins sn —ll th—t follows we sh—ll use the terms –™urviline—r fourEdimension—l ™oordin—te sysE tem9 —nd –fr—me of referen™e9 inter™h—nge—˜lyF „husD in iinstein9s writingsD wh—tever equiv—len™e is est—˜lished ˜y gener—l ™ov—ri—n™e of —r˜itr—ry ™oordin—te systems is —lso ™onferred upon —r˜itr—ry fr—mes of referen™e —ndD if we re™—ll the ™onne™tion ˜etween — fr—me of referen™e —nd — st—te of motionD the powerful suggestion is th—t this is —ll th—t is needed to extend the prin™iple of rel—tivity to —r˜iE tr—ry motionsF „he ™onne™tion is ™ompli™—ted slightly ˜y the f—™t th—t some ™oordin—te 28„he pro˜lem is even more ™ompli™—ted th—n iinstein indi™—tedF en inerti—l fr—me of referen™e in — winkowski sp—™etime is n—tur—lly —sso™i—ted with iu™lide—n sp—™esD with —re the sp—ti—l hyE persurf—™es everywhere orthogon—l to the worldlines of the fr—me9s elementsF „he worldlines of the elements of — rot—ting disk —dmit no su™h orthogon—l hypersurf—™esF ƒin™e the sp—™etime of spe™i—l rel—tivity rem—ins —tD we m—y well —sk in wh—t sp—™e does the geometry ˜e™ome nonEiu™lide—nF „he most dire™t —nswer is th—t this geometry is indu™ed onto the –rel—tive sp—™e9 formed ˜y the worldlines of the elements of the diskF „his sp—™e ™—n ˜e dened pre™isely —s in xorton @IWVSD se™tion QAF por further dis™ussion of the role of the rot—ting disk thought experiment in iinstein9s thoughtD see ƒt—™hel @IWVH—AF General covariance and general relativity 837 tr—nsform—tions ™le—rly do not rel—te dierent st—tes of motionD su™h —s the tr—nsform—tion ˜etween sp—ti—l ™—rtesi—n —nd pol—r ™oordin—tesF rowever some su˜group of the gener—l group of ™oordin—te tr—nsform—tions is the —ppropri—te oneD —s iinstein @IWITD se™tion QA m—kes ™le—r when he writesX st is ™le—r th—t — physi™—l theory whi™h s—tises this postul—te ‘of gener—l ™ov—ri—n™e“ will —lso ˜e suit—˜le of the gener—l postul—te of rel—tivityF por the sum of all su˜stitutions in —ny ™—se in™ludes those whi™h ™orrespond to —ll rel—tive motions of threeEdimension—l systems of ™oEordin—tesF wore re™entlyD to negoti—te the o˜vious —m˜iguities of iinstein9s tre—tmentD the notion of fr—me of referen™e h—s re—ppe—red —s — stru™ture distin™t from — ™oordin—te systemF sf one ™on™eives of — fr—me of referen™e —s — sp—™e lling system of hypotheti™—l instruments moving with —r˜itr—ry velo™itiesD then the minimum inform—tion needed to pi™k out the fr—me is the spe™i™—tion of —n —r˜itr—ry fr—me of referen™e|—nd the one s sh—ll use here| is th—t it is — ™ongruen™e of ™urvesD th—t isD — set of ™urves su™h th—t every event in the sp—™etime m—nifold lies on ex—™tly one of its ™urvesF @„orretti IWVQD pPVD xortonD IWVSD se™tion QD †l—dimirov et al IWVUD pWSAF sf the notion of timelike is denedD we would —lso require the ™urves ˜e timelike to ensure th—t they —re the worldlines of physi™—l elementsF sn the ™—se of the semiE‚iem—nni—n sp—™etimes of rel—tivity theoryD wh—tever further inform—tion one might need is supplied ˜y the theory9s metri™—l stru™tureF prom it we ™—n re—d the time el—psed —s re—d ˜y proper ™lo™ks moving with the fr—meD or ™h—nges in the dire™tions —nd sp—ti—l dist—n™es of neigh˜ouring elements of the fr—meF †—rious —ltern—tive denitions of fr—me of referen™e —re possi˜leF ƒin™e — smooth ™onE gruen™e of ™urves ™—n ˜e spe™ied —s the unique set of integr—l ™urves of — smoothD nonE v—nishing timeElike ve™tor eldD one ™ould t—ke — fr—me of referen™e to ˜e su™h — timelike ve™tor eld @i—rm—n IWURD pPUHD tonesD IWVID pITQAF eg—inD one ™—n employ ri™her stru™E turesF „he timelike ve™tor eld ™ould ˜e supplemented ˜y — tri—d of sp—™elike ve™tors pointing to the worldlines of neigh˜ouring elements of ve™tors over the sp—™etime m—nifoldF @ƒynge IWTHD ™h sssFSD †l—dimirov et alIWVUD pWSAF pin—lly — ™oordin—te system is —d—pted to — fr—me of referen™e if the ™urves of the fr—me ™oin™ide with the ™urves of ™onst—nt sp—ti—l ™oordin—tesF „herefore we ™ould t—ke — fr—me to ˜e the equiv—len™e ™l—ss of —ll ™oordin—te systems —d—pted to some ™ongruen™e @i—rm—n IWURD PPUHAF „his denition h—s the —dv—nt—ge of ˜ringing us ™losest to the tr—dition—l ™orresponden™e ˜etween fr—mes of referen™e —nd ™oordin—te systemsF sn spe™i—l rel—tivityD —n inerti—l fr—me of referen™e is — ™ongruen™e of timelike geodesi™sF en inerti—l ™oordin—te system is — ™oordin—te system —d—pted to —n inerti—l fr—me of referE en™eF 6.4. Relativity principles and the equivalence of frames ‡ith the notion of fr—me of referen™e ™l—riedD it proves possi˜le to give — more pre™ise tre—tment of the prin™iple of rel—tivity in so f—r —s it —sserts —n equiv—len™e of v—rious st—tes of motionD th—t isD of v—rious fr—mes of referen™eF iinstein9s origin—l tre—tment of the prin™iple of rel—tivity in spe™i—l rel—tivity —mounted to requiring th—t the l—ws of physi™s —dopt the s—me form when expressed in —ny inerti—l ™oordin—te systemF „his type of formul—tion of the prin™iple w—s quite servi™e—˜le in the ™ontext of — vorentz ™ov—ri—nt spe™i—l theory of rel—tivityF es we h—ve seenD howeverD there h—ve ˜een signi™—nt ™h—llenges to the ide— th—t form inv—ri—n™e of l—ws ™—n ™—pture —ny physi™—l prin™iple when we —re prep—red to employ m—them—ti™—l te™hniques powerful enough to render virtu—lly —ny theory gener—lly ™ov—ri—ntF e pre™ise formul—tion of the relev—nt notion of equiv—len™e of fr—me h—s ˜een developed 838 J D Norton within work th—t in™ludes i—rm—n @IWURAD priedm—n @IWVQD espe™i—lly ™h s†FSA —nd tones @IWVIAF „heir propos—ls explore m—ny v—ri—nt denitions —nd do so within the ™ontext of — wide r—nge of theoriesD in™luding v—ri—nts of xewtoni—n sp—™etime theoryF „he essenE ti—l ide—s they sh—re ™—n ˜e illustr—ted ˜y the following tre—tment of spe™i—l —nd gener—l rel—tivityF „he essen™e of the prin™iple of rel—tivity in the spe™i—l theory is the indistinguish—˜ility of —ll the inerti—l st—tes of motionF „hus iinstein9s IWHS spe™i—l rel—tivity p—per h—d ˜een motiv—ted ˜y the re—liz—tion th—t no experiment in me™h—ni™sD opti™s or ele™trodyn—mi™s ™ould reve—l the uniform motion of the e—rth through the —etherF „h—t isD sp—™e —nd time –look the s—me9 experiment—lly to o˜servers in —ny st—te of inerti—l motionF iinstein9s t—sk w—s to devise — theory in whi™h they looked the s—me theoreti™—lly —s wellF „his ™ondition ™—n ˜e ˜roken up into — kind of pseudoEexperimentF ‡e ˜egin with —n inerti—l o˜serverD who performs — r—nge of experiments in kinem—ti™s —nd other ˜r—n™hes of physi™sF „he o˜server is then ˜oosted into uniform motion with respe™t to his origin—l st—te of motion —nd ™—rries —long with him — ™omplete re™ord of —ll the experiments —nd their out™omesF „hese experiments —re now repe—ted —nd the out™omes ™omp—red with those of the origin—l setF „he prin™iple of rel—tivity requires th—t ˜oth sets of out™omes must ˜e the s—me —nd — theory s—tisfying the prin™iple of rel—tivity must predi™t th—t this will ˜e soF @por — ™omp—rison of this sense of the prin™iple —nd the one th—t requires form inv—ri—n™e of l—wsD see enderson @IWTRD ppIUT{VPAFA „his pseudoEexperiment—l ™ondition ™—n ˜e tr—nsl—ted into — theoreti™—l ™ondition th—t —mounts to the prin™iple of rel—tivity in spe™i—l rel—tivityF „he theoreti™—l di—log of the inerti—l o˜server is the inerti—l fr—me of referen™eF „he —n—log of the setting of the o˜server into uniform motion is — vorentz tr—nsform—tion of the fr—me of referen™eF „he setting up —nd out™ome of —ll experiments performed ˜y the o˜server will ˜e determined fully ˜y the sp—™etime stru™tures of the theoryF „herefore the ™—rrying —long of the ™omplete des™ription of the o˜server9s experiments —nd out™omes tr—nsl—tes into the ™—rrying —long under vorentz tr—nsform—tion of the sp—™etime stru™tures of the theoryF29 „he prin™iple of rel—tivity now simply —mounts to the requirement th—t the vorentz tr—nsform—tion m—p sp—™etime stru™tures —llowed ˜y the theory into sp—™etime stru™tures —llowed ˜y the theoryF ‡ithout further —ssumption it follows th—t spe™i—l rel—tivity s—tises the prin™iple of rel—tivity —s f—r —s —ll kinem—ti™—l experiments —re ™on™ernedF „hese —re ide—lized experE iments in whi™h the fr—me dire™tly –sees9 the metri™—l stru™ture of the sp—™etime without —ssist—n™e from further m—teri—l systemsF „heir out™ome is determined solely ˜y th—t metE ri™—l stru™tureF „he s—tisf—™tion of the prin™iple of rel—tivity follows immedi—tely from the f—™t th—t —n —r˜itr—ry vorentz tr—nsform—tion h is — symmetry of the winkowski metri™ gab F1 th—t isD into —n it s—tises po™k9s inerti—l fr—me F2D ™ondition @IIAF then the metri™ „hereforeD seen ˜y F1 if h tr—nsforms —n —nd ™—rried —long inerti—l to F2D h fr—me £ gabD is the s—me —s the metri™ gab seen ˜y F2F sn the more re—listi™ ™—seD the experiments will involve further sp—™etime stru™turesD su™h —s ele™trom—gneti™ elds —nd ™h—rgesF „he prin™iple of rel—tivity will ˜e s—tised only if these further sp—™etime stru™tures s—tisfy the following ™onditionD whi™h is the geometri™ st—tement of the vorentz ™ov—ri—n™e of the theories of these furthers stru™turesF vet the theory h—ve models hM gab; @O1Aab:::; @O2Aab:::; : : :i @IPA 29„his tre—tments —ssumes th—t there —re no sp—™etime stru™tures th—t elude experiment—l testD su™h —s the —˜solute sp—™etime rigging of — xewtoni—n sp—™etimeD whi™h introdu™es — st—te of rest th—t ™—nnot ˜e reve—led in —ny experiment @see priedm—n IWVQD ™h sssAF General covariance and general relativity 839 where M is —n R4 dierenti—˜le m—nifoldD gab — winkowski metri™D —nd @O1Aab:::; @O2Aab:::; : : : the extr— sp—™etime stru™turesF sf h is —ny vorentz tr—nsform—tion —nd @IPA is — model of the theoryD then hMgab; h £ @O1Aab:::; h £ @O2Aab:::; : : :i must —lso ˜e — model of the theoryF „he s—tisf—™tion of the prin™iple of rel—tivity now folE lowsF vet F1 ˜e —n inerti—l fr—me of referen™e in whi™h —re ™ondu™ted experiments —sso™i—ted with stru™tures @O1Aab:::; @O2Aab:::; : : :F sf we tr—nsform vi— vorentz tr—nsform—tion h to —ny other inerti—l fr—me —nd out™omesF „h—t Fis2wD ewreeqreuqirueirteh—tth—thtetthheetohryeo—rlylo—wdsmtriut ™ptruer™eissehl£y@Oth1eAasb—::m:; he£e@xOp2eAraibm:::e;n:t:s: „his is pre™isely wh—t the geometri™ version of vorentz ™ov—ri—n™e —llowsF „his —n—lysis gives us — pre™ise sense in whi™h the equiv—len™e of inerti—l fr—mes of referen™e is re—lized within the spe™i—l theory of rel—tivityF „he ˜—si™ mor—l of the work of i—rm—nD priedm—n —nd tones is th—t there is no n—tur—l sense in whi™h this equiv—len™e o˜t—ins in the sp—™etimes of gener—l rel—tivity —nd th—t there is ™ert—inly no extension of it to —™™eler—ted fr—mes of referen™eF sn this senseD there is no prin™iple of rel—tivity in the gener—l theory of rel—tivityF „his mor—l follows immedi—tely from the f—™t th—t spe™i—l rel—tivity —dmits — nonEtrivi—l symmetry groupD the vorentz groupD whi™h m—ps inerti—l fr—mes of referen™e into one —notherF „he sp—™etimes of gener—l rel—tivity in gener—l —dmit no symmetriesF sn gener—l rel—tivityD the ™losest —n—log of —n inerti—l fr—me of referen™e is — fr—me in free f—llF st is represented ˜y — ™ongruen™e of timelike geodesi™sF sn gener—lD — tr—nsform—tion of th—t m—ps one freely f—lling fr—me of referen™e into —nother will not ˜e — symmetry of the metri™—l stru™tureF „herefore sp—™etime o˜servers of the rst fr—me will see dierent metri™—l properties in sp—™etime th—n will those of the se™ondF „he indistinguish—˜ility required for the equiv—len™e of fr—mes does not o˜t—inF gonsidering —r˜itr—ry fr—mes of referen™e r—ther th—n those in free f—ll ™le—rly does not ™h—nge this resultF „h—t this sense of equiv—len™e of fr—mes f—ils to o˜t—in in gener—l rel—tivity is not so surprising —nd it is di™ult to im—gine th—t iinstein ever expe™ted th—t it wouldF „he re—l puzzleD thenD is to determine the sense in whi™h iinstein ˜elieved the equiv—len™e to ˜e extended ˜y gener—l rel—tivityF „here is one re—ding in this geometri™ l—ngu—ge th—t does —llow — gener—l equiv—len™e of fr—mes @xorton IWVSD se™tion SAF ƒo f—r it h—s ˜een —ssumed th—t the ˜—™kground sp—™etime is represented ˜y the ™om˜in—tion of m—nifold —nd metri™F sf inste—d one t—kes the m—nifold —lone —s the ˜—™kground sp—™etimeD then one immedi—tely h—s —n equiv—len™e of —ll fr—mes of referen™eF porD ™onsidering just R4 m—nifolds for simpli™ityD —n —r˜itr—ry —utomorphism is — symmetry of the m—nifoldF ƒin™e —ny fr—me of referen™e ™—n ˜e m—pped into —ny other ˜y —n —utomorphismD it follows th—t e—™h fr—me –sees9 the s—me sp—™etime ˜—™kground so th—t they —re equiv—lent in —t le—st th—t senseF sf this equiv—len™e is to ˜e extended to the sort of equiv—len™e of the prin™iple of rel—tivity of spe™i—l rel—tivityD then the metri™ tensor eld of gener—l rel—tivity must ˜e tre—ted in — simil—r f—shion to the stru™tures @O1Aab:::; @O2Aab::: of the —˜ove dis™ussion of spe™i—l rel—tivityF „hen — simil—r sense of equiv—len™e of arbitrary fr—mes follows dire™tly from the —™tive gener—l ™ov—ri—n™e of gener—l rel—tivityF vet F1 ˜e —ny fr—me of referen™e whi™h sees — metri™—l eld gab —nd other elds @O1Aab:::; @O2Aab:::; : : :F „h—t isD the theory h—s — model hM gab; @O1Aab:::; @O2Aab:::; : : :i @IPA „henD if F2 is —ny other fr—me of referen™eD the theory must —llow — model in whi™h F2 sees —n identi™—lly ™ongured set of eldsF „h—t isD if h is —n —utomorphism th—t m—ps F1 into 840 J D Norton F2D then F2 must see the elds h £ @O1Aab:::; h £ @O2Aab::: so th—t the theory must —lso h—ve — model hMgab; h £ @O1Aab:::; h £ @O2Aab:::; : : :i „h—t it does follows dire™tly from its —™tive gener—l ™ov—ri—n™e @se™tion SFR —˜oveAF „he di™ulty with this propos—l is th—t it —llows —n equiv—len™e of —r˜itr—ry fr—mes of referen™e in —ll theories th—t —re gener—lly ™ov—ri—ntF ƒu™h theories in™lude versions of spe™i—l rel—tivity —nd xewtoni—n sp—™etime theoryF „husD if this gener—lized equiv—len™e of fr—mes is to ˜e distin™tive to gener—l rel—tivityD there must ˜e some prin™ipled w—y of releg—ting the metri™ tensor to the ™ontents of sp—™etime in gener—l rel—tivityD where—s in other sp—™etime theoriesD su™h —s spe™i—l rel—tivityD this metri™—l stru™ture is to ˜e p—rt of the ˜—™kground sp—™etimeF ‡h—t m—kes su™h — division pl—usi˜le is the f—™t th—t the metri™ tensor of gener—l rel—tivity in™orpor—tes the gr—vit—tion—l eldF „hus its st—te is —e™ted ˜y the disposition of m—sses in the s—me w—y —s — w—xwell eld is —e™ted ˜y the disposition of ™h—rgesF „he —n—logy ™—n ˜e pressed furtherF sn spe™i—l rel—tivity one ™—n ™ondu™t —n ele™tri™—l experiment with some ™ongur—tion of ™h—rges in —n inerti—l fr—me of referen™eF „he prinE ™iple of rel—tivity requires th—tD if we were to re™re—te th—t s—me ™ongur—tion of ™h—rges in —nother inerti—l fr—meD then we would produ™e the identi™—l elds —nd experiment—l out™omesF „his is the sense in whi™h —ll inerti—l fr—mes of referen™e —re equiv—lentF ƒimE il—rlyD one ™ould ™onsider some ™ongur—tion of m—sses —nd the metri™ eld they produ™e in rel—tion to —n —r˜itr—ry fr—me of referen™e in gener—l rel—tivity —s — kind of gr—vit—tion—l experiment in th—t fr—meF „he —™tive gener—l ™ov—ri—n™e of gener—l rel—tivity then tells us th—t we ™ould h—ve l—id out the s—me ™ongur—tion of m—sses —nd elds in —ny other fr—me of referen™eD so th—t the gr—vit—tion—l experiment would h—ve pro™eeded identi™—lly in —ny fr—me of referen™eF „his gives us — sense in whi™h —r˜itr—ry fr—mes of referen™e —re equiv—lent in gener—l rel—tivityF „he su™™ess of this gener—lized equiv—len™e depends fully on our ˜eing —˜le to ™on™eive of the metri™ eld —s —p—rt of the ™ontents of sp—™etime in gener—l rel—tivity ˜ut not in other theories like spe™i—l rel—tivityF iinstein9s IWIV version of w—™h9s prin™iple —llowed this ™on™eption sin™e it required th—t the metri™ eld ˜e fully determined ˜y the m—tter distri˜utionD so th—t this eld would h—ve the s—me sort of st—tus —s the m—tter distri˜utionF ƒin™e w—™h9s €rin™iple in this form f—ils in m—ny of the sp—™etimes of gener—l rel—tivityD it ™—nnot ˜e used to justify — gener—lized equiv—len™e of fr—mes in th—t theoryF „he only other well developed —n—lysis th—t —llows this ™on™eption of the metri™ eld ™on™erns the distin™tion ˜etween —˜solute —nd dyn—mi™ o˜je™tsD to ˜e dis™ussed in the se™tion V ˜elowF es — dyn—mi™—l o˜je™tD the metri™ of gener—l rel—tivity is n—tur—lly ™l—ssied —s p—rt of the ™ontent of sp—™etimeF es —n —˜solute o˜je™tD the winkowski metri™ of spe™i—l rel—tivity is n—tur—lly ™l—ssied —s p—rt of the ˜—™kground sp—™etimeF General covariance and general relativity 841 7. General relativity without principles 7.1. General relativity without general relativity iinstein9s own developments —nd dis™ussion of the gener—l theory of rel—tivity pl—™e so mu™h import—n™e on gener—l ™ov—ri—n™e —nd the extension of the prin™iple of rel—tivity th—t most —™™ounts of the theory seem ™ompelled to t—ke — position on their import—n™eF w—ny essenti—lly —gree with iinstein —s we h—ve seen in se™tion RF w—ny othersD —s we h—ve seen in se™tions S —nd TD dis—gree with iinstein9s viewsY they develop gener—l rel—tivity without ™l—iming gener—l ™ov—ri—n™e —s — fund—ment—l physi™—l postul—te —nd they expl—in why they do soF „here is — third ™—tegory of exposition of gener—l rel—tivityF „hese —re the expositions th—t t—ke no spe™i—l noti™e of gener—l ™ov—ri—n™e —t —llF yf ™ourse the develop gener—l rel—tivity in — gener—lly ™ov—ri—nt form—lismD —s is the st—nd—rd pr—™ti™eF rowever the expositions —re ™onspi™uous for the —˜sen™e of —ny st—tement of fund—ment—l prin™iple ™on™erning ™ov—ri—n™e or rel—tivityF „here is no –prin™iple of gener—l ™ov—ri—n™e9D no –gener—l prin™iple of rel—tivity9 —nd no pronoun™ement th—t the theory h—s extended the equiv—len™e of fr—mes of referen™e to —™™eler—ted fr—mesF end there is no expl—n—tion of why these prin™iples —re not dis™ussedF st is di™ult to know wh—t signi™—n™e to re—d into su™h formul—tions of gener—l rel—E tivity without gener—l rel—tivityF w—ny of these expositions —re m—them—ti™—lly orientedF ƒo we might suppose th—t their —uthors simply de™ided not to ™ontend with the question of the physi™—l found—tions in f—vor of other more m—them—ti™—l —spe™ts of the theoryF st is h—rd to im—gineD howeverD th—t —n —uthor of writing on gener—l rel—tivity ™—n ˜e ™ompletely un—w—re of iinstein9s viewsD if not —lso the disputes over themF „herefore when th—t —uthor writes text˜ook length exposition of gener—l rel—tivity whi™h f—ils to in™lude su™h phr—ses —s –gener—l prin™iple of rel—tivity9 or –prin™iple of gener—l ™ov—ri—n™e9D one must suppose th—t the —uthor is m—king — st—tement ˜y omissionF @„he omissions —re typi™—lly so ™omplete th—tD if the text h—s —n indexD these terms will not ˜e listed in itFA ‡e h—ve —lre—dy seen th—t ƒynge —nd po™k o˜je™t to –gener—l rel—tivity9 —s — misnomerF „hus it seems o˜vious th—t simil—r sentiments drive su™h —uthors —s th—t of Time and space, Weight and Inertia: A Chronogeometrical Introduction to Einstein's Theory @pokker IWTSA who displ—y rem—rk—˜le ingenuity in —voiding the term –gener—l rel—tivity9F pin—llyD even if no st—tement is ˜eing m—de ˜y omissionD the very possi˜ility —nd freE quen™y of su™h —™™ounts of gener—l rel—tivity do indi™—te th—t the pl—™e of these prin™iples in the theory might not ˜e so str—ightforw—rdF sf the prin™iples —re fund—ment—l physi™—l —xiomsD they would ˜e h—rd to —voidD even —s ™onsequen™es in —n —ltern—te —xiom—tiz—E tionF yne is h—rd pressed to im—gine — formul—tion of thermodyn—mi™s without the l—w of ™onserv—tion of energy —s — fund—ment—l —xiom or one of the e—rliest —nd most import—nt theorems3 „he su˜tlety of the situ—tion is ™—ptured ˜y „r—utm—nD who o˜served well into his exposition @IWTRD pIPPA of gener—l rel—tivity F F F we h—ve m—n—ged to o˜t—in gener—l rel—tivity ˜y — @we hopeA f—irly ™onvin™ing ™h—in of re—soning without ever mentioning su™h — prin™iple ‘of gener—l ™ov—ri—n™e“F re did pro™eedD howeverD to list sever—l senses of the prin™iple —nd their nonEtrivi—l rel—E tionships to the theoryF „hus one ™—n nd gener—l ™ov—ri—n™e relev—nt without mentioning it in development of the theoryF ‡ith these interpret—tive ™—utionsD we ™—n pro™eed to note th—t the tr—dition of exE position of gener—l rel—tivity without gener—l rel—tivity extends ˜—™k to the e—rliest of the theoryF „here —re m—ny exposition of rel—tivity theory with this ™h—r—™ter from the IWPHsF „hey in™lude f—uer @IWPPAD firkho @IWPUAD h—rmois @IWPUAD gh—zy @IWPVA —nd he honder 842 J D Norton @IWPSA @˜ut he honder @IWPID ppIH{ISA h—d emph—sized the —r˜itr—riness of ™oordin—tes in gener—l rel—tivity —nd the inv—ri—n™e of its fund—ment—l equ—tionsAF iddington @IWPRD ™h sD se™tion l A l—˜ours in det—il the notion th—t one ™—n use —r˜itr—ry –sp—™eEtime fr—mes9 for deE s™ri˜ing phenomen—D ˜ut without ever mentioning — prin™iple of ™ov—ri—n™e or — gener—lized prin™iple of rel—tivityF ris e—rlier iddington @IWPHD pPHA h—d —llowed th—t — gener—liz—tion of the prin™iple of rel—tivity in the theory in so f—r —s he ™on™eded –it will ˜e seen th—t this prin™iple of equiv—len™e is — n—tur—l gener—liz—tion of the prin™iple of rel—tivity9F „his rem—rk w—s not repe—ted in iddington @IWPRAF „he le—n ye—rs —fter the IWPHs s—w sever—l exposition of gener—l rel—tivity without gener—l rel—tivityX ‚—ini™h @IWSHA —nd the synopsis of gener—l rel—tivity ˜y —tskis @IWSSAF „he reviv—l of interest in gener—l rel—tivity in the IWTHs ˜rought more su™h expositions —nd they h—ve in™luded some of the most import—nt expositions of the theoryX pokker @IWTSAD ƒ™hild @IWTUA @—lthough he mentions @pPHA th—t gener—l rel—tivity –shows there —re no inerti—l fr—mes —t —ll9AD ‚o˜ertson —nd xoon—n @IWTVAD ihlers @IWUIAD r—wking —nd illis @IWUQAD hir—™ @IWUSAD p—lk —nd ‚uppel @IWUSA @—lthough the notion of — gener—lized prin™iple of rel—tivity is —lluded to ˜rieyD eFgFD pQPQAD ƒ—™hs —nd ‡u @IWUUAD gl—rke @IWUWA @—lthough se™tion QFIFQ does emph—size the loss of glo˜—l inerti—l systems —nd the novelty of —r˜itr—ry ™oordin—te systems in gener—l rel—tivityAD pr—nkel @IWUWAD ƒ™hutz @IWVSA @elthough it is —llowed @p QA th—t gener—l rel—tivity is more gener—l in —llowing ˜oth inerti—l —nd —™™eler—ted o˜serversAD w—rtin @IWVVAD rughston —nd „od @IWWHAD ƒtew—rt @IWWHAF 7.2. The principle of equivalence as the fundamental principle ‡hile m—ny of these —™™ounts of gener—l rel—tivityD —void mention of prin™iples of gener—l ™ov—ri—n™e —n gener—lized rel—tivityD m—ny of them do nd — spe™i—l pl—™e for just one of the three fund—ment—l prin™iples listed ˜y iinstein in IWIVD the prin™iple of equiv—len™eF yf ™ourseD the version used is typi™—lly not iinstein9s ˜ut one or other v—ri—nt of —n innitesE im—l prin™iple of equiv—len™eF „he prin™iple is not used in iinstein9s m—nner —s — stepping stone to — gener—lized prin™iple of rel—tivityF ‚—ther it is used to est—˜lish — notion ™l—imed —s — fund—ment—l prin™iple of gener—l rel—tivityD th—t spe™i—l rel—tivity holds innitesim—lly in the theoryY orD less ™ommonlyD it is just t—ken to ˜e —s mu™h of the gener—lized prin™iple of rel—tivity —s gener—l rel—tivity will —dmitF ƒu™h tre—tmentsD whi™h employ only the prin™iple of equiv—len™e —s — fund—ment—l prin™ipleD in™ludeX ƒil˜erstein @IWPPD p IPAD iddington @IWPRD se™tion IUA @—lthough emph—E sizing @pRIA th—t the prin™iple is to ˜e derived r—ther th—n postul—ted in the expositionAD firkho @IWPUD pp IRH{RAD v—nd—u —nd vifshitz @IWSID ™h IHA pokker @IWTSD se™tion TFWA @with the prin™iple in iinstein9s origin—l formAD ‚o˜ertson —nd xoon—n @IWTVD se™tion TFWAD ƒ™hild @IWTUAD p—lk —nd ‚uppel @IWUSD se™tion QPAD gl—rke @IWUWD ™h QAD pr—nkel @IWUWD ™h PAD ‚—ine —nd reller @IWVID ™h TFVA ƒ™hutz @IWVSD p IVRAD w—rtin @IWVVD se™tion IFTD SFIIAD ƒtew—rt @IWWHD se™tion IFIQAF ‡e h—ve the exposition of „onnel—t @IWSWAD who t—kes the prin™iple of equiv—len™e to ˜e — –prin™iple of gener—lized rel—tivity9 @p QPUA —nd ‡—sserE m—n @IWWPAD who —lso rem—rks ˜riey @pQRPA th—t the prin™iple of equiv—len™e extends the prin™iple of rel—tivity to in™lude —™™eler—ted fr—mes of referen™eF 7.3 Challenges to the principle of equivalence yne might well wonder if we h—ve not —t l—st found the un™ontroversi—l ™ore of iinstein9s —™™ounts of the found—tion—l prin™iples of gener—l rel—tivity in these expositionsF „h—t ™ore would now just ˜e the prin™iple of equiv—len™eD even if it is in —n —ltered form iinstein never endorsedF rowever not even the popul—r versions of the prin™iple of equiv—len™e h—ve es™—ped telling —tt—™kF „he ˜est known ™h—llenge h—s ˜een st—ted most ™le—rly ˜y ƒyngeF ris ™on™ern is th—t General covariance and general relativity 843 the presen™e or —˜sen™e of — gr—vit—tion—l eld must ˜e ™h—r—™terized geometri™—llyD th—t isD in inv—ri—nt termsF re —sserts th—t the presen™e of — gr—vit—tion—l eld ™orresponds just with nonEv—nishing ™urv—ture of sp—™etimeF ƒu™h —n inv—ri—nt ™riterion is un—e™ted ˜y ™oordin—te tr—nsform—tionD ˜y ™h—nge of fr—me of referen™e or ˜y — ™h—nge of the st—te of motion of the o˜serverF „herefore none of these ™h—nges will ˜e —˜le to tr—nsform —w—y — gr—vit—tion—l eld or ˜ring one into existen™eD ™ontr—ry to m—ny versions of the prin™iple of equiv—len™eF re is unimpressed with the requirement th—t the sp—™etime metri™ ˜e™ome di—g@IDIDIDEIA —t some nomin—ted eventD there˜y mimi™king spe™i—l rel—tivity —t le—st in some innitesim—l senseF ƒynge deems this trivi—l sin™e it merely —mounts to the requirement th—t the metri™ h—ve vorentz sign—tureF „hus he wrote his f—mous l—ment @IWTHD p ixA —˜out rel—tivists who F F F spe—k of the €rin™iple of iquiv—len™eF sf soD it is my turn to h—ve — ˜l—nk mindD for s h—ve never ˜een —˜le to underst—nd this prin™ipleF hoes it me—n th—t the sign—ture of the sp—™etime metri™ is CP @or EP if you prefer the other ™onventionAc sf soD it is import—ntD ˜ut h—rdly — prin™ipleF hoes it me—n th—t the ee™ts of — gr—vit—tion—l eld —re indistinguish—˜le from the ee™ts of —n o˜server9s —™™eler—tionc sf so it is f—lseF sn iinstein9s theoryD either there is — gr—vit—tion—l eld or there is noneD —™™ording —s the ‚iem—nn tensor does not or does v—nishF „his is —n —˜solute propertyD it h—s nothing to do with —ny o˜server9s worldlineF ƒp—™etime is either —t or ™urvedD —nd in sever—l pl—™es in the ˜ook s h—ve ˜een —t ™onsider—˜le p—ins to sep—r—te truly gr—vit—tion—l ee™ts due to ™urv—ture of sp—™etime from those due to ™urv—ture of the o˜server9s worldEline @in most ™—ses the l—tter predomin—teAF „he €rin™iple of iquiv—len™e performed the essenti—l o™e of midwife —t the ˜irth of gener—l rel—tivityF futD —s iinstein rem—rkedD the inf—nt would never get ˜eyond its longE™lothes h—d it not ˜een for winkowski9s ™on™eptF s suggest th—t the midwife ˜e now ˜uried with —ppropri—te honors —nd the f—™ts of —˜solute sp—™eEtime f—™edF „he ide— th—t the presen™e of — gr—vit—tion—l eld is —sso™i—ted with the inv—ri—nt property of ™urv—ture ™—n ˜e tr—nsl—ted in to o˜serv—tion—l termsF „he nonEv—nishing of the ‚iem—nn ™urv—ture tensor ent—ils the existen™e of tid—l for™es —™ting on ˜odies in free f—llF „he go—l of restri™ting versions of the prin™iple of rel—tivity to innitesim—l regions of sp—™etime is to elimin—te these tid—l for™esF rowever they ™—nnot ˜e so elimin—tedY for ex—mpleD the tid—l ˜ulges on — freely f—lling droplet rem—in —s the droplet ˜e™omes —r˜itr—rily sm—llD ignoring su™h ee™ts —s surf—™e tensionY see yh—ni—n @IWUTD ™h ID IWUUA —nd fondi @IWUWAF ƒee —lso xorton @IWVSD se™tion IHA for —n —ttempt to ™h—r—™terize the impre™ise restri™tion to innitesim—l regions —s — restri™tion on —™™ess to ™ert—in orders of qu—ntities dened —t — pointF pollowing — suggestion from iinsteinD it turns out th—t —n innitesim—l prin™iple of equiv—len™e ™—n hold only —t the expense of — restri™tion so severe th—t it trivi—lizes the prin™ipleF ƒee —lso xorton @IWVSD se™tion IIA for iinstein9s response to the ide— th—t v—nishing sp—™etime ™urv—ture is to ˜e —sso™i—ted with the —˜sen™e of — gr—vit—tion—l eldF 8. Eliminating the absolute 8.1. Anderson's absolute and dynamical objects rowever else he m—y h—ve ™h—nged his viewpointsD we h—ve seen @se™tion QFWA th—t iinE stein m—int—ined throughout the lifetime of his writings on gener—l rel—tivity th—t it w—s distinguished from e—rlier theories ˜y — single —™hievementD it h—d elimin—ted — ™—us—l —˜E soluteD the inerti—l systemF sf we —re to h—ve —n —™™ount th—t truly ™—ptures iinstein9s underst—nding of gener—l ™ov—ri—n™eD then we should expe™t this r—ther impre™isely notion to pl—y — prominent roleF „his notion surely lies ˜ehind €—uli —nd ‡eyl9s emph—sizing th—t the metri™ tensor is determined ˜y the m—tter distri˜ution through eld equ—tions —nd th—t 844 J D Norton this justies @‡eylA the n—me –gener—l theory of rel—tivity9 @see se™tion SFS —˜oveAF iinstein9s notion surf—™es more ™le—rly in fergm—nn9s @IWSUD ppII{IPA ™on™eption of we—k —nd strong ™ov—ri—n™eF ‡e—k ™ov—ri—n™e is the type we see in when we use m—ny dierent ™oordin—te systems to des™ri˜e the one phenomenon in v—gr—ngi—n me™h—ni™sF „he fund—ment—lly trivi—l n—ture of this –we—k ™ov—ri—n™e9 derives from the rigidity of the ™l—ssi™—l metri™F „his is quite distin™t from the strong ™ov—ri—n™e of gener—l rel—tivity where30 it is one9s t—sk to calculate the metric F F F —s — dyn—mi™ v—ri—˜leF ‡e ™—n t—ke one ™oordin—te system or —nother for this jo˜D ˜ut —ll th—t we ™—n know is the rel—tion of one fr—me to the otherX we do not know the rel—tion of either to the worldF –ƒtrong ™ov—ri—n™e9D thereforeD ™ont—ins not only — referen™e to the stru™tur—l simil—rity of —n equ—tion —nd its tr—nsform—tionD it implies —s well th—t one fr—me is —s good — st—rting point —s —nother|th—t we do not need prior knowledge of its physi™—l me—ning F F F whi™h is gener—ted —t the endF w—ny import—nt themes —re tou™hed on hereD —s h—s ˜een indi™—ted ˜y ƒt—™hel @forth™omingD footnote QAF „his distin™tion ˜etween we—k —nd strong ™ov—ri—n™e —mount to th—t ˜etween p—ssive —nd —™tive ™ov—ri—n™eF ‡h—t ™on™erns us hereD howeverD is th—t ™ontr—sting of the –rigidity of the ™l—ssi™—l metri™9 with the metri™ of gener—l rel—tivity –—s — dyn—mi™ v—ri—˜le9F „he most pre™ise ™ontext so f—r for the st—tement of iinstein9s ™—us—l ™on™erns h—s ˜een provided ˜y enderson @IWTRD IWTUD ™h RD IWUIA @˜ut see —lso enderson @IWTPA for — denition of —˜solute ™h—nge within gener—l rel—tivityAF sn l—ying out his systemD enderson uses — somewh—t idiosyn™r—ti™ nomen™l—tureF re l—˜els the set of —ll possi˜le v—lues of the geometri™ o˜je™ts of — theory the –kinem—ti™—lly possi˜le tr—je™tories9F „hose s—n™tioned ˜y the –dyn—mi™ l—ws9 or –equ—tions of motion9 of the theoryD he ™—lls the –dyn—mi™—lly possi˜le tr—je™tories9F „he prin™iple novelty of enderson9s development is the distin™tion ˜etween –—˜solute9 —nd –dyn—mi™—l9 o˜je™tsF „h—t distin™tion will ˜e used to strengthen the prin™iple of gener—l ™ov—ri—n™e into — more restri™tive –prin™iple of gener—l inv—ri—n™e9F elthough —llowing for — time th—t ˜oth spe™i—l —nd gener—l ™ov—ri—n™e prin™iples —re devoid of physi™—l ™ontent @IWTRD pIVRAD enderson @IWTUD se™tion RFPD IWUID ppITP{TSA then ™—me to urge th—t the requirement of gener—l ™ov—ri—n™e is not physi™—lly v—™uousF re —llowed th—t one ™—n t—ke — physi™—l theory —nd gener—te su™™essive formul—tions of wider —nd wide ™ov—ri—n™eF rowever there is — point in the hier—r™hy —t whi™h we —re for™ed to introdu™es elements whi™h —re uno˜serv—˜le or tr—ns™end me—surementF ƒin™e we —re prohi˜ited from pro™eeding to this point in the hier—r™hyD ™ov—ri—n™e requirements h—ve physi™—l for™eF @„his str—tegy for inje™ting physi™—l ™ontent into ™ov—ri—n™e prin™iples is essenti—lly the one used ˜y €—uli —nd others in se™tion SFS —˜oveFA „he —˜solute o˜je™ts of — sp—™etime theory —re distinguished ˜y pre™isely the ™—us—l ™riterion th—t —llowed iinstein to design—te the inerti—l systems of spe™i—l rel—tivity —s —˜soluteF enderson —nd q—utre—u @IWTWD pITSUA summ—rizeX ‚oughly spe—kingD —n —˜solute o˜je™t —e™ts the ˜eh—viour of other o˜je™ts ˜ut is not —e™ted ˜y these o˜je™ts in turnF 30„he two ellipses –F F F 9 —nd emph—sis —re fergm—nn9sF General covariance and general relativity 845 „he rem—ining o˜je™ts —re dyn—mi™—lF „hus the winkowski metri™ of spe™i—l rel—tivity is —n —˜solute o˜je™tF sn spe™i—l rel—tivisti™ ele™trodyn—mi™sD the winkowski metri™ —e™ts the w—xwell eld —nd ™h—rge ux in determiningD for ex—mpleD whi™h —re the inerti—l tr—je™tories of ™h—rgesF rowever neither w—xwell eld nor ™h—rge uxD the dyn—mi™—l o˜je™ts of the theoryD —e™t the winkowski metri™F ‡h—tever their formD the winkowski metri™ st—ys the s—meF „his is the sense in whi™h it —e™ts without ˜eing —e™tedF ƒin™e the winkowski metri™ indu™es the inerti—l fr—mes on sp—™etimeD enderson9s identi™—tion of the winkowski metri™ —s —n —˜solute o˜je™t ts ex—™tly with iinstein9s identi™—tion of inerti—l fr—mes —s —˜solutesF „his loose denition must ˜e m—de more pre™ise —nd enderson @IWTUD pVQ{RA @see —lso enderson @IWUID pITTA gives — more pre™ise denitionF r—ving elimin—ted irrelev—nt o˜je™ts from the set of geometri™ o˜je™ts yA —llowed in the theory ‡e now pro™eed to divide the ™omponents of yA into two setsD  —nd za where the  h—ve the following two propertiesX @IA the  ™onstitute the ˜—sis of — f—ithful re—liz—tion of the ™ov—ri—n™e group of the theoryF @PA eny  th—t s—tises the equ—tions of motion of the theory —ppe—rsD together will —ll its tr—nsforms under the ™ov—ri—n™e groupD in every equiv—len™e ™l—ss of dpt @dyn—mi™—lly possi˜le tr—je™toriesA „he  if they existD —re the ™omponents of the —˜solute o˜je™ts of the theoryF „he rem—ining p—rt of yAD the za —re then the ™omponents of the dyn—mi™—l o˜je™ts of the theoryF gondition @IA is —n import—nt ˜ut essenti—lly te™hni™—l ™ondition th—t the tr—nsform—E tion ˜eh—viour of the  respe™t the group stru™ture of the theory9s ™ov—ri—n™e group @ eF gF the  ought to tr—nsform ˜—™k into themselves under —n identity tr—nsform—tion of the ™ov—ri—n™e groupFA gondition @PA essenti—lly just s—ys th—t the —˜solute o˜je™ts  —re the s—me in every dyn—mi™—lly possi˜le tr—je™tory @iFeF modelA of the theoryF „he ™ondiE tionD howeverD must —llow th—t —n —˜solute o˜je™tD su™h —s — winkowski metri™D g ™—n ˜e m—nifested on m—ny dierent forms —s it tr—nsforms under the mem˜ers of the ™ov—ri—n™e groupF „herefore the se™ond ™ondition ™olle™ts the dyn—mi™—lly possi˜le tr—je™tories into equiv—len™e ™l—sses of intertr—nsform—˜le mem˜ersF ƒin™e e—™h ™l—ss is ™losed under tr—nsE form—tions of the ™ov—ri—n™e groupD the one set of —˜solute o˜je™ts —nd —ll their tr—nsforms will —ppe—r in e—™h ™l—ssF „hus ™ondition @PA requiresD in ee™tD th—t the —˜solute o˜je™ts th—t —ppe—r in —ll models —re the s—me up to — tr—nsform—tion of the theory9s ™ov—ri—n™e groupF ‡ith this distin™tion in pl—™eD enderson now denes the symmetry group or –inv—ri—n™e group of — physi™—l theory9 @enderson IWUID pITTA —s th—t su˜group of the ™ov—ri—n™e group of the theory whi™h le—ves inv—ri—nt the —˜solute o˜je™ts of the theoryF sn p—rti™ul—rD if there —re no —˜solute o˜je™tsD the inv—ri—n™e group —nd the ™ov—ri—n™e groups —re the s—me groupF „he –le—ves inv—ri—nt9 is to ˜e understood in the sense of — symmetry tr—nsform—tion su™h —s given in @IHA —nd @IIA —˜oveF „here is —n —n—logous denition for the –symmetry group of — physi™—l system9 @enderson IWTUD pVUA enderson9s ™entr—l ™l—im @eFgF enderson IWTUD pQQVA is th—t this symmetry group is wh—t iinstein re—lly h—d in mind when he —sso™i—ted the vorentz group with spe™i—l relE —tivity —nd the gener—l group with gener—l rel—tivityF por — requirement on — symmetry groupD not — ™ov—ri—n™e groupD is the ™orre™t w—y to express — rel—tivity prin™ipleF iven if we formul—te our theories in gener—lly ™ov—ri—n™e f—shionD they ™ontinue to ˜e ™h—r—™terized ˜y the groups expe™ted if we look to their symmetry groupsF „he symmetry group of — gener—lly ™ov—ri—n™e spe™i—l rel—tivity is the vorentz groupF eg—inD ™onsider — gener—lly ™oE 846 J D Norton vw—hrei—rentthfoermgru—lv—itti—otnioonf—xl eweltdonisi—nnotspin—™™oetripmoer—ttheedorinytwo irthaFsp„—h™eentimtheessteruth™rteuereos˜tjae;™htsab——rnedtrhae —˜solute o˜je™ts of the theory —nd their symmetry group is the q—lile—n groupF pin—llyD gener—l rel—tivity h—s no —˜solute o˜je™tsF sts symmetry group is the gener—l groupF yne ™—n gr—sp the pi™ture urged if one im—gines th—t the ˜—™kground sp—™etime of — theory is the sp—™etime m—nifold together with the theory9s —˜solute o˜je™ts|—lthough –˜—™kground sp—™etime9 is not — notion dis™ussed ˜y endersonF sn the ™—ses of spe™i—l rel—tivity —nd the —˜ove version of xewtoni—n sp—™etime theoryD ˜oth —dmit — f—mily of preferred inerti—l fr—mes of referen™e whi™h rem—in un™h—nged under the vorentz group or q—lile—n group respe™tivelyF sn the ™—se of gener—l rel—tivityD the ˜—™kground sp—™etime is just the m—nifold whose symmetry group is the group of —r˜itr—ry tr—nsform—tionsF e™™ording to endersonD wh—t iinstein re—lly intended with his prin™iple of gener—l ™ov—ri—n™e is wh—t enderson ™—lls the –prin™iple of gener—l inv—ri—n™e9F „his prin™iple requires th—t the symmetry group of — theory ˜e the gener—l group of tr—nsform—tions orD —s enderson ™—lls themD the –m—nifold m—ppings group9F „his prin™iple rules out the possi˜ility of —ny nonEtrivi—l —˜solute o˜je™ts in the theoryD th—t isD those whi™h h—ve more th—n merely topologi™—l propertiesF sn this senseD the prin™iple of gener—l inv—ri—n™e —mounts to — noE—˜soluteEo˜je™t requirement —nd oers — pre™ise re—ding for iinstein9s ™l—im th—t gener—l ™ov—ri—n™e h—s elimin—ted —n —˜solute from sp—™etimeF 8.2. Responses to Anderson's viewpoint enderson9s ide—s on —˜solute —nd dyn—mi™—l o˜je™ts h—ve found — limited ˜ut f—vor—˜le response in the liter—tureF wisner et al @IWUQD se™tion IUFTA present — requirement of no —˜solute o˜je™ts in terms of the requirement of –no prior geometry9 whereX fy –prior geometry9 one me—ns —ny —spe™t of the geometry of sp—™etime th—t is xed immut—˜lyD iFeF th—t ™—nnot ˜e ™h—nged ˜y ™h—nging the distri˜ution of gr—vit—ting sour™esF „hey des™ri˜e iinstein —s seeking ˜oth this requirement —s well —s — –geometri™D ™oordin—te independent formul—tion of physi™s9 when he required gener—l ™ov—ri—n™e|—nd th—t this h—s ˜een responsi˜le for h—lf — ™entury of ™onfusionF enderson9s prin™iple of gener—l inv—ri—n™e —ppe—rs in „r—utm—n @IWUQAD —s does the distin™tion ˜etween —˜solute —nd dyn—mi™—l o˜je™ts in uop™zynski —nd „r—utm—n @IWWPD ™h IQAF yh—ni—n @IWUTD ppPSP{RA uses enderson9s prin™iple of gener—l inv—ri—n™e to respond to urets™hm—nn9s o˜je™tion th—t gener—l ™ov—ri—n™e is physi™—lly v—™uousF re does insistD howeverD th—t the prin™iple is not — rel—tivity prin™iple —nd th—t the gener—l theory of rel—tivity is no more rel—tivisti™ th—n the spe™i—l theory @pPSUAF enderson9s ide—s seem —lso to inform fu™hd—hl9s @IWVID le™ture TA notion of –—˜solute form inv—ri—n™e9F „he distin™tion ˜etween —˜solute —nd dyn—mi™—l o˜je™ts h—s ˜een re™eived —nd develE oped most w—rmly ˜y philosophers of sp—™e —nd timeD so th—t in pl—™e of @TAD the gener—l model of sp—™etime theory is given —s h i M; A1; A2; : : : ; D1; D2; : : : where A1; A2; : : : —re the —˜solute o˜je™ts —nd D1; D2; : : : the dyn—mi™—lF rowever they do not gener—lly —llow th—t enderson9s re—soning h—s vindi™—ted iinstein9s ™l—im th—t the gener—l theory of rel—tivity extends the prin™iple of rel—tivity of spe™i—l rel—tivityF ƒee i—rm—n @IWURD IWVWD ™h QAD priedm—n @IWUQD IWVQAD —nd riskes @IWVRAF i—rm—n @IWVWD se™tion QFRA investig—tes the possi˜ilities of the symmetry group of the —˜solute o˜je™ts of — theory diering from the symmetry group of the dyn—mi™—l o˜je™tsF 8.3. No gravitational eld|no spacetime points General covariance and general relativity 847 ƒt—™hel @IWVTD se™tions SD TA h—s provided —n interesting extension of the viewpoint —dv—n™ed ˜y endersonF ƒt—™hel9s ™on™ern is th—t our formul—tions of gener—l rel—tivity —re still not in — position to expli™—te iinstein9s ide— th—t sp—™etime ™—nnot exist without the gr—vit—tion—l eld @see se™tion QFS —˜oveAF ƒt—™hel f—ults our representing of physi™—l sp—™etime events ˜y the m—them—ti™—l points of the sp—™etime m—nifoldF ‚e—d n—ivelyD this denition tells us th—t — m—nifold without metri™—l eld represents — physi™—l sp—™etime of events with topologi™—l properties ˜ut with no metri™—l rel—tionsF ƒt—™hel9s propos—l —pplies to sp—™etime theories without —˜solute o˜je™tsD whi™h he ™—lls –gener—lly ™ov—ri—nt9D —nd ™—n ˜e reviewed only inform—lly hereF „o form the models of su™h theories one —ssigns v—rious geometri™ o˜je™ts|tensor eldsD for ex—mple|to e—™h point of the m—nifold in the usu—l w—yF sn prin™ipleD m—ny dierent su™h elds ™ould ˜e —ssignedF sn the ™—se of gener—l rel—tivityD we h—ve — host of possi˜le metri™—l elds of —ll sorts of dierent ™urv—tureF „he loose notion of the sp—™e of —ll su™h possi˜le elds is given pre™ise formul—tion ˜y ƒt—™hel —s — ˜re ˜undle E over the m—nifold MF „he p—rti™ul—r elds th—t —re ™hosen for in™lusion in the theory9s models —re pi™ked out though ™rossEse™tions of the ˜re ˜undle EF voosely spe—kingD — ™ross se™tion  —mounts to —n —sso™i—tion of — point of the m—nifold M with the geometri™ o˜je™ts —ssigned to it in some model of the theoryF @wore pre™iselyD — ™rossEse™tion  is — m—p th—t goes from — point p of the m—nifold M to — mem˜er @pA of the ˜re ˜undle ED where @pA must ˜e —sso™i—ted with p ˜y the ˜undle9s proje™tion m—p D so th—t @pA a pFA „he ™ore of ƒt—™hel9s propos—l is th—t the physi™—l events of sp—™etime —re represented ˜y the inverse of this m—p F „h—t is|loosely spe—king|the physi™—l events —re not represented dire™tly ˜y the points of the sp—™etime m—nifoldY r—therD in their pl—™eD we use the —sso™i—tion of the points of the m—nifold with the geometri™ stru™tures dened on themF ‡e now —utom—ti™—lly h—ve the property of sp—™etime th—t iinstein —nnoun™edF sf we t—ke —w—y the gr—vit—tion—l eldD th—t is the metri™ eldD from — sp—™etime in gener—l rel—tivityD then we h—ve t—ken —w—y the ˜re ˜undle —nd with it the m—p th—t represents the physi™—l sp—™etime eventsF sn — theory with —˜solute o˜je™tsD howeverD physi™—l events —re represented dire™tly ˜y points of the ˜—se m—nifoldF „herefore their ˜eh—viour is quite dierentF ƒee ƒt—™hel @IWVTA for further det—ils of how theories with —˜solute o˜je™ts —re tre—ted —nd of the m—™hinery needed to —llow th—t one physi™—l situ—tion is represented ˜y —n equiv—len™e ™l—ss of dieomorphi™ modelsF 8.4. What are absolute objects and why should we despise them? „here —re two —re—s of di™ulty —sso™i—ted with the gener—l theory of —˜solute —nd dyE n—mi™—l o˜je™tsF „he rst is th—t question of how we dene —˜solute o˜je™tsF enderson9s denition w—s th—t —n o˜je™t w—s —˜solute if the s—me o˜je™t @up to ™oordin—te tr—nsforE m—tionA —ppe—red in —ll the theory9s modelsF sn the ™oordin—te freeD geometri™ l—ngu—ge how —re we to underst—nd the –s—me9c „he o˜vious ™—ndid—te is th—t two o˜je™ts —re the s—me if they —re isomorphi™F qlo˜—l isomorphism is the ™riterion used in i—rm—n9s @IWURD pPVPA denition of —˜solute o˜je™ts to pi™k out when one h—s the s—me o˜je™t in —ll modE elsF priedm—n @IWUQD pQHV{WD IWVQD pSV{THA uses only the requirement th—t the o˜je™ts ˜e lo™—lly dieomorphi™F31 „he rst di™ulty with this ™riterion of dieomorphi™ equiv—len™e —s s—meness w—s pointed out ˜y qero™h @priedm—n IWVQD pSWAF „he ™riterion deems —s the s—me —ll timelike 31wore pre™iselyD in the IWVQ version of the denitionD wh—t priedm—n ™—lls –dEequiv—len™e9 is thisX sf — theory for every p h—s PM mthoedreels—rheMn;e¨ig1h;˜:o: :u;r¨honoid—snAd hM; —nd B©1o;f: :: p ; ©niD —nd — then ©i —nd ¨i —re dEequiv—lent ofD dieomorphism h X A 3 B su™h th—t ©i a h £ ¨iF 848 J D Norton nonEv—nishing ve™tor eldsD so th—t however su™h — eld —rises in — theoryD it will ˜e one of its —˜solute o˜je™tsF „husD in st—nd—rd –dust9 ™osmologiesD the velo™ity eld Ua of the dust ˜e™omes —n —˜solute o˜je™tF „o —void the pro˜lemD priedm—n suggests — r—ther ™ontrived es™—peX formul—te the theory of dust with the dust ux Ua where  is the m—ss densityD inste—d of  —nd Ua sep—r—telyF @priedm—n is relying her on the possi˜ility th—t  v—nishes somewhereF e ˜etter ™hoi™e would h—ve ˜een the stressEenergy tensor for pressureless dust U aU bFA wore seriouslyD modifying slightly —n ex—mple of „orretti @IWVRD pPVSAD we ™ould im—gine the following hy˜rid ™l—ssi™—l ™osmologyF „he sp—™etime stru™ture is given exactly ˜y —ny of the ‚o˜ertsonE‡—lker sp—™etime metri™sF „he metri™s —re posited a priori —nd not governed ˜y the presumed inhomogeneous m—tter distri˜ution through gr—vit—tion—l eld equ—tionsD „herefore the ™urv—ture of the metri™ is un—ltered in the vi™inity of m—ssive ˜odiesF sn this ™—seD we would judge the metri™—l sp—™etime stru™ture to —™t on the m—tter distri˜ution without the m—tter distri˜ution —™ting ˜—™k on itF roweverD sin™e models of the theory would —llow metri™s of dierent ™urv—tureD we ™—nnot use existing denitions to identify the sp—™etime metri™ —s —n —˜solute o˜je™tF „orretti9s ™ounterex—mple shows us th—t the ˜—si™ notion of –s—meness9 does not fully ™—pture the notion of things th—t —™t ˜ut —re not —™ted uponF „he se™ond —re— of di™ulty —sso™i—ted with the gener—l theory of —˜solute —nd dyE n—mi™—l o˜je™ts is — presumption of enderson —nd iinstein @—ssuming th—t he is ™orre™tly interpreted ˜y the theoryAF „hey presume th—t there is some ™ompulsion to elimin—te —˜E solute o˜je™tsF yf ™ourse they —re right in the sense th—t our ˜est theory of sp—™e —nd time h—ppens not to employ —˜solute o˜je™tsF „hus sever—l of enderson9s —rguments of the prin™iple of gener—l inv—ri—n™e ™—n form — premise of —rguments th—t le—d to empiri™—lly ™onrmed results @enderson IWTUD se™tion IHFQD IWUID pITWAF rowever —˜solutes —re supE posed to ˜e defe™tive in — deeper senseF st is not just th—t we h—ppen not to see —˜solutes in n—tureY x—ture is somehow supposed to —˜hor things th—t —™t ˜ut —re not —™ted uponF „he di™ulty is to ™l—rify —nd justify this deeper senseF enderson @IWTUD pQQWD IWUID pITWA sees in n—ture — –gener—lized l—w ‘prin™iple in IWU“ of —™tion —nd re—™tion9F fut the prin™iple is so v—gue th—t it is un™le—r wh—t the prin™iple re—lly s—ys —nd where it ™—n ˜e —ppliedF hoes €l—n™k9s ™onst—nt h or the gr—vit—tion ™onst—nt G –—™t9 on m—tter without suering –re—™tion9c ‡ith this v—gueness how ™—n we tell if the l—w is true or even whether we should hope for it to ˜e truec ss itD perh—psD — du˜ious guilt ˜y —sso™i—tion with eristotle9s …nmoved woverc iinstein ™omes ™loser to —n expl—n—tion with his —n—logy @se™tion QFW —˜oveA to pots of w—terD one ˜oilingD one notF „here h—s to ˜e — su™ient re—son for the dieren™eF en—logouslyD the di™ulty with —˜solute o˜je™ts is th—t there is no su™ient re—son for them to ˜e one w—y r—ther th—n —notherF xow we might —llow th—t su™h — prin™iple of su™ient re—son —pplies to tempor—rily su™™essive st—tes of systemsD —lthough qu—ntum theory ™—lls even th—t into dou˜tF fut why should we require this sort of prin™iple to hold for —spe™ts of the universe —s — wholec sn —nswerD we might t—ke forn exp—nsion of iinstein9s @IWITD se™tion PA denun™i—tion of —n —˜soluteD inerti—l sp—™e —s —n ad hoc ™—useF forn @IWPRD pQIIA expl—ins sfD howeverD we —sk wh—t —˜solute sp—™e is —nd in wh—t other w—y it expresses itselfD no one ™—n furnish —n —nswer other th—n th—t —˜solute sp—™e is the ™—use of ™entrifug—l for™es ˜ut h—s no other propertiesF „his ™onsider—tion shows th—t sp—™e —s the ™—use of physi™—l o™™urren™es must ˜e elimin—ted from the world pi™tureF st is h—rd to symp—thize with forn9s ™ompl—intF „he —˜solute winkowski metri™ of — spe™i—l rel—tivisti™ world h—s —n extremely ri™h ™olle™tion of properties —ll of whi™h ™—n ˜e ™onrmed ˜y possi˜le experien™esF st is di™ult not to see the very o˜je™tion of forn —nd iinstein —s ad hocF „hey seek to use v—gue —nd spe™ul—tive met—physi™s to ™onvert General covariance and general relativity 849 something th—t h—ppens to ˜e f—lse into something th—t h—s to ˜e f—lseF „hese seem to ˜e ƒ™hli™k9s IWPHD pRHA sentiments when he o˜serves F F F we ™—n F F F ™onsider the expression –—˜solute sp—™e9 to ˜e — p—r—phr—se of the mere f—™t th—t these ‘™entrifug—l“ for™es existF „hey would then simply ˜e immedi—te d—t—Y —nd the question why they —rise in ™ert—in ˜odies —nd —re w—nting in others would ˜e on the s—me level with the question why — ˜ody is present —t one pl—™e in the world —nd not —t —notherF F F F s ˜elieve xewton9s dyn—mi™s is quite in order —s reg—rds the prin™iple of ™—us—lityF ƒpe™i—l rel—tivity h—s suered too long form the ™r—nk myth th—t it not just h—ppens to ˜e true ˜ut it h—s to ˜e true —nd th—t proper medit—tion on ™lo™ks —nd light sign—lling reve—ls itF vet us not ™re—te — simil—r myth for gener—l rel—tivityF 9. Boundaries and puzzles 9.1. Is general covariance too general? Or not general enough? ‡hile most h—ve ˜een s—tised with gener—l rel—tivity —s — gener—l ™ov—ri—nt theoryD po™k @IWSUD IWSWD pp xv{xviD se™tion WQA h—s proposed th—t the four ™oordin—te degrees of freedom of the gener—lly ™ov—ri—nt theory ˜e redu™ed ˜y —ppli™—tion of — ™oordin—te ™onditionF po™k9s –h—rmoni™ ™oordin—tes9 —re pi™ked out ˜y the ™ondition  x @p g gA a H po™k —pplies this ™ondition to the ™—se of sp—™etimes whi™h —re winkowski—n —t sp—ti—l inE nity —nd nds th—t the resulting equ—tions —re the n—tur—l gener—liz—tion of the st—nd—rd q—lile—n ™oordin—tes of sp—ti—l rel—tivity —nd —re xed up to — vorentzi—n tr—nsform—tionF po™k sees the physi™—l import—n™e of h—rmoni™ ™oordin—tes in su™h pro˜lems —s the justiE fying of goperni™—n of the €tolem—i™ ™osmologyF sn h—rmoni™ ™oordin—tesD the e—rth or˜its the sun —nd not vi™e vers—F po™k9s propos—l proved ™ontroversi—lF griti™ism of po™k9s propos—l w—s —ired —t — ™onE feren™e in ferne in tuly IWSS for the ju˜ilee of rel—tivity theory @po™k IWSTAF snfeld —rgued th—t — restri™tion to h—rmoni™ ™oordin—tes is —™™ept—˜le —s — ™onvenien™eF –fut to —dd it —lw—ys @or —lmost —lw—ysA to the gr—vit—tion—l equ—tion —nd to ™l—im th—t its virtue lies in the f—™t th—t the system is only vorentz inv—ri—ntD me—ns to ™ontr—di™t the prin™iple ide— of rel—tivity theoryF9 „r—utm—n @IWTRD pIPQA —nd uop™zynski —nd „r—utm—n @IWWPD pIPRA h—ve —lso o˜je™ted th—t po™k9s propos—l —mount to the postul—tion of new sp—™etime stru™tures for whi™h no physi™—l interpret—tion ™—n ˜e givenF 850 J D Norton sn so f—r —s po™k intended to redu™e perm—nently the ™ov—ri—n™e of gener—l rel—tivity —nd introdu™e further stru™tureD then these ™riti™—l —tt—™ks —re w—rr—ntedF „he h—rmoni™ ™oordin—te ™ondition is un—™™ept—˜le —s — new physi™—l prin™ipleF fut po™k @IWSWA seems to hold — milder positionF re emph—sized @ppQSH{IA th—t the introdu™tion of h—rmoni™ ™oE ordin—tes is intended in — spirit no dierent from th—t whi™h introdu™es preferred q—lile—n ™oordin—tes into — gener—lly ™ov—ri—nt formul—tion of spe™i—l rel—tivityF „hus –the existen™e of — preferred set of ™oordin—tes F F F is ˜y no me—ns trivi—lD ˜ut ree™ts intrinsi™ properE ties of sp—™eEtime9F sn the ™—se of — sp—™etime winkowski—n —t sp—ti—l innityD h—rmoni™ ™oordin—tes simply reve—l — stru™ture —lre—dy —ssumed —s p—rt of the ˜ound—ry ™onditionF „heir use does not —mount to —n unw—rr—nted postul—tion of new stru™ture|unless one deems the ˜ound—ry ™onditions themselves unw—rr—ntedF por further dis™ussion see qorelik @forth™omingA „he issue surrounding po™k9s propos—l w—s whether — restri™tion of the ™ov—ri—n™e of gener—l rel—tivity ™ould ˜e justiedF erzelies @IWTIA h—s proposed — modi™—tion of gener—l rel—tivity whi™h —mounts to — kind of exp—nsion of its ™ov—ri—n™eF re urges th—t iinstein9s theory h—s still not s—tised the requirements of the gener—lized prin™iple of rel—tivity —nd th—t the tr—nsform—tions it —llows should ˜e extended in the following senseF sf we st—rt with — ™oordin—te system Xi thenD under ™oordin—te tr—nsform—tionD the ™oordin—te dierenti—l dXi tr—nsform into new ™oordin—te dierenti—ls dxiF st is ™ustom—rily —ssumed th—t the ™oordin—te dierenti—ls dxi —re ex—™tD so th—t they ™—n ˜e integr—ted into the new ™oordin—te systems xiF erzelies proposed th—t this restri™tion ˜e droppedF „his would ™ert—inly g£eikndeXr—klizneeetdhenogrloonugperof˜etrr—ensstfroi™rtmed—t˜ioyntshseinr™eequtihreemfeunnt™toifonexs —£™tikneosfsFth„eheeqmu—otdiion™s—tdioxni a of extremely f—r re—™hingD howeverD in so f—r —s it le—ds to the loss of m—ny f—mili—r theoremsF por ex—mpleD it will now ˜e possi˜le to tr—nsform the line elements of nonE—t metri™s to the form ds2 a @dx1A2 C @dx2A2 C @dx3A2 C @dx4A2 over — neigh˜ourhood @not just — pointAD where this w—s formerly only possi˜le if the metri™ w—s —tF 9.2 The Einstein puzzle „here is — presumption in mu™h modern interpret—tion of iinstein9s pronoun™ements on the found—tions of the gener—l theory of rel—tivityF st is th—t mu™h of wh—t he s—ys ™—nnot ˜e t—ken —t f—™e v—lueF @‡hy does iinstein m—ke su™h — fuss —˜out introdu™ing —r˜itr—ry sp—™etime ™oordin—tesc ‡e h—ve —lw—ys ˜een —˜le to l—˜el sp—™etime events —ny w—y we ple—se3A „hus we —re either to tr—nsl—te wh—t he re—lly me—nt into some more pre™ise ™ontextD —s does endersonD or to dismiss it —s ™onfusedF „he propos—l of xorton @IWVWD IWWPA is th—t our modern di™ulty in re—ding iinstein liter—lly —™tu—lly stems from — ™h—nge of ™ontextF @por rel—ted ™on™erns see xorton @IWWQAFA „he relev—nt ™h—nge lies in the m—them—ti™—l tools used to represent physi™—lly possi˜le sp—™etimesF sn re™ent work in sp—™etime theoriesD we ˜egin with — very rened m—them—ti™—l entityD —n —˜str—™t dierenti—˜le m—nifoldD whi™h usu—lly ™ont—ins the minimum stru™ture to ˜e —ttri˜uted to the physi™—l sp—™etimesF ‡e then judi™iously —dd further geometri™ o˜je™ts only —s the physi™—l ™ontent of the theory w—rr—ntsF woreoverD we h—ve two levels of represent—tionF ‡e rst represent the physi™—lly possi˜le sp—™etimes ˜y the geometri™ models of form @TA —nd then these geometri™ models —re represented ˜y the ™oordin—te ˜—sed stru™tures @UAF qener—l ™ov—ri—n™e is usu—lly understood —s p—ssive gener—l ™ov—ri—n™e —nd therefore —rises —s — m—them—ti™—l denitionD —s we h—ve seenF General covariance and general relativity 851 sn the IWIHsD m—them—ti™—l pr—™ti™es in physi™s were quite dierentF „he two levels of represent—tion were not usedF ‡hen one represented — gener—l sp—™e or sp—™etimeD one used num˜er m—nifolds|Rn or Cn for ex—mpleF „hus winkowski9s –world9 w—s not — dierenti—˜le m—nifold th—t w—s merely topologi™—lly R4F st w—s liter—lly R4D th—t is it w—s the set of —ll qu—druples of re—l num˜ersF xow —nyone seeking to ˜uild — sp—™etime theory with these m—them—ti™—l tools of the IWIHs f—™es — very dierent pro˜lems from the ones we see nowF wodern dierenti—˜le m—nifolds h—ve too little stru™ture —nd we must —dd to themF xum˜er m—nifolds h—ve f—r too mu™h stru™tureF „hey —re fully inhomogeneous —nd —nisotropi™F „he origin hH; H; H; Hi is quite dierent from every other pointD for ex—mplesF end —ll this stru™ture h—d ™—nonE i™—l physi™—l interpret—tionF sf one took the x4 —xis —s the time —xisD then x4 ™oordin—te dieren™es were physi™—lly interpreted —s dieren™es of ™lo™k re—dingsF „imelike str—ights would ˜e the inerti—l tr—je™tories of for™e free p—rti™lesF „he pro˜lem w—s not how to —dd stru™ture to the m—nifoldsD ˜ut how to deny physi™—l signi™—n™e to existing p—rts of the num˜er m—nifoldsF row do we rule out the ide— th—t hH; H; H; Hi represents the preferred ™enter of the universe —nd th—t the x4 ™oordin—te —xis — preferred st—te of restc pelix ulein9s Erlangen progr—m provided pre™isely the tool th—t w—s neededF yne —ssigns — ™h—r—™teristi™ group to the theoryF sn winkowski9s ™—seD it is the vorentz groupF ynly those —spe™ts of the num˜er m—nifold th—t rem—in inv—ri—nt under this group —re —llowed physi™—l signi™—n™eF „hus there is no physi™—l signi™—n™e in the preferred origin hH; H; H; Hi of the num˜er m—nifold sin™e it is not inv—ri—nt under the tr—nsform—tionF fut the ™olle™tion of timelike str—ights of the m—nifold —re inv—ri—ntY they represent the physi™—lly re—l ™olle™tion of —ll inerti—l st—tes of motionF es one in™re—ses the size of the groupD one strips more —nd more physi™—l signi™—n™e out of the num˜er m—nifoldF ‡e ™—n put this in —nother w—yF e sp—™etime theory ™oordin—tes — physi™—lly possi˜le sp—™etime with the num˜er m—nifoldF „he ™h—r—™teristi™ group of the theory tells us th—t m—ny dierent su™h ™oordin—tions —re —llowed —nd equ—lly goodF ‡h—t is physi™—lly signiE ™—nt is re—d o —s th—t p—rt of e—™h ™oordin—tion ™ommon to —ll of themF „his ™oordin—tion of physi™—l events with qu—druples of num˜ers in is wh—t w—s me—nt ˜y –™oordin—te system9 —nd the equiv—len™e of two su™h systems w—s f—r from — m—them—ti™—l trivi—lityF st w—s the essen™e of the physi™—l ™ontent of the theoryF st is in this tr—dition th—t iinstein worked in the IWIHsF ris proje™t w—s to exp—nd the group of his theory —s f—r —s possi˜leF fut he h—d to pro™eed ™—refully sin™e su™h exp—nsions ™—me with — stripping of physi™—l signi™—n™e from the num˜er m—nifoldF „hus iinstein @IWITD se™tion QA needed to pro™eed very ™—utiously in expl—ining how the gener—l ™ov—ri—n™e of his new theory h—d stripped the ™oordin—tes of their dire™t rel—tionship to the results of me—surements ˜y rod —nd ™lo™kF „he proje™t is ™le—rly —lso — proje™t of rel—tiviz—tion of motionF „he imposition of the vorentz groups stripped the x4 —xis of the physi™—l signi™—n™e —s — st—te of restD implementing — prin™iple of rel—tivity for inerti—l motionF „he tr—nsition to the gener—l groups stripped the set of timelike str—ights of physi™—l signi™—n™e —s inerti—l motionD extending the prin™iple to —™™eler—ted motionF sf this w—s —ll th—t iinstein h—d doneD then his whole proje™t would h—ve rem—ined within the Erlangen progr—m tr—dition —nd there would ˜e no de˜—tes tod—y over whether iinstein su™™eeded in extending the prin™iple of rel—tivityF futD in the tr—nsition from the vorentz tot the gener—l groupD iinstein —dded —n element th—t ™—rried him out of the tr—dition of the irl—ngen progr—mF re —sso™i—ted — ‚iem—nni—n qu—dr—ti™ dierenti—l form with the sp—™etimeF @„hus g—rt—n @se™tion TFP —˜oveA ™—ptures pre™isely the ™ru™i—l pointFA ‡hile iinstein ™ould ™orre™tly s—y th—t he h—d gener—lized the prin™iple of rel—tivity insof—r —s he h—d stripped physi™—l signi™—n™e from the timelike str—ights of the num˜er m—nifoldD wh—t rem—ined to ˜e seen w—s whether he h—d reintrodu™ed essenti—lly this s—me stru™ture 852 J D Norton ˜y me—ns of the qu—dr—ti™ dierenti—l formF sn ee™t this question h—s ˜e™ome the fo™us of the de˜—te over the gener—lized prin™iple of rel—tivityF pin—llyD it is helpful to ˜e—r in mind th—t wh—t iinstein me—nt ˜y –™oordin—te system9 is not the s—me —s the modern –™oordin—te ™h—rts9 of — dierenti—˜le m—nifoldF „he l—tter rel—te stru™tures of @TA —nd @UA —nd the equiv—len™e of e—™h represent—tion is — m—tter of m—them—ti™—l denitionF iinstein9s ™oordin—te systems —re —™tu—lly —kin the represent—E tion rel—tion ˜etween physi™—lly possi˜le sp—™etimes —nd the models of form @TAF „h—t two models represent the one physi™—lly possi˜le sp—™etime is — physi™—l —ssumption th—t —mounts to —ssuming th—t their m—them—ti™—l dieren™es h—ve no physi™—l signi™—n™eF gorrespondinglyD within the ™ontext of iinstein9s formul—tion of sp—™etime theoriesD th—t two ™oordin—te system represent — physi™—lly possi˜le sp—™etime is on™e —g—in — physi™—l —ssumption —nd for the s—me re—sonF „h—t isD iinstein9s ™ov—ri—n™e prin™iples —re most —kin to modern —™tive ™ov—ri—n™e prin™iplesF sn sumD there is no re—l puzzle in mu™h th—t of wh—t iinstein s—idF ‚—ther it now only seems puzzling sin™e he is solving pro˜lems we longer h—ve ˜e™—use of the gre—ter sophisti™—tion of our m—them—ti™—l toolsF sndeedD in good me—sure we owe to iinstein9s inspir—tion the development —nd widespre—d use of m—them—ti™—l tools th—t —utom—ti™—lly solve pro˜lems over whi™h he l—˜oured so h—rdF 10. Conclusion „he de˜—te over the signi™—n™e of gener—l ™ov—ri—n™e in iinstein9s gener—l theory of rel—E tivity is f—r from settledF „here —re essenti—lly three viewpoints now ™urrentF pirst is the viewpoint routinely —ttri˜uted to iinsteinF st holds th—t the —™hievement of gener—l ™ov—ri—n™e —utom—ti™—lly implements — gener—lized prin™iple of rel—tivityF sn view of the ™onsider—˜le weight of ™riti™ismD this view is no longer ten—˜leF ‚el—tivity prin™iples —re symmetry prin™iplesD the requirement of gener—l ™ov—ri—n™e is not — symmetry prin™ipleF „he requirement of gener—l ™ov—ri—n™eD t—ken ˜y itselfD is even devoid of physi™—l ™ontentF st ™—n ˜e s—lv—ged —s — physi™—l prin™iple ˜y supplementing it with further requirementsF „he most popul—r —re — restri™tion to simple l—w forms —nd — restri™tion on the —ddition—l stru™tures th—t m—y ˜e used to —™hieve gener—l ™ov—ri—n™eF rowever neither supplement—ry ™ondition h—s ˜een developed system—ti™—lly ˜eyond the st—ge of f—irly ™—su—l rem—rksF „he se™ond viewpoint h—s ˜een developed ˜y enderson —nd is ˜—sed on his distin™tion ˜etween —˜solute —nd dyn—mi™—l o˜je™tsF ris –prin™iple of gener—l inv—ri—n™e9 ent—ils th—t — sp—™etime theory ™—n h—ve no nonEtrivi—l —˜solute o˜je™tsF enderson —rgues th—t the prin™iple is — rel—tivity prin™ipleD sin™e it is — symmetry prin™ipleD —nd th—t it is wh—t iinstein re—lly intended with his prin™iple of gener—l ™ov—ri—n™eF sn this —ppro—™hD gener—l rel—tivity is —˜le to extend the symmetry group of spe™i—l rel—tivity form the vorentz group to the gener—l groupF „his extension depends on the metri™ ˜eing — dyn—mi™—l o˜je™tD whi™h is no longer required to ˜e preserved ˜y the symmetry tr—nsform—tions of the theory9s rel—tivity prin™ipleF „he third viewpoint holds th—t the dyn—mi™—l ™h—r—™ter of the metri™ is irrelev—nt in this ™ontext —n th—t the metri™ must ˜e preserved under the theory9s symmetry groupD if th—t group is to ˜e —sso™i—ted with — rel—tivity prin™ipleF ƒin™e the metri™s of gener—l rel—tivisti™ sp—™etimes h—veD in gener—lD no nonEtrivi—l symmetriesD there is no nonEtrivi—l rel—tivity prin™iple in gener—l rel—tivityF ‡h—tever m—y h—ve ˜een its role —nd pl—™e histori™—llyD gener—l ™ov—ri—n™e is now —utom—ti™—lly —™hieved ˜y routine methods in the formul—tion of —ll seriously ™onsidered sp—™etime theoriesF „he found—tions of gener—l rel—tivity do not lie in one or other prin™iple —dv—n™ed ˜y iinsteinF ‚—therD they lie in the simple —ssertion General covariance and general relativity 853 th—t sp—™etime is semiE‚iem—nni—nD with gr—vity represented ˜y its ™urv—ture —nd its metri™ tensor governed ˜y the iinstein eld equ—tionsF Acknowledgements „he rese—r™hing —nd writing of this review w—s supported ˜y the x—tion—l ƒ™ien™e pound—E tion under qr—nt xoF ƒfiEWIPIQPTF s th—nk the found—tion for its support —nd —lso te—n iisenst—edtD hon row—rdD el t—nisD —nd g—rlo ‚ovelli for helpful dis™ussionsF s —m —lso very gr—teful to hon row—rd for —ssist—n™e with qerm—n tr—nsl—tionsF References edler ‚F f—zin w —nd ƒ™hier w IWUU Introduction to General Relativity Pnd ed @„okyoX w™qr—wErill uog—kush—A eguirre w —nd ur—use t IWWI International Journal of Theoretical Physics 30 RWS{SHW enderson t v IWTP Recent Developments in General Relativity ed –iditori—l gommittee9 @xew ‰orkX w—™will—nA | IWTR Gravitation and Relativity eds r ‰ ghiu —nd ‡ p rom—nn @xew ‰orkX fenj—minA ™h W | IWTT Perspectives in Geometry and Relativity ed f rom—nn @floomington sxX sndi—n— …niversity €ressA pp IT{PU | IWTU Principles of Relativity Physics @xew ‰orkX e™—demi™A | IWUI General Relativity and Gravitation 2 ITI{UP | —nd q—utre—u ‚ IWTW Phys. 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Phys. 4 IUQ{PHR row—rd h IWWP Studies in the Hislory of General Relativity ed t iisenst—edt —nd e uox @fostonX firkh—userA pp ISR{PRQ row—rd h —nd xorton t h @forth™omingA The Attraction of Gravitation: New Studies in History of General Relativity ed t i—rm—nD w t—nssen —nd t h xorton @fostonX firkh—userA General covariance and general relativity 857 rughston v € —nd „od u € IWWH An Introduction to General Relativity @g—m˜ridgeX g—m˜ridge …niversity €ressA tones ‚ IWVI After Einstein ed € f—rker —nd g q ƒhug—rt @wemphisD „xX wemphis ƒt—te …niversity €ressA tord—n € IWSS Schwerkraft und Weltall @fr—uns™hweigX †iewegA uenyon s ‚ IWWH General Relativity @yxfordX yxford …niversity €ressA uenz˜erg € IWVW Einstein and the History of General Relativity ed h row—rd —nd t ƒt—™hel @fostonX firkh—userA pp QPS{ST ulein p IVUP Gesammelte Mathematische Abhnadlungen. vol ID ed ‚ pri™ke —nd e ysE trowski @ferlinX ƒpringerA pp RTH{RWU uop™zynski ‡ und „r—utm—n e IWWP Spacetime and Gravitation @ghi™hesterX ‡ileyA uop e IWPQ Grundzuge der Einsteinschen Relativitatstheorie @veipzigX rirzelA uottler p IWPP Encyklopadie der mathematischen wissenschaften mit Einschluss an ihrer Anwendung vol †sD p—rtPD PP— estronomie ed u ƒ™hwertzs™hildD ƒ yppenheim —nd ‡ v hyke @veipzigX „eu˜nerA pp ISW{PQU ur—tzer e IWST Relativitatstheorie @wunsterX es™hendorfs™heA urets™hm—nn i IWIS Ann. Phys. 48 WHU{WVP | IWIU Ann. Phys. 53 SUS{TIR uu™h—r u † IWVV Highlights in Gravitation and Cosmology @g—m˜ridgeX g—m˜ridge …niE versity €ressA v—nd—u v h —nd vifshitz i w IWSI The Classical Theory of Fields tr—nsl w r—mermesh @yxfordX€erg—monA v—ue w IWII Das Relativitatsprinzip @fr—uns™hweigX †iewegA | IWPI Die Relativitatstheorie. Vol 2: Die Allgemeine Relativitatstheorie und Einsteins Lehre von der Schwerkraft @fr—uns™hweigX †iewegA v—wden h p IWTP An Introduction to Tensor Calculus and Relativity @vondonX wethuenA ven—rd € IWPI Uber Relativitatsprinzip, Ather, Gravitation Qrd edn @veipzigX rirzelA veviEgivit— „ IWPT The Absolute Di erential Calculus tr—nsl w vong @xew ‰orkX hoverA vevinson r g —nd eisler i f IWPW The Law of Gravitation in Relativity @ghi™—goX …niE versity of ghi™—go €ressA Pnd edn vorentz r e et al IWPQ Principle of Relativity @vondonX wethuenAY @IWSP xew ‰orkX hoverA w—™h i IVWQ Science of Mechanics tr—nsl IWTH „ t w™gorm—™k @v— ƒ—lleD svX ypen gourtA w—l—ment h IWVT From Quark to Quasars: Philosophical Problems of Modern Physics ed ‚ q golodny @€ins˜urghD €eX …niversity of €itts˜urgh €ressA pp IVI{PHI w—nin t v IWVV General Relativity: A Guide to its Consequences for Gravity and Cosmology @ghi™hesterX illis rorwoodA w—shoon f IWVT Found. Phys. 16 TIW{QSF w—udlin „ IWVV Proc. 1988 Biennial Meeting of the Philosophy of Science Association vol. 2D ed e pine —nd t veplin @i—st v—nsingF wsX €hilosophy of ƒ™ien™e esso™i—tionA pp VP{WI | IWWH Stud. Hist. Philos. Sci. 21 SQI{TI w—vrides ƒ IWIQ L'Univers Relativiste @€—risX w—ssonA w™†ittie q g IWTS General Relativity and Cosmology @…r˜—n— svX …niversity of sllinois €ressA wehr— t IWUR Einstein. Hilbert and The Theory of Gravitation @hordre™htX ‚eidelA 858 J D Norton winkowski r IWHV Koniglichen Gesellschaft der Wissenschaften zu Gottingen, MathematischePhysikalische Klasse. NachrichtenD SQ{III | IWHW Physikalische Zeitschrift 10 IHR{III @tr—nsl vorentz IWPQD pp US{WIA wisner g ‡D „horne u ƒ —nd ‡heeler t e IWIQ Gravitation @ƒ—n pr—n™is™oX preem—nA woller g IWSP The Theory of Relativity @yxfordX gl—rendonA wundy f IWWP Proc. 1992 Biennial Meeting of the Philosophy of Science Association volF I ed h rullD w por˜es —nd u ykruhluk @i—st v—nsingD wsX €hilosophy of ƒ™ien™e esso™i—tionA pp SIS{SPU xewton s ITVU Mathematical Principle of Natural Philosophy tr—nsl IWQR p g—jori @ferkeE leyD geX …niversity of vos engeles €ressA xorth t h IWTS The Measure of the Universe: A History of Modern Cosmology @yxE fordXgl—rendonA xorton t h IWVR Historical Studies in the Physical Sciences 14 PSQ{QITX reprinted in Einstein and the History of General Relativity: Einstein StudiesD †ol I eds h row—rd —nd t ƒt—™hel @fostonX firkh—userD IWVWA pp IHI{ISW | IWVS Studies in History and Philosophy of Science 16 PHQ{PRTX reprinted in Einstein and the History of General Relativity: Einstein StudiesD †ol I eds hrow—rd —nd t ƒt—™hel @fostonX firkh—userD IWVWA pp Q{RUF | IWVU Measurement, Realism and Objectivity ed s porge @hordre™htX ‚eidelA pp ISQ{IVV | IWVV Proceedings of the 1988 Biennial Meeting of the Philosophy of Science Association †olF P eds e pine —nd t veplin @i—st v—nsingD wsX €hilosophy of ƒ™ien™e esso™FD IWVWA pp ST{TR | IWVW Found Phys 19 IPIS{IPTQ | IWWP Studies in the History of General Relativity: Einstein StudiesD †olF Q t iisenst—edt —nd e uox eds @fostonXfirkh—userA pp PVI{QIS | IWWP— Introduction to the Philosophy of Science w r ƒ—lmon et —l @inglewood glisD xtX €renti™eEr—llA pp IUW{PQI | IWWQ Semantical aspects of Spacetime Theories ed … w—jer —nd r s ƒ™hmidt @w—nnheimX fsA yh—ni—n r g IWUT Gravitation and Spacetime @xew ‰orkX xortonA €—ge v IWPH The Principle of General Relativity and Einstein's Theory of Gravitation @xew r—venF g„X gonne™ti™ut e™—demy of erts —nd ƒ™ien™esA €—inleve € IWPI C. 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Paris 173 VUQ{VVU €—p—petrou e IWUR Lectures on General Relativity @hordre™htX ‚eidelA €—thri— ‚ u IWUR The Theory of Relativity Pnd edn @yxfordF€erg—monA €—uli ‡ IWPI Encyklopadie der mathematischen Wissenschaften, mit Einschluss an ihrer Anwendung. vol 5 PhysikD €—rt P ed e ƒommerfeld @veipzigX „eu˜nerA pp SQW{UUSY IWSV Theory of Relativity tr—nsl q pield @vondonX €erg—monA €ost i t IWTU Delaware Seminar in the Foundations of Physics ed w funge @xew ‰orkX ƒpringerA pp IHQ{IPQ €r—s—nn— e ‚ IWUI Lectures on General Relativity and Cosmology, Matscience Report No 71 @w—dr—sX snstitute of w—them—ti™—l ƒ™ien™esA ‚—ine h t —nd reller w IWVI The Science of Space-Time @„u™sonD eX €—™h—rt €u˜lishing rouseA ‚—ini™h q ‰ IWSH Mathematics of Relativity @xew ‰orkX ‡ileyA ‚—y g IWVU The Evolution Of Relativity @fristol —nd €hil—delphi—X ed—m rilgerA General covariance and general relativity 859 ‚ei™hen˜—™h r IWPR Axiomatization of the Theory of Relativity tr—nsl IWTW w ‚ei™hen˜—™h @ferkeleyD geX …niversity of g—liforni— €ressD IWTWA ‚einh—rdt w  Naturf 28a SPW{QU ‚i™™i q —nd veviEgivit— „ IWHI Math. Ann. 54 IPS{PHIX reprinted in veviEgivit— IWSR Opere MatematicheD volF I @fologn—A pp RUW{SSW ‚indler ‡ IWTW Essential Relativity: Special, General, and Cosmological @xew ‰orkX †—n xostr—ndE‚einholdA ‚o˜ertson r € —nd xoon—n „ ‡ IWTV Relativity and Cosmology @€hil—delphi—D €eX ƒ—unE dersA ‚oll € qD urotkov ‚ —nd hi™ke ‚ r IWTR Ann. Phys. 26 RRP{SIU ‚osser ‡ q † IWTR An Introduction to the Theory of Relativity @vondonX futterworthsA ‚ovelli g IWWI Class. Quantum Grav. 8 PWU{QIT ‚yn—siewi™z ‚ @forth™oming @—AA 9‚ingsD roles —nd ƒu˜st—ntiv—lismX yn the €rogr—m of vei˜niz elge˜r—9 Phil. Sci. ‚yn—siewi™z ‚ @forth™oming @˜AA „he vessons of the role ergument ‚y™km—n „ e IWWP Stud. History Phil. Sci. 23 RUI{RWU ƒ—™hs ‚ u —nd ‡u r IWUU General Relativity for Mathematicians @xew ‰orkX ƒpringerA ƒ™hei˜e i IWVI Scienti c Philosophy Today eds t eg—ssi —nd ‚ ƒ gohen @hordre™htX ‚eidelA pp QII{QIY reprinted IWVQ Space, Time, and Mechanics ed h w—y —nd q ƒussm—n @hordre™htX ‚eidelA pp IPS{RU | IWWI Causality, Method, and Modality ed q q fritt—n tr @uluwerEe™—demi™A pp PQ{RH ƒ™hild e IWTU Relativity Theory and Astrophysics Vol. 1 Relativity and Cosmology ed t ihlers @€roviden™eD ‚sX emeri™—n w—them—ti™—l ƒo™ietyA pp I{tIT ƒ™hli™k w IWPH Space and Time in Conremporary Physics tr—nsl r v frose @xew ‰orkX yxford …niversity €ressA ƒ™hrodinger i IWSH Space-Time Structure @g—m˜ridgeX g—m˜ridge …niversity €ressA ƒ™hutz f p IWVS A First Course in General Relativity @g—m˜ridgeX g—m˜ridge …niversity €ressA ƒesm—t e IWQU Systemes de Reference et Mouvements @€hysique ‚el—tivisteA @€—risX rerE m—nnA ƒexl ‚ … —nd …r˜—ntke r u IWVQ Gravitation und Kosmologie: Eine Einfuhrung in die Allgemeine Relativitatstheorie @w—nnheimX fi˜liogr—phis™hes snstitutA ƒil˜erstein v IWPP The Theory of General Relativity and Gravitation @…niversity of „oronto €ressA ƒkinner ‚ IWTW Relativity @‡—lth—mD weX fl—isdell €u˜lishing gyA ƒommerfeld e IWIH Ann. Phys. QP URW{UUTY Ann. Phys. 33 TRW{VW ƒt—™hel t @forth™omingA Philosophical Problems of the Internal and External Worlds: Essays on the Philosophy of Adolf Grunbaum eds t i—rm—nD e t—nisD q w—ssey —nd x ‚es™her @€itts˜urghD €eX …niversity of €ins˜urgG…niversity of uonst—nzA | IWVH 9iinstein9s ƒe—r™h for qener—l gov—ri—n™e9 9th Int. Conf. on General Relativity and Gravitation. JenaY reprinted in Einstein and the History of General Relativity; Einstein StudiesD volF I eds h row—rd —nd t ƒt—™hel @fostonX firkh—userD IWVWA pp TQ{IHH | IWVH— General Relativity and Gravitation: A Hundred Years After the Birth of Einstein ed e reld @xew ‰orkX €lenumAX reprinted in Einstein and the History of General Relativity: Einstein StudiesF †olF I h row—rd —nd t ƒt—™hel eds @fostonX firkh—userD IWVWA pp RV{TP 860 J D Norton | IWVT €ro™F 4th Marcel Grossmann Meeting on General Relativity ed ‚ ‚uni @emsE terd—mX ilsevierA pp IVSU{IVTP | IWWP Studies in the History of General Relativity: Einstein StudiesD †olF Q eds t iisenE st—edt —nd e uox @fostonX firkh—userA pp RHU{RIV ƒteph—ni r IWUU General Relativity: An Introduction for the Theory of the Gravitational Field ed t ƒtew—rtD tr—nsl w €ollo™k —nd t ƒtew—rtF @g—m˜ridgeX g—m˜ridge …niversity €ressA ƒtew—rt t IWWH Advanced General Relativity @g—m˜ridgeX g—m˜ridge …niversity €ressA ƒtr—um—nn x IWVR Generai Relativity and Astrophysics @ferlinX ƒpringerA ƒynge t v IWTH Relativity: The General Theory @emsterd—mX xorthEroll—ndA | IWTR Relativity, Groups and Topology eds g he ‡itt —nd f he ‡itt @xew ‰orkX qordon —nd fre—™hA pp R{WH | IWTT Perspectives in Geometry and Relativity ed f rom—nn @floomingtonD sxX sndi—n— …niversity €ressA pp U{IS ƒzekeres q IWSS Phys. 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Sci. 35 PVH{PWP „r—utm—n e IWTR Lectures on General Relativity: Brandeis Summer Institute in Theoretical Physics Vol I 1964 @inglewood glisD xtX €renti™e r—llD IWTSA pp I{PRV | IWTT Perspectives in Geometry and Relativity: Essays in Honor of Vaclav Hiavaty ed f rom—nn @floomingtonF sxX sndi—n— …niversity €ress pp RIQ{PS | IWUQ The Physicist's Conception of Nature ed t wehr— @hordre™htX ‚eidelA pp IUV{WV „reder rEtD von forzeszkowski rErD v—n der werwe e —nd ‰ourgr—u ‡ IWVH Fundamental Principles of General Relativity Theories: Local and Global Aspects of Gravitation and Cosmology @xew ‰orkX €lenumD IWVHA †e˜len y —nd ‡hitehe—d t r g IWQP The Foundations of Diferential Geometry @g—mE ˜ridgeX g—m˜ridge …niversity €ressA †l—dimirov ‰uD witskevis™h x —nd rorski t IWVU Space, Time and Gravitation tr—nsl e q il˜erm—nD ed p s pedorov @wos™owX wirA ‡—ld ‚ IWVR General Relativity @ghi™—goD svX…niversity of ghi™—go €ressA ‡—sserm—n ‚ r IWWP Tensors and Manifolds with Applications to Mechanics and Relativity @xew ‰orkX yxford …niversity €ressA ‡e˜er t IWTI General Relativity and Gravitational Waves @xew ‰orkX snters™ien™eA ‡ein˜erg ƒ IWUP Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity @xew ‰orkX ‡ileyA ‡eyl r IWPI Space Time Matter tr—nsl IWSP r v frose @xew ‰orkX hoverA General covariance and general relativity 861 ‡hit—ker i „ IWSI A History of Theories of Aether and Electricity. Vol. I The Classical Theories, Vol 2 the Modern Theories 1900-1926 @xew ‰orkX hoverA ‰ilm—z r IWTS Introduction to the Theory of Relativity and Principles of Modern Physics @xew ‰orkX fl—idsdellA —h—r i sWVW Einstein's revolution: A study in Heuristic @v— ƒ—lleD svX ypen gourtA —tzkis r IWSS in Fundamental Formulas of Physics ed h r wenzel @xew ‰orkX hoverD IWTHA †ol PD ™h U