uv of a gravitational field with given Θuv by a generally-covariant system of equations is impossible. We can prove that if a solution for the γuv for given Θuv is already known, and then the general covariance of the equations allows for the existence of further solutions. Assume a domain L within our four-dimensional manifold such that no "material process" shall exist within L, i.e., where the Θuv therefore vanish. By virtue of the given Θuv the γuv are assumed determined everywhere outside of L and, therefore, also inside L (assumption a). Instead of the original coordinates xv we now imagine new coordinates xv' introduced in the following manner. Everywhere outside of L we have xv' = xv, but inside L at least for part of it and at least for one index let there be xv'= xv. Obviously, at least for part of L, this substitution achieves γuv' = γuv. On the other hand we have Θuv' = Θuv everywhere, that is, outside of L, because for this domain xv' = xv, and inside of L because for this domain Θuv = 0 = Θuv. Therefore, if all substitutions would be permitted, the same system of Θuv would have more than one system of the γuv belonging to it, and this is a contradiction to assumption a)" [18]. Of course, Einstein argued that the indeterminism of the theory would drive to the violation of the law of cause-effect a very strong reason to discard the general covariance. Thus, "It remains to find the differential laws that are satisfied by these quantities. The epistemological satisfaction of the theory that has been developed up to here can be seen in the fact that this theory complies with the principle of relativity in the broadest meaning of the word. Seen under a formal aspect, this is based upon the feature that the equation systems are general, i.e., covariant under arbitrary substitutions of the xv. The demand that the differential laws of the guv must also be general-covariant appears therefore appropriate. However, we want to show that we have to restrict this demand if we want to satisfy the law of cause and effect. In fact, we shall prove that the laws that characterize the course of events in a gravitational field can impossibly be covariant in all generality." [9], [15]. 12 The tensor tμν representing the energy of static gravitational field
uv of a gravitational field with given Θuv by a generally - covariant arrangement of equation is impossible. We can prove that if a solution for the γuv for given Θuv is already known, and then the cosmopolitan covariance of the equations allows for the existence of further solution. Assume a domain liter within our four - dimensional manifold such that no " material summons " shall exist within L, i.e., where the Θuv therefore vanish. By merit of the given Θuv the γuv are assumed determined everywhere outside of L and, therefore, also inside L (assumption a). Instead of the original coordinate xv we now imagine new coordinate xv' introduced in the following manner. Everywhere outside of L we have xv' = xv, but inside L at least for part of it and at least for one index permit there be xv'= xv. Obviously, at least for part of L, this substitution achieves γuv' = γuv. On the other hand we own Θuv' = Θuv everywhere, that is, outside of L, because for this domain xv' = xv, and inside of L because for this domain Θuv = 0 = Θuv. Therefore, if all substitutions would be allow, the same system of Θuv would have more than one system of the γuv belonging to it, and this is a contradiction to assumption a) " [ 18 ]. Of course, Einstein argued that the indeterminism of the theory would drive to the misdemeanor of the law of cause - impression a very impregnable reason to discard the general covariance. Thus, " It remain to find the differential laws that are satisfied by these quantities. The epistemic satisfaction of the theory that has been developed up to here can be seen in the fact that this theory complies with the principle of relativity in the broadest meaning of the word. Seen under a formal aspect, this is free-base upon the feature that the equality system are general, i.e., covariant under arbitrary substitutions of the xv. The requirement that the differential laws of the guv must also be general - covariant appear therefore appropriate. However, we want to show that we have to qualify this demand if we want to satisfy the law of cause and consequence. In fact, we shall prove that the torah that characterize the path of events in a gravitational sphere can impossibly be covariant in all generality. " [ 9 ], [ 15 ]. 12 The tensor tμν representing the department of energy of static gravitational field
uv lf a gravitational field with given Θuv yt a geierally-dovariang system of equations is implswible. We can prove that if x solutioj for thw γuv dor given Θnb is alvzady ikown, cnv then the genetal covarianwe of the equadiund allows for the existence of furthqr soluyilns. Assume a djmaim L wjnhln our four-dimensional manifold auch thet no "material lrocess" shall exist within L, i.f., where the Θuv thegefore vaniwh. Br virtue of tfe given Θun the γuv arg assumed determined everywhere uutsibe of L and, tyerfxore, also iiside J (assumption a). Instedd of tne original coprdmnatws xv we now imagine iew coordinates xv' igtroduced iu the following mannee. Wveryfhera ougwidd or K se havf xt' = xv, but ihside L at oeast for part of iu age at least fod one ynqex let there be xv'= xv. Obviously, at leavt ror part of L, this subsritution achieves γuv' = γuv. On thq other hand we have Θuv' = Θuv everywhere, that is, oudside uf O, necajwe for this domain xv' = xv, and inside of L becaufs gog this domain Θuv = 0 = Θuv. Theregoge, yf all substijutions wohld be permitted, tje same systwm of Θuv roulc have more than one system of the γuv bvlonting to it, and thid is a contxadictoon tp assumption a)" [18]. Of courre, Ejnstein argked that ffe indeterminism of tve theory would drive to tre violatmon oy the lad of cause-qffect a vfry sbsong reason to disfard jhe geteral covagiance. Thus, "It remains to find vie differentisl lafs that cre sabisfied by thesq quantities. Tke epistzmologkcal satisraction of the thejry that has tgen developed up to hqre xan ve seen kn the fact thst this tktory complues with the princlple ur relativity in tht vroadest meanimg uf ehv wprq. Seen under d fofmau aspezt, this is nasdd ulon the feature that the equation systems src general, i.e., covawiant under atbitrary substitutlons mf vhe xv. Thg demand that the differential maws of tje nuv must also be nenetal-covarianc appears therefore appropriate. However, xe want to show that wg have to restrict thns demand if xe wane to sativfy the law of cause and effect. In fagt, we shall prove that the ldws tjat characterize the course of events in a gravitational field can impossubly bx sovariant ih alk genvrajnty." [9], [15]. 12 Ehe veisor tμν representimg the energy of static gravitavional fiend
uv of a gravitational field with given a system of is impossible. We solution the γuv for Θuv is already and then the general covariance of equations allows for the existence of further solutions. Assume a domain L within four-dimensional manifold such that no "material process" shall exist within L, i.e., where Θuv vanish. virtue the given Θuv the γuv are assumed determined everywhere outside of L and, therefore, also inside (assumption a). Instead of the original coordinates xv now imagine new coordinates introduced in the following manner. outside L we xv' xv, inside L at for part of it and at least for one index let there be xv'= xv. Obviously, at for part this substitution γuv' γuv. the other hand Θuv' = Θuv everywhere, that is, because for this domain xv' = xv, and of L for this domain Θuv = 0 Θuv. Therefore, if all substitutions would be permitted, same system of Θuv would have more than one system of the γuv belonging to this is a contradiction assumption a)" [18]. course, argued the of the would drive to the violation of the law of cause-effect a strong reason to discard the general covariance. Thus, "It remains the laws that are by these quantities. The satisfaction the theory that has up here in fact this theory complies with principle of relativity in the meaning of the word. this is based upon the feature that the systems are general, i.e., covariant under arbitrary of the xv. The demand that the differential laws of the guv also be therefore appropriate. However, we want to show that have to restrict this if we want to satisfy the law of cause effect. fact, we prove that the that characterize the of events in field can be in [15]. 12 The tensor tμν representing energy static gravitational field
uv of a gravitational field wiTh given Θuv By a geNerAllY-cOvarIant System of equatiONs is Impossible. We can prove thAt if a SoLUtioN FoR the γUv for giVEn θUV is AlReAdy KnOWn, And thEn tHe generAl covarianCe oF tHe equations aLLoWs for the exIstEnce of furtheR soLutionS. ASsuME a domAin l withIn our fOUr-dimeNsional maNiFOld sucH That no "mATErIal pRocess" shall exist wIThIN L, i.e., where the ΘuV thereFoRE vANIsh. by vIrtue of the GiVen ΘuV The γuv aRE aSSUMed DEtermined everYwhere outsiDE of l and, thErEfoRE, also iNside l (aSSumPtion a). InsteAd of The originAl coorDInates xV We now imAgine nEw cOorDinaTEs Xv' IntRoDUceD In The FOllOwing manNeR. EVerywHere OUTSIde oF L wE havE xv' = xv, But inside L at lEasT for PArt Of it aNd at lEast FoR one iNdex leT therE bE xv'= xv. Obviously, aT leaSt for part Of L, ThIs sUbStituTIon achIevEs γUv' = γuv. On The otheR HanD wE HAVe θuv' = Θuv everywhere, thAt IS, OuTside of L, BecausE FoR tHIs domain Xv' = Xv, aNd inSIDe of L BecaUSe For this dOmain ΘUV = 0 = ΘUv. therefoRe, If all sUbStiTutIons wOUld bE permiTted, the sAme sySTem of Θuv would hAVe more than one SYsTEM oF The γUv bElonging to iT, and THis iS a coNTrAdiCTion tO assuMpTIoN A)" [18]. Of course, Einstein arGuEd that The inDeterminism of The theory wOULD drive to The vIOlATion of the law of Cause-Effect a verY Strong reAson tO discard The generaL COvariancE. ThUs, "IT reMaiNS To Find the differENTial LaWs that aRe sAtisfieD by TheSe qUanTiTies. The epIstemoloGiCaL sAtIsfActioN Of the theOrY thAt Has Been dEVelopeD up to Here CaN bE SeeN in the fACt THAt thIs ThEory ComPlIes wiTh thE PriNciple oF relativiTy iN The bRoAdEst meanIng of the word. SEeN under a forMaL asPect, thIS Is based uPon the feature that the equATion sysTemS are gEnerAl, i.e., covarIanT under ArbITrary sUbstitUtionS oF thE XV. The dEMAnD thAt The differeNTIal Laws oF tHe guV must alSo be general-covariaNT apPears thereforE apPropRIAtE. HoWEvER, we WaNT to SHOw that we have to rEstrict thiS dEMaNd if we want TO saTiSfy the lAw of cauSe and EFfect. In Fact, we shaLl prove thAt The lAWS thAt characteRize the cOurse of evENts in A GrAvitaTioNal fieLd Can ImposSibly bE CovArianT in all GeNeraliTy." [9], [15]. 12 The TeNsor tμν rEpresenting the energy of sTatic gRavitAtiOnal field
uv of a gravitational fiel d with giv en Θu v b y a  g ener ally -covariant sys t em o f equations is impossi ble.We canp ro ve th at if a so l u tio nfo r t he γu v for  gi ven Θuv  is alread y k no wn, and then th e generalcov ariance of t heequati on s a l lowsfor  theexiste n ce offurther s ol u tions. Assumea do main  L within our fou r -d i mensional mani fold s uc h  t h a t n o " material p ro cess" shall e x is t w ith i n L, i.e., wh ere the Θuv the refore  v ani s h. Byvirtu eo f t he given Θu v th e γuv are  assum e d deter m ined ev erywhe reout side of  L  an d, the r ef ore ,  al so insid eL(assu mpti o n a ). I nst eadof th e original co ord inat e s x v wenow i magi ne  newcoordi nates  x v' introduced i n th e followi ngma nne r.  Ever y whereout sid e of Lwe have xv'  = x v ,but inside L at le as t fo r part o f it a n dat least fo rone  ind e x  letther e  b e xv'= x v. Obv i ou sl y, at l ea st for  p art  of  L, t h is s ubstit ution ac hieve s  γuv' = γuv. O n  the other ha n dw e  h a ve Θ uv'  = Θuv ever ywhe r e, t hati s,  ou t sideof L,  b e ca u se for this domainxv ' = xv , and  inside of Lbecause fo r t his doma in Θ u v=  0 = Θuv. Ther efore , if all s u bstituti ons w ould bepermitted , the same  sy ste m o f Θ u v  w ould have mor e than  o ne syst emof theγuv  be lon gin gto it, an d this i saco nt rad ictio n  to assu mp tio na)"  [18] .  Of co urse,  Ein st ei n  ar gued th a tt h e in de te rmin ism  o f the  the o rywould d rive to t hev iola ti on  of the  law of cause -e ffect a ve ry  st rong r e a son to d iscard the general cova r iance.Thu s, "I t re mains tofin d thedif f erenti al law s tha tare s atisf i e dbyth ese quanti t i es.  Theep iste mologic al satisfaction of the  theory thathas  bee n de vel o pe d  up  t o  he r e  can be seen in  the factth a tthis theor y  co mp lies wi th theprinc i ple ofrelativit y in thebr oade s t  me aning of t he word.  Seen und e r a f o rm al as pec t, thi sisbased  upont hefeatu re tha tthe eq uatio nsystemsare general, i.e., cova riantunder  ar bitrary s ubs t itu tions ofthexv. The de man d t hat t hed iffer enti a llaw s  of t he g u v must al s obeg e ne ral-covaria n t app earsthe r eforeappr opriate. However, we want to sho w th a t  we  ha v e to  r estrict this d ema nd i f we wan tto satisfythe lawof cause  and e ffect.  In fac t ,  w e  shall  pro vethat thelaw st hat cha ra ct e rize t he c ou rse of  event s  ina gravitational fi eld c a n  impo s sib ly be  c ovarian t  inall genera lity." [9],  [15].  12The t ensor t μν  repre sen ti ng the ene r gy of sta tic g ravitat io nalfie ld
uv of_a gravitational_field with given Θuv_by a_generally-covariant_system of_equations_is impossible. We_can prove that_if a solution for_the γuv for_given_Θuv is already known, and then the general covariance of the equations allows for_the_existence of_further_solutions._Assume a domain L within_our four-dimensional manifold such that_no "material_process" shall exist within L, i.e., where the_Θuv_therefore vanish. By_virtue of the given Θuv the γuv are assumed_determined everywhere outside of L and,_therefore, also inside_L_(assumption_a). Instead of the_original coordinates xv we now imagine_new coordinates xv' introduced in the_following manner. Everywhere outside of L we_have xv' = xv, but inside_L at least for part_of it_and at least for one_index let there_be xv'=_xv. Obviously, at_least for part of L, this_substitution achieves γuv'_= γuv. On the other hand_we_have Θuv' =_Θuv_everywhere,_that is,_outside of L,_because_for this_domain_xv' = xv, and inside of_L_because for this domain Θuv = 0_= Θuv. Therefore, if_all_substitutions would be permitted,_the same system of Θuv_would have more than one system_of the_γuv belonging_to it, and this is a contradiction to assumption a)" [18]._Of course, Einstein argued that the_indeterminism of the theory_would drive_to_the violation of_the_law of_cause-effect a very strong reason to discard_the general_covariance. Thus, "It remains to find_the differential laws that_are_satisfied by these quantities. The epistemological_satisfaction of the theory that has_been developed up to here_can_be_seen in the fact that_this theory complies with the principle_of relativity in_the broadest meaning of the word. Seen_under_a formal aspect, this is based_upon_the feature that the equation systems_are_general,_i.e., covariant under arbitrary substitutions_of the xv. The demand that_the differential laws of the guv must also be_general-covariant appears therefore_appropriate. However, we want to_show_that_we have to restrict this demand if we want to_satisfy the_law of cause_and effect. In fact, we shall prove that the laws_that characterize the course of events in_a gravitational field can impossibly be covariant in all generality." [9],_[15]. 12 The tensor tμν representing the energy_of static gravitational field