A Furter eerlzed r Desty d Its Specl Cses F-Pe Ce Deprtet of Pyscs Dl Uversty of ecoloy Dl 64 C E-l: cefp@dluteduc Abstrct By surz d eted te r destes of te eerl reltvty d te Kbble s ue teory of rvtto furter eerlzed r desty for rvttol syste s obted d lyzed reter detl wc c be used for study ore etesve re of rvtto y specl cses c be derved fro ts eerlzed r desty ter eerl crcters d peculrtes wll be brefly descrbed PACS ubers: 4Cv 49e Keywords: r desty coupls betwee felds coservto lws eery-oetu tesor desty sp desty Itroducto I te teory of specl reltvty te r of tter feld c be deoted by te fuctol for: were s te ordry dervtve of It s well ow tt te reltvstc teores of rvtto te r desty of tter feld ust be deoted by te fuctol for : were s te covrt dervtve of : j 3 j For te Kbble s ue teory of rvtto te fre coecto j s depedet feld vrles d te torso ust pper te spce-te I ts cse te Eq c be eerlzed s j 4
For te reltvstc teores of rvtto te spce-te wtout torso j sould ot be depedet feld vrles d t wll be prove te Apped tt b d d j j j d d j j 5 I ts cse te Eq c be eerlzed s 6 Sce te ret jorty of te fudetl tter felds re spors t s ecessry to use tetrd feld e etrc feld s epressed s j j fro wc we ve j j etc I te reltvstc teores of rvtto 6 π 7 s lwys dopted s te r desty of rvttol feld For te Kbble s ue teory of rvtto Eq7 c be eerlzed s j j 8 For te reltvstc teores of rvtto te spce-te wtout torso e eerl reltvtyfter us Eq5 Eq7 c be eerlzed s 9 9 I ts pper order to coduct dept study o te eerl crcter d te peculrty of r destes for soe reltvstc teores of rvtto Eqs46 wll be eteded to te follow epresso: j j d Eqs89 wll be eteded to te follow epresso: j j
3 We wll e were d re deoted by Eq d Eq respectvelys furter eerlzed r desty represets te tter feld d j represet te rvttol felds s furter eerlzed r desty s sfctly ore eerl t te r destes deoted by Eqs46 d Eqs89 It ust be dcted tt prt fro descrb rvttol syste wt torso ts furter eerlzed r desty e Eqs c be used lso to descrbe rvttol syste wtout torso If Eqs re used to descrbe rvttol syste wtout torso t ust be oted tt j s fucto of d j s fucto of So te Eq c be epressed s d te Eq c be epressed s 3 Evdetly Eq6 s specl cse of Eq we Eq9 s equvlet to Eq3 For te reltvstc teores of rvtto te spce-te wt torso besdes Eq4 te follow r destes j 4 j j 5
re lso te specl cses of Eq By es of study te furter eerlzed r desty d ts specl cses ter eerl crcter d peculrty c be sow clerly e furter eerlzed r desty surzes y propertes of vrous teores of rvtto Below we sll prove tt Eq d Eq c be rewrtte s d j 6 j 7 j s te curvture tesor wt ed des j j j j j 8 s te torso tesor wt ed dees { } 9 e pyscl e of Eq6 s tt te rvttol felds could ct o te tter feld oly trou covrt dervtve curvture of spce-te d torso of spce-te erefore te fors of coupls betwee te rvttol felds d tter feld t be j j j j or etc e coupl j coted te covrt dervtve j s clled te l coupl wc s well ow te eerl reltvty d te ue teory of rvtto Eq6 tells us tt ddto to te l coupl tere t be oter coplcted coupls teory e pyscl e of Eq7 s tt te r of rvttol feld s coposed of curvture tesor feld d torso tesor feld Becuse s bot coordte sclr d fre sclr te possble ters volved 4
re sclrs costructed fro j Hece te study of furter eerlzed r destes sfctly eted te re of te studes of rvtto If Eq7 s used to descrbe rvttol syste wtout torso te d {} te possble ters volved re oly te sclr curvture {} {} j j d ts power suc s j Cosder oter requreets 6π s cose eerl reltvty We sll dscuss te ove probles te follow sectos e syetry of te r destes for rvttol syste Syetres est uverslly pyscl systes We suppose tt oe fudetl syetry of rvttol syste s tt te cto terls I d I d 4 4 d I I I d 4 I stsfy I I d respectvely uder te follow two sulteous trsfortos 4: te ftesl eerl coordte trsforto ξ te locl oretz trsforto of tetrd fre j e e e ε e j e suffcet codto of cto terl trsfortos s : I 4 d be I uder ove 5
6 ξ were represets te vrto t fed vlue of For te ost eerlzed r desty we ve j j j j 3 j j j j 4 et or or becuse of te depedet rbtrress of ε ε ε ξ ξ d ξ It s ot dffcult to derve te follow dettes 3 f : 5
7 6 7 j j j j 8 j j j j j 9 j j 3 Fro Eq3 d Eq7 t s foud tt tere ust est oter detty: 3 It wll be sow bellow tt y propertes of rvttol syste c be derved fro te ove dettes We Eqs re used to descrbe rvttol syste wtout torso fro Eq we ve
8 3 were deote te prtl dervtve t te costt vlues of d Hece we et 33 34 35 36 37 Fro Eq3 we ve
9 38 Hece we et 39 4 4 O te oter d t s evdet tt d relt to Eqs3 stsfy lso ξ were or or Ow to te depedet rbtrress of ε ε ε ξ ξ ξ d ξ we obt oter set of dettes 9 f :
4 43 44 45 46
47 48 I ddto tere re te reltos: 49 5 Utlz tese reltos d tose Eqs33-3739-4 d crry out soe coplcted clcultos t c be prove tt te dettes Eqs5-3 re equvlet to te dettes Eqs4-48 3 Possble fors of te rs uder te syetry of trsfortos Eqs I ts secto we wll prove tt due to te requreet of te cto terls of rvttol syste be vrt uder te trsfortos Eqs te possble fors of te r destes Eq d Eq t be: j 6 d j 7 respectvely e proof of Eq6 s ve te follow: Eq7 es tt j ppers oly trou curvture tesor feld j
becuse Eq3 es tt ppers j j oly trou torso tesor feld becuse Eq6 es tt ppers oly trou covrt dervtve d curvture tesor feld j d torso tesor feld becuse j j Hece te tter r desty sould te te for deoted by Eq6 O te oter d f tere ests te relto Eq6 we ust ve: j j erefore fro Eq5 j j j j 5
3 j j j j j j j j j j j j 5 us we ve 53 54 55 56
4 j j j j 57 j j 58 Us Eqs53-58 d Eqs5-3 we ve te follow dettes: 59 6
5 6 j j j j j j 6 ρ ρ ρ ρ j j j j j j j 63
6 j j 64 65 O te oter d fro Eq6 we lso ve: 66 were 67 b b b b 68 } { 69 Substtut Eqs67-69 to Eq66 d us ξ becuse of te depedet rbtrress of ε ε ε ξ ξ d ξ fter soe lety clcultos we c obt te dettes Eqs59-64 oe by oe Hece te dettes obted drectly fro
7 re just te se s tose derved fro j j j j j j e ove lyss prove tt te relto j j j ust est Wt te se etod we c lso prove te follow reltos: j j j Prevously we tred to prove Eq6 ef However tere pper to be soe errors tt pper I te ove lyss we beleve we ve corrected te flws 4 Coservto lws for rvttol syste wt our furter eerlzed r desty Below we sll derve te coservto lws for rvttol syste wt our furter eerlzed r desty deoted by Eqs fro te dettes Eqs5-3 d equtos of felds: 7 7
8 j j j j j j 7 Fro te we c et te follow reltos: j j 73 74 j j j j j _ 75 j j j j j _ 76
9 77 78 Eq73 t be rerded s coservto lws of eery-oetu tesor desty for te rvttol syste: t t 79 were j j t 8 d j j t 8 t be terpreted s te eery-oetu tesor desty of tter feld d of rvttol feld respectvely But we ust dcted tt t d t re ot tesor destes d Eq79 lcs te vrt crcter t sould ve te teores of reltvstc rvtto However f we use Eq 75 to defe
j j j j j _ 8 d use Eq 76 to defe j j j j j _ 83 te we et 84 85 At ere d re tesor destes d Eq85 s covrt relto Hece we wll te Eqs8485 to be te coservto lws of eery-oetu tesor desty for te rvttol syste wt our furter eerlzed r destes Hstorclly Este d proposed oter coservto lws of eery- -oetu tesor desty for rvttol syste : ~ t 86 were u t ~ u u e vrtues d defects out Eq85 d Eq86 ve bee dscussed torouly efs9 Becuse Eqs8485 ve ore locl bss d rc pyscl cotets te utor beleves tt te coservto lws Eqs8485 t be better t Este s coservto lws Eq86 9 d could be tested by future eperets d observtos Eq74 t be rerded s coservto lws of sp desty for te rvttol syste: s s 87 were
s 88 d s 89 t be terpreted s te sp desty of tter feld d of rvttol feld respectvely But we ust dcted tt s s d Eq87 lc t ll te vrt crcter t sould ve te sprt of eerl reltvty However f we use Eq 77 to defe S 9 d use Eq 78 to defe S 9 te we et S S 9 S S 93 At ere S S d Eq93 ll ve te vrt crcter ece we wll te Eqs993 to be te coservto lws of sp desty for te rvttol syste wt our furter eerlzed r destes
5 Soe specl cses of Eq d Eq We ve dcted tt j j j j re ll te specl cses of Eq d It s evdet tt j j re ll te specl cses of Eq Wt te se etod to prove j j j we c lso prove te follow reltos: j 94 j j j 95 j 96
3 j j j 97 98 It s ot dffcult to verfy tt for te ove specl cses of Eq d Eq te coservto lws for rvttol syste ll ve te se tetcl for: d S S S S Of course te deftos of eery-oetu tesor desty d sp desty for dfferet r destes re dfferet Fro te dscussos te ove sectos we ve see tt our furter eerlzed r desty c be used for descrb y teores of rvtto er eerl crcters re: te rvttol felds could ct o te tter feld oly trou covrt dervtve curvture of spce-te d torso of spce-te te r destes of rvttol feld re coposed of curvture tesor feld d torso tesor feld te coservto lws for rvttol syste ll ve te se tetcl for er peculrtes re: te cocrete fors of r destes for tter d rvttol feld re dfferet so te coupls betwee te rvttol felds d tter feld re dfferet te deftos of eery-oetu tesor desty d sp desty for dfferet r destes re dfferet Apped Ⅰ Proof of te relto Eq5 for te spce-te wtout torso e olooc coecto feld s relted to d j by 3 j j A
were j j I ddto j j EqA c be derved fro te requreet: A s requreet urtees tt lets d les re preserved uder prllel dsplceet 4 e torso tesor s defed by 5 A3 ere ests te relto 5: { } A4 were { } A5 s te Crstoffel sybol EqA4 c be derved fro EqsAA3A5 I te spce-te wtout torso fro EqA4 t s obvously { } I ts cse te relto j j j c be obted fro EqsAA5 d d b d d j j 5 Ⅱ Soe useful reltos of dfferetl eoetry At ere we troduce soe useful reltos of dfferetl eoetry wc wll be used ts pper e curvture tesor relted to coecto { } s defed by 8 {} ρ ρ { } { } { }{ } { }{ } A6 ρ ρ Slrly te curvture tesor relted to coecto s defed by ρ ρ ρ ρ A7 4
EqA4 suests tt {} for te spce-te wt torso d {} oly for te spce-te wtout torso We c lso defe te curvture tesor relted to te fre coecto j 3: j j j j j A8 Us EqA t c be verfed tt torso tesor c lso be verfed: j j Us EqA te follow relto for j j j j A9 efereces Kbble WB 96 oretz Ivrce d te rvttol Feld J t Pys Weber S 97 rvtto d Cosoloy Wley New Yor 3 Ce F P 99 eerl equtos of oto for test prtcles spce-te wt torso Iter J eor Pys 9 6 4 Hel F W vo der Heyde P d Kerlc D 976 eerl reltvty wt sp d torso: foudtos d prospects ev od Pys 48 393 5 Scoute J A 954 cc Clculus Sprer Berl 8 du d fstz E 975 e Clsscl eory of Felds rslted by Heres Pero Press Oford 9 Ce F P 7 Feld equtos d coservto lws derved fro te eerlzed Este s r desty for rvttol syste d ter plctos to cosoloy Iter J eor Pys to be publsed Corso E 953 Itroducto to esors Spors d eltvstc Wve Equtos Blce & So odo F P Ce 993 e eerlzed rs of rvttol teory wt torso d ter vrce uder ξ trsfortos Scece I C Seres A 366 ε Ce F P e restudy o te debte betwee Este d ev-cvt d te eperetl tests Spcete & Substce 3 6 5