
richi vighadanam

remanian, suedo-remanian jyaamithiyude ganithashaasthra mekhalakalil, oru remanian allengil suedo-remanian palamadangu riman vakrata tensorine prathyeka beejaganita gunangalulla kashanangalayi vibhajikkunnathinulla oru maargamaan rickie vighadanam. eee vighadanam remanian, suedo-remanian jyamithiyil adisthaana praadhaanyamullathaanu.
vighadanathinte nirvachanam.
("m", "g") oru remanian allengil suedo-remanian "n"-manifold ayirikkatte. athinte riman vakrathaye (0,4)-tensor feeldaayi kanakkaakkuka. eee lekhanam saine kanvenshane pinthudarum
multilenear aayi ezhuthiyirikkunnathu ithaanu kanvenshan
eee kanvenshanil, richi tensor oru (0,2)-tensor feeldaanu, ithu "rjk" = "gilrijkl" ennum scaler vakrata "r" = "gjkrjk" ennum nirvachikkunnu."(richi tensorinte saadhaarana chinnangal kuravaanithu enna kaaryam shradhikkuka; onnum moonnum allengil randaamatheyum naalaamatheyum soochikakal churukki athine nirvachikkunnath kooduthal saadhaaranamaanu, ithu vipareeta chihnamullah richi tensor nalkunnu. kooduthal saadhaaranamaaya aaa kanvenshanu keezhil, richi tensorinteyum skailerinteyum adayaalangal chuvadeyulla samavakyangalil mattanam.) trassless richi tensor nirvachikkuka
thudarnnu moonnu (0,4)-tensor feeldukal "s", "e", "w" enniva nirvachikkuka.
richi decomposition" aanu prasthavana
paranjathupole, ithu "dablyu" enna nirvachanatthinte punasamghadanayaayathinaal ithu shoonyamaanu. iii", "dablyu" ennee moonnu puthiya tensorukalude gunangalilaanu vighadanathinte praadhaanyam.
padavali kuripp. tensorine "dablyu" ennu vilikkunnu. ganithashaasthra saahithyathil "dablyu" enna notteshan staanderdaanu, athesamayam bhauthikashaasthra saahithyathil "sii" kooduthalaayi kaanapedunnu. r" enna notteshan randilum staanderd aanu, athesamayam "s", "z", "e" ennivaykku staanderd notteshan illa.
adisthaana swathukkal.
kashanangalude savisheshathakal.
tensorukal, iii, dablyu ennivaykku oronninum riman tensorinte athe beejaganita samamithikalundu. athu ithaanu
koode
weyl tensorinu poornnamaayum traceless aanenna adhika samamiti undu
praadeshika anuroopa parannathil ninnu oru remanian allengil suedo-remanian maanifoldinte vyathiyaanam alakkunna sradheyamaaya savisheshatha "dablyu" nuu undennu herman weyl kaanichu; athu poojyamaanengil, "jii" kku "jij = ef" δ "aiji" enna roopamulla chaarttukalal "emm" cover cheyyaan kazhiyum.
vighadanathinte gunangal.
richi vighadanam orthogonalaano ennu parisodhikkaam
pothuvaaya nirvachana soothravaakyam _ 10 oormmikkumbol ithinu anantharafalamundu, athu nerittu theliyikkaan kazhiyum,
padavali kuripp. eee orthogonaalitti parayunnathupole avatharippikkunnath pratheekaathmakamaayi shudhamaayirikkum.
koode
ennirunnaalum, formula _ 14 multilenear mapp formula _ 15 aayi kaanunnundo atho leanier mapp formula _ 16 aayi kaanunnundo ennathine aashrayich atharam notteshanil ozhivaakkaanaavaatha avyakthathayund, eee saahacharyathil anubandha maanadandangalum aantharika uthppannangalum oru sthiramaaya gadakathaal vyathyaasappedum. ithu melpparanja samavakyangalil poruthakkedukalonnum undaakkillengilum, ellah padangalum ore gadakathaal maarumennathinal, kooduthal ulppetta sandarbhangalil ithu aashayakkuzhappathilekk nayichekkaam. ekkaaranathaal, soochika notteshan palappozhum manassilakkaan eluppamaanu.
bandhapetta soothravaakyangal.
oraalkk "saadhaarana soothravaakyangal" kanakkakkaam."
koodaathe "trace soothravaakyangal"
vighadanathinte ganithaparamaya vishadeekaranam.
ganithashaasthraparamaayi, riman tensorinte samamithikalulla ellah tensorukaludeyum idam oorthoogoonal grouppinte pravarthanatthinaayi athinte irreducible praathinidhyangalilekku vighadippikkunnathaanu richi vighadanam. v" oru metrik tensor (orupakshe mixed signature) ulla oru "n"-dimensional vekttar space ayirikkatte. ivide "v" oru pooyintil kottaanjant spacil maathrkayaakkappedunnu, athinaal oru vakrata tensor "r" (ellah soochikakalum kurachirikkunnu) tensor ulppannamaaya "v" ′v "′v" ′v "vii" yude oru gadakamaanu. vakrata tensor athinte aadyatheyum avasaanatheyum rund entrikalil vakra samamithiyaann
interchange simetri anusarikkukayum cheiyunnu
ellah "x", "y", "z", "w" ω "v" ω. thalphalamaayi, "r" ennathu subspace formula _ 21-leeii oru gadakamaanu, ithu "v"-nte randaamathe baahyashakthiyude randaamathe samamiti sakthiyaanu. oru vakrata tensor bianchi identityum thripthippeduthanam, athaayath ithu nalkiyirikkunna rekheeya bhoopada soothravaakyathinte kernalilaanu _ 22
s" 2λ2 "v" leeii space beejaganita vakrata tensorukalude space aanu. eee idam irterducible gadakangalaayi vighadippikkunnathaanu richi vighadanam. richi sangocha mapping
nalkia
ithu oru beejaganita vakrata tensorumaayi oru samamiti 2 roopathe bandhappeduthunnu. neremarichu, "h", "k" ennee rund samamiti roopangal nalkiyaal, "h", "k" ennivayude kulkarni-nomisu uthppannam"
ithu oru beejaganita vakrata tensor srishtikkunnu.
n" ≤4 aanenkil, (advitheeya) irreducible subspaceukalaayi oru oorthoogoonal vighadanam undu.
evide
thannirikkunna oru reeman tensor "r"-nte richi vighadanathinte bhagangal "s", "e", "c" enniva eee asthira gadakangalilulla "r"-nte oorthoogoonal projakshanukalaanu. prathyekichum,
eee ardhathil oru oorthoogoonal vighadanamaanu
eee vighadanam yathaakramam scaler sub module, richi sub module, weyl sub module ennivayude nerittulla thukayayi riman samamithikalulla tensorukalude idam prakadippikkunnu. eee moduleukal oronnum oorthoganal groupinu irreducible praathinidyamaanu, athinaal oru semi simple lye groupinaayi oru modyooline athinte irreducible gadakangalaayi vibhajikkunnathinte oru prathyeka kesaanu richi decomposition. alavu 4 ll, weyl module prathyeka oorthoogoonal groupinaayi oru jodi irreducible gadakangalaayi vighadikkunnu selph-duval, antiselph-duval bhagangal "dablyu" +, "dablyu" −.
bhouthikamaaya vyaakhyaanam.
ainsteente pothu aapekshikatha sidhaanthathil richi vighadanathe bhouthikamaayi vyaakhyaanikkaan kazhiyum, avide ithine chilappol "jeheniu-debivar vighadanam" ennu vilikkunnu. eee sidhaanthathil, ainsteen feeld samavaakyam
ellah dravyangaludeyum alavineyum chalanatheyum vivarikkunna stress-energy tensoraanu formula 34, ellah nongraavitteshanal feeld energy momentavum, richi tensor-allengil thathulyamaayi, ainsteen tensor-guruthwaakarshana mekhalayude aaa bhaagathe prathinidheekarikkunnuvennu prasthaavikkunnu. dravyamo nongraavitteshan feeldukalo illaatha oru pradeshathiloode guruthwaakarshana tharamgamaayi vyapikkaan kazhiyunna guruthwaakarshana mandalatthinte bhaagatheyaanu weyl tensor prathinidheekarikkunnath. weyl tensor aprathyakshamaakunna bahirakasha samayathinte pradeshangalil guruthwaakarshana vikiranam adangiyittilla, avayum anuroopamaayi parannathaanu.

