samocch tensor

vibhedak bhoomititil samocch tensor haah tyateel tortions aanee naan jodata asalelya sambandhatil farak aahe. heey samanyatah firki jodanyanchya abhyasat disun yete. asha prakare, udaharanarth, jevha valane nisaradanyachya sthitichya adheen asate tevha firki jodanisah eka whilbean, einstein gurutvakarshanache varnan dete. atisamitisathi, teech maryada, lupt hott asalelya valanachya, 11-aayami atee gurutvakarshan dete (kshetra samikarane). mhanjech, jodanisah, samarupata tensor, siddhantatil gatishil vastunpaiki eka banato, jyamule metrikala duyyam, vyutpann bhoomikesathi padbhrasht kele jaate.
jodanitil valane kadhoon taknyala 'valanache shoshan' asey mhanatat aanee bhaumitik sanrachananchi samanata sthapit karnyasathi cartonchya samatulyata paddhatichya paddhatichya eka aahe.
metrik bhoomitimadhye vyakhya.
metrik bhoomitimadhye, contortion tensor christophel chinh sootra _ 1 saha metrik-susangat sanlagn sambandh aanee tyach metriksathi adviteey tortion-mukt levy-sivita sambandh yanchyatil farak vyakt karato.
samocch tensor sootra _ 2 heey tortion tensor sootra _ 3 chya sandarbhat paribhashit kele aahe (chinhaparyant, khalee pahaa)
jethe metrikchya sandarbhat nirdeshank vadhavale aanee qamee kele jaat aaheth
contortion tensorchya vyakhyetil spasht nasalelya berijache kaaran mhanje metrik susangatata lagoo lagoo berries-berries farkamule aahe. pahilya done nirdeshankanmadhye samocch tensor heey pratisamitiy aahe, tarr shevatchya done nirdeshankanmadhye tortion tensor swatach pratisamitiy aahe; heey khalee dakhavale aahe.
purn metrik susangat sanlagn jodani ashee lihili jaau shakateh 
jethe sootra _ 11 heey tortion-mukt levy-nagri jodani aaheh
affine bhoomitimadhye vyakhya.
affine bhoomitimadhye, ekhadyas metrik kinwa metrik connection nasate aanee mhanoon maganinusar nirdeshank vadhavanyasathi aanee qamee karnyasathi konihi swatantra nasato. soulder formcha vapar karoon, bandalla tyachya bases spacever kaay ghadat aahe tyachyashi sambandhit houu deuun, ajunahi ekhada samaan parinam sadhya karoo shakato. haah eka spashtapane bhaumitik drushtikon aahe, jyamadhye tensor aata keval bases spacever paribhashit kelelya anukramit bijganitiy vastoo honyaeivaji, fiber bandalchya ubhya aanee aadvya bandalmadhil bhaumitik vastoo aahet. yaa prakaranat, sparshika bandalvar eka-roop mhanoon rahnara eka samocch tensor tayar karoo shakato. 
lakshat thewa qee jodani sootra _ 13 chey tortion asey vyakt kele jaau shakate 
jethe sootra _ 15 heey soulder swaroop aahe (toutological vann-form). heey tortion tensor jodanitun prapt jhaale hote yachee keval aathvan karoon denare '_ 13' heey vargani sootra aahe.
vareel vibhagavaril tortion tenservaril nirdeshank qamee karnyachya sadrushyane, ekhadi vyakti soulder forms samaan operation karoo shakate aanee tensor tayar karoo shakate. 
yethe sootra _ 18 heey scalar utpadan aahe. haah taana asha prakare vyakt keela jaau shakato
praman sootra _ 20 heey samocch swaroop aahe aanee tortion-mukt levy-sivita jodani midvinyasathi aniyantrit jodanimadhye jodanyasathi "agadi" aavashyak aahe. mhanjech, ehressman jodani sootra _ 13 dile tarr, aanakhi eka jodani sootra _ 22 aahe jeey tortion-mukt aahe.
nantar valane nahise hone heey asane samatulya aahe 
kinwa 
heey contortion tensorchya jodanichya gatishilateshi sambandhit kshetra samikaran mhanoon pahile jaau shakate.
vyutpatti.
metrik susangat ephene jodani patkan midvanyacha eka marg mhanje levy-nagri jodanichya vyutpattimadhye vaparlya jaanarya berries-berries farak kalpanechi punaravrutti karane parantu tortion shoonya naan ghene. khalee eka vyutpatti aahe.
vyutpattisathiche niyam (asha prakare jodani gunank paribhashit karane nivada. gej siddhantatil connection-vann formchi prerana aahe):
aapan metrik susangat sthitipasun suruvat karatoh 
aata aapan berries-berries farak vaparato (ativar nirdeshank chakr):
aata aapan jodani punha lihinyasathi khaleel tortion tensor vyakhya (holonomic framesathi) vapartoho
lakshat ghya qee jodani gunankanchya khalchya anukramanika kramvarisathi vareel prachalit sootra _ 32 vapartana, yaa tortionchya vyakhyet nehamichya vyakhyepramane viruddha chinh aahe, i.eae. jyamitivaril khaleel vibhagat samanvay-mukt vyakhya sootra _ 33 mhanoon yachya viruddha chinh aahe. hii visangati sudharlyas (jee sahityat samanya asalyache disate) ulat chinhasah eka tiraskaran taana nirman hoil.
tortion tensorchi vyakhya aaplyapashi jeey aahe tyat badalvah
teey swachchh qaraa aanee padampramane ekatra qaraa 
tortion sandnya ekatra yeuun tensorial paddhatine rupantarit honari vastoo tayar karatat. yaa sandnya metrik susangat paddhatine ekatra yet asalyane, tyanna eka naav dile jaate, contortion tensor, jeey metrik susangat ephene jodanicha tirkas-samamiteey bhag nirdharit karate. 
vareel samikaranachya davya bajuchya nirdeshankanshi teey julate yaa preranene aapan yethe tyachi vyakhya karoo.
tortion tensorchi samamiti vaparoon saaf kelyane aapan contortion tensor mhanoon paribhashit karooh
heey aaplya abhivyaktimadhye parat samavisht karoon, aamchyakade aaheh
aata jodani gunank vegale qaraa aanee tortionchya atincha ektrit gut karah
lakshat thewa qee aanshik vyutpann asaleli pahili sandnya mhanje sapekshatavadindvare anekada vaparali jaanari levy-sivita jodani abhivyakti.
pudhilpramane, khaleel goshti tortion-free levy-sivita jodani mhanoon paribhashit karah
mug aamchyakade purn metrik susangat sanlagn jodani aata asha prakare lihili jaau shakateh 
teleparalismshy sambandh.
teleparalism siddhantat, ekhadyala eka jodani, wetzenbock jodani, jee sapat (riman vakrata nahishi hott aahe) asate parantu tyat naan gayab honare valane asate. sapatpana mhanje samantar chaukatichi kshetre tayar karanyachi parvanagi dete. yaa kalpana supermanifoldsparyant vadhavalya jaau shakatat.

