resha ghatak

bhoomitimadhye, resha ghatak kinwa lambi ghatak anaupacharikpane metrik spacemadhil anant visthapan vectorshee sambandhit resha vibhag mhanoon vichar keela jaau shakato. reshechya ghatakachi lambi, jee eka vibhedak kamanichi lambi manali jaau shakate, heey metrik tensorche karya aahe aanee teey "sootra _ 1" dware darshavile jaate.
resheche ghatak bhautikshastrat vaparale jaatat, visheshtah gurutvakarshanachya siddhantanmadhye (visheshtah samanya sapekshata) jethe antaralkal yogya metrik tensors vakr syudo-remanian manifold mhanoon tayar keela jaato.
samanya rachana.
reshechya ghatakachi aanee arclenthchi vyakhya.
reshechya ghatakachya vargabhagachi 'dii. ess.' chee 'aine.'-aayami remanion kinwa syudo remanion manifoldmadhil (bhautikshastrat samanyatah laurentzian manifold) samanvay-swatantra vyakhya hii anant visthapan sootra _ 2 (syudo remanion manifoldmadhye bahudha nakaratmak) chaa 'lambicha varg' aahe jyache vargamool vakr lambichya gananesathi vaparale pahijeh 
sootra _ 3 
jethe "g" haah metrik tensor aahe, teey antargat utpadan darshavate aanee "d" q heey (syudo) remanion manifoldvaril eka anant visthapan darshavate. vakr sootra _ 4 chey parametriesing karoon, aapan sootra _ 5 aanee sootra _ 6 darmyanchya vakr lambichya karvachi kamani lambi avibhajya mhanoon paribhashit karoo shakatoh
sootra _ 7
chhadm remanian manifoldmadhil vakranchi sanvedanshil lambi mojanyasathi, asey gruheet dharane changle aahe qee anant visthapan sarvatra samaan chinh aahe. eae.jee. bhautikshastrat kalareshechya vakrareshevaril reshechya ghatakacha varg (sootra _ 8 swakshari paramparet) nakaratmak asel aanee vakrareshevaril reshechya ghatakachya vargatil nakaratmak vargamool vakrareshevar vakrareshevar nirikshakasathi yogya vel nighun jaane mojel.
yaa drushtikonatun, metrik reshechya ghatakanvyatirikt prushthbhag aanee aakarman ghatak ii. dekhil paribhashit karate.
reshechya ghatakachya vargala metrik tensors olakhane.
sootra _ 2 heey aniyantrit "kamani lambicha varg" sootra _ 10 heey metrikchi purnapane vyakhya karate, mhanoon samanyatah sootra _ 10 sathichya abhivyakticha vichar metrik tensorchi vyakhya mhanoon karane sarvottam aahe, jeey soochak parantu naun tensorial sanketanat lihile jaateh
sootra _ 12 
chaap lambichya sootra _ 10 chya vargala metricks olakhane heey "aine"-aayami samanya vakrarekheey nirdeshankat pahane aanakhi sope aahe, jithe teey metrik tensorshee julanare samamiteey shreni 2 tensor mhanoon lihile jaateh
sootra _ 14
yethe "i" aanee "j" heey nirdeshank 1,2,3, hii moolye ghetat..., "aine" aanee einstein summation convention vaparale jaate. (chhadm) remanian jaganchya samanya udaharananmadhye trimiteey jaaga (velechya nirdeshankancha samavesh nahee) aanee kharokharach chaar-aayami antaralkal yancha samavesh hoto.
euclidean jaaget resheche ghatak.
metrikmadhun resheche ghatak kase aadhaltat yachee udaharane khalilpramane aahet.
cartasian nirdeshank.
sarvat sopa resha ghatak cartasian nirdeshankat aahe-asha paristhitit metrik fakt kronekar delta aaheh
sootra _ 15
(yethe jagesathi "i, j" = 1,2,3) kinwa matricks swarupat ("i" mhanje pankti, "j" mhanje stambh):
sootra _ 16
samanya vakrarekheey nirdeshank cartasian nirdeshankat qamee hotatah
sootra _ 17
tyamule
sootra _ 18
orthogonal vakrarekheey nirdeshank.
serve orthogonal nirdeshankansathi metrik khalilpramane dile jaateh
sootra _ 19
kuthe?
sootra _ 20
i" = 1,2,3 heey praman ghatak aahet, mhanoon reshechya ghatakacha varg aaheh
sootra _ 21
yaa nirdeshankatil reshechya ghatakanchi kahee udaharane khalee dilee aahet.
samanya vakrarekheey nirdeshank.
pariman sootra _ 22 chya jaagecha aniyantrit aadhar dilyas, metrikchi vyakhya besis vectorche antargat utpadan mhanoon kelee jaate.
sootra _ 23
jethe sootra _ 24 aanee aateel utpadan sabhovatalachya jageshi sambandhit aahe (samanyatah tyache sootra _ 25)
samanvay aadhar sutramadhye _ 26
samanvay aadhar haah eka vishesh prakarcha aadhar aahe joou vibhedak bhoomitimadhye niyamitpane vaparla jaato.
4 dii spacetime resheche ghatak.
minkowskian spacetime.
minkowski metrik aaheh
sootra _ 27
jethe eka kinwa dusare chinh nivadle jaate, tethe donhi parampara vaparlya jaatat. heey keval sapat antaral kalasathi lagoo hote. nirdeshank 4-sthanadvare dile jaatatah
sootra _ 28
tarr reshecha ghatak aaheh
sootra _ 29
schwarzschield nirdeshank.
schwarzschield nirdeshank nirdeshank sootra _ 30 aahet, jeey formche samanya metrik aaheh
sootra _ 31
31 golakar dhruviy nirdeshankatil metricks samya lakshat ghya).
tarr reshecha ghatak aaheh
sootra _ 32
sarvasadharan antaralkal.
antaralkalatil reshechya ghatakachya vargabhagachi samanvay-swatantra vyakhya ashee aaheh
sootra _ 33 
nirdeshankanchya babatitah
sootra _ 34
jethe yaa prakaranat nirdeshank aanee spacetimesathi 0,1,2,3 peksha jaast aahet.
heey antaralkalin antar aahe-antaralkalatil done manasvipane jawalchya ghatananmadhil vibhajanache mojmap. vishesh sapekshatemadhye teey laurentz rupantaranankhali aparivartaniy asate. samanya sapekshatemadhye teey aniyantrit ulat karata yenyajogya bhinn samanvay parivartanankhali aparivartaniy aahe.

