
sahsambandh kaary

ack sahsambandh falan ack aisaa falan haye joe yaadrichhik charon key beach saankhyikiya sahsambandh detaa haye, joe una charon key beach sthaanik yaa laukik doori parr nirbhar kartaa haye. yadi koee dou alag-alag binduon parr maapi gaee samaan matra kaa pratinidhitva karane vaale yaadrichhik charr key beach sahsambandh kaary parr vichaar kartaa haye, tou ise aksar ack swa-sambandh kaary key roop mein sandarbhit kiya jaataa haye, joe swa-sambandh seey banaa hotaa haye. vibhinn yaadrichhik charon key sahsambandh karyon koo kabhi-kabhi cross-sahsambandh kaary kahaa jaataa haye taaki iss baat parr jore diyaa jaa sakey kii vibhinn charr parr vichaar kiya jaa rhaa haye aur kyonki way cross-sahsambandhon seey baney hotey hain.
sahsambandh kaary samay yaa sthaan mein doori key kaary key roop mein nirbharta kaa ack upayogi sanketak hain, aur unakaa upayog moolyon koo prabhaavi roop seey asambandhit honey key liye namuna binduon key beach aavashyak doori kaa aaklan karane key liye kiya jaa sakata haye. isake alawa, way una binduon parr moolyon koo prakshepit karane key liye niyamon kaa aadhar banaa sakate hain jinake liye koee avalokan naheen hain.
khagol vigyaan, vittiya vishleshan, arthshastra aur saankhyikiya yantriki mein upayog kiye jaane vaale sahsambandh kaary kewal una vishesh yaadrichhik prakriyaon mein bhinn hotey hain jinn parr way lagu hotey hain. quantum kshetra siddhant mein quantum vitaran parr sahsambandh kaary hain.
paribhasha.
sambhavatah kuchh sthaan key vibhinn binduon "s" aur "t" parr alag yaadrichhik charr "x" ("s") aur "y" ("t") key liye, sahsambandh falan haye
jahan sootra _ 2 kaa varnan sahsambandh parr lekh mein kiya gayaa haye. iss paribhasha mein, yeh maanaa gayaa haye kii yaadrichhik charr scalar-valued hain. yadi way naheen hain, tou adhik jatil sahsambandh karyon koo paribhaashit kiya jaa sakata haye. udaaharan key liye, yadi "x" ("s") "n" tatvon key saath ack yaadrichhik sadish haye aur "y" (t) "q" tatvon key saath ack sadish haye, tou sahsambandh karyon key ack "n" × "q" matricks koo sootra _ 3 tatva key saath paribhaashit kiya jaataa haye.
jaba "n" = "q", kabhi-kabhi iss matricks kaa nishan iss parr kendrit hotaa haye. yadi sambhaavyata vitaran mein koee lakshya sthaan samroopata haye, tou i.ee. yaadrichhik charr (jise aantarik samroopata bhee kahaa jaataa haye) key moolya sthaan mein samroopata, taba sahsambandh matricks mein prerit samroopata hogi. isi tarah, yadi sthaan (yaa samay) kshetra kii samaruupataaen hain jinamen yaadrichhik charr maujood hain (jinhen sthaankaal samroopata bhee kahaa jaataa haye), tou sahsambandh falan mein sambandhit sthaan yaa samay samaruupataaen hongi. mahatvapurna antariksh kaal samroopata key udaaharan hain -
uchch crum sahsambandh karyon koo aksar paribhaashit kiya jaataa haye. crum "aine" kaa ack vishisht sahsambandh kaary haye (konn koshthak apeksha moolya kaa pratinidhitva karate hain)
yadi yaadrichhik sadish mein kewal ack ghatak charr haye, tou soochkaank sootra _ 3 anaavashyak hain. yadi samaruupataaen hain, tou sahsambandh falan koo samroopata key aparivartaniya pratinidhitva mein vibhajit kiya jaa sakata haye-aantarik aur antariksh kaal donon.
sambhaavyata vitaran key guna.
inn paribhashaon key saath, sahsambandh karyon kaa adhyayan sambhaavyata vitaran key adhyayan key samaan haye. kaee yaadrichhik prakriyaon koo poori tarah seey unake sahsambandh karyon dwara chihnit kiya jaa sakata haye; sabse ullekhaniya udaaharan gausi prakriyaon kaa varg haye. 
ack seemit sankhya mein binduon parr paribhaashit sambhaavyata vitaran koo hamesha samanya kiya jaa sakata haye, lekin jaba inhen nirantar sthaanon parr paribhaashit kiya jaataa haye, tou atirikt dekhbhal kii aavashyakta hoti haye. iss tarah key vitaran kaa adhyayan yaadrichhik chalane key adhyayan key saath shuru huaa aur isane ito cullon kii dhaarana koo janm diyaa.
euclidean sthaan mein fenman path abhinn anga ise saankhyikiya yantriki key liye ruchi kii anya samasyaon key liye saamaanyikrit kartaa haye. koee bhee sambhaavyata vitaran joe sahsambandh karyon parr ack shart kaa paalan kartaa haye jise pratibimb sakaratmakta kahaa jaataa haye, vaha wick rotation key baad ack sthaaniya quantum kshetra siddhant kii or leey jaataa haye joe minkovsky spacetime (austerwalder-scredder swayamsiddh dekhein) key liye hotaa haye. punarsanchayit karane kaa sanchaalan sambhavana vitaran key sthaan seey swayam mein maanchitran kaa ack nirdisht samooh haye. ack quantum kshetra siddhant koo punarnaveenikaran yogya kahaa jaataa haye yadi iss maanchitran mein ack nishchit bindu haye joe ack quantum kshetra siddhant detaa haye.

