twaran (vishesh saapekshata)

vishesh saapekshata (aiss. aar.) mein twaran, jaisa kii neutonian yantriki mein, samay key sambandh mein wegg key vibhedan dwara hotaa haye. lorentz parivartan aur samay vistaar key kaaran, samay aur doori kii avadhaaranaaen adhik jatil hoe jaati hain, joe "twaran" kii adhik jatil paribhashaon kii or bhee leey jaati hain. aiss. aar. key roop mein sapaat minkowski spacetime kaa siddhant twaran kii upasthiti mein maanya rahata haye, kyonki samanya saapekshata (jee. aar.) kewal tabhi aavashyak hoti haye jaba oorjaa-momentum tensor (joe mukhya roop seey dravyamaan dwara nirdhaarit kiya jaataa haye) key kaaran spacetime kii vakrata hoti haye. haalaanki, choonki prithvi yaa usake aspas key kshetra mein antariksh kaal vakrata kii matra vishesh roop seey adhik naheen haye, isaliye aiss. aar. adhikansh vyaavhaarik uddeshyon key liye maanya rahata haye, jaise kii kann twarak mein prayog.
teen sthaanik aayamon (teen-twaran yaa samanvaya twaran) mein samanya twaran key liye parivartan sootra praapt kiye jaa sakate hain jaisa kii sandarbh key ack baahari jadatviya frame mein maapa jaataa haye, saath hii saath ack sahavarti twaranmaapak dwara maapa jaane vaale uchit twaran key vishesh mamle key liye bhee. ack anya upayogi aupchaariktaa chaar-twaran haye, kyonki isake ghatakon koo ack lorentz parivartan dwara vibhinn jadatviya frames mein jodaa jaa sakata haye. gati key sameekaran bhee taiyar kiye jaa sakate hain joe twaran aur bull koo jodate hain. nikaayon key twaran key kaee roopon key liye sameekaran aur unaki ghumaavdaar vishva rekhaaen ekikaran dwara inn sootron kaa anusaran karati hain. gyaat vishesh mamle nirantar anudairdhya uchit twaran yaa samaan golakar gati key liye atee-maukhik gati hain. antatah, vishesh saapekshata key sandarbh mein twarit frame mein inn ghatnaaon kaa varnan karna bhee sambhav haye, uchit sandarbh frame (sapaat antariksh samay) dekhein. aise frames mein, prabhaav utpann hotey hain joe sajaatiya gurutvaakarshan kshetron key anuroop hotey hain, jinamen samanya saapekshata mein ghumaavdaar antariksh kaal key vaastavik, sajaatiya gurutvaakarshan kshetron key saath kuchh aupchaarik samaantaaen hoti hain. hyperbolic gati key mamle mein koee bhee rindler nirdeshank kaa upayog curr sakata haye, samaan golakar gati key mamle mein koee bhee janmajaat nirdeshank kaa upayog curr sakata haye.
aitihaasik vikas key sambandh mein, twaran vaale saapeksh sameekaran pahale seey hii saapekshata key shuruati varshon mein paae jaa sakate hain, jaisa kii max wonn laa (1911,1921) yaa wolfgang pauli (1921) dwara praarambhik pathyapustakon mein sankshep mein bataaya gayaa haye. udaaharan key liye, gati aur twaran parivartan key samikaranon koo hendrick antoon lorentz (1899,1904), heinry poinkere (1905), albert einstein (1905), max plank (1906), aur chaar-twaran, uchit twaran, hyperbolic gati, twaran sandarbh frame, janm kathorta key patron mein vikasit kiya gayaa thaa, jinka vishleshan einstein (1907), herman minkowski (1907,1908), max borne (1909), gustave herglotz (1909), arnold somerfeld (1910), wonn lyu (1911), friedrich cottler (1912,1914), itihas parr khand dekhein.
teen-twaran.
neutonian yantriki aur aiss. aar. donon key anusaar, teen-twaran yaa samanvaya twaran sootra _ 1 samay key samanvaya key sambandh mein wegg sootra _ 2 kaa pahala vyutpann haye yaa samay key samanvaya key sambandh mein sthaan sootra _ 3 kaa doosara vyutpann haih
haalaanki, vibhinn jadatviya frames mein maapa gayaa teen-twaranon key beach sambandh key sandarbh mein unaki bhavishyavaniyon mein siddhant teji seey bhinn hain. neutonian yantriki mein, galilean parivartan key anusaar sootra dwara samay nirpeksh haye, isaliye isase praapt teen-twaran sabhi jadatviya frames mein bhee barabar haih
isake vipareet aiss. aar. mein, sootra _ 7 aur sootra _ 8 donon lorentz parivartan parr nirbhar karate hain, isaliye teen-twaran sootra _ 9 bhee aur isake ghatak alag-alag jadatviya frame mein bhinn hotey hain. jaba frames key beach saapeksh wegg sootra _ 10 dwara x-disha mein nirdeshit kiya jaataa haye, sootra _ 11 key saath lorentz kaarak key roop mein, lorentz parivartan kaa roop hotaa haye.
yaa parimaan sootra key manmaane wegg sootra key liye _ 13:
\baaen (\gamma _ {v}-1 \dain)-t\gamma _ {v} \dain]\\
t ^ {\prime} & = \gamma _ {v} \left (t-\frac {\mathbf {r\cdot v}} {c ^ 2} \rite)
 & \start {sanrekhit} \mathbf {r} & = \mathbf {r} '+ \mathbf {v} \left [\frac {\left (\mathbf {r' \cdot v} \rite)} {v ^ {2}} \left (\gamma _ {v}-1 \rite) + t '\gamma _ {v} \rite]\\
t & = \gamma _ {v} \left (t '+ \frac {\mathbf {r' \cdot v}} {c ^ 2} \rite)
teen-twaran key parivartan kaa pataa lagaane key liye, sootra _ 8 aur sootra _ 17 key sambandh mein lorentz parivartan key sthaanik nirdeshank sootra _ 7 aur sootra _ 15 koo alag karna hoga, jisase sootra _ 18 aur sootra _ 19 key beach teen-wegg (jise wegg-vriddhi sootra bhee kahaa jaataa haye) kaa parivartan hotaa haye, aur antatah sootra _ 8 aur sootra _ 17 key sambandh mein ack anya antar dwara sootra _ 9 aur sootra _ 23 key beach teen-twaran kaa parivartan hotaa haye. () seey shuru hocker, yeh prakriya parivartan detee haye jahan twaran wegg key samanantar (x-disha) yaa lambavat (y-, z-disha) hainh
a _ {y} ^ {\prime} & = \frac {a _ {y}} {\gamma _ {v} ^ {2} \left (1-\frac {u _ {x} v} {c ^ {2}} \rite) ^ {2}} + \frac {a _ {x} \frac {u _ {y} v} {c ^ {2}} {\gamma _ {v} ^ {2} \left (1-\frac {u _ {x} v} {c ^ {2}} \rite)}}\\
 & \start {sanrekhit karein} a _ {x} & = \frac {a _ {x} ^ {\prime}} {\gamma _ {v} ^ {3} \left (1 + \frac {u _ {x} ^ {\prime} v} {c ^ {2}} \rite) ^ {3}}\\
a _ {y} & = \frac {a _ {y} ^ {\prime}} {\gamma _ {v} ^ {2} \left (1 + \frac {u _ {x} ^ {\prime} v} {c ^ {2}} \rite)-\frac {a _ {x} ^ {\prime} \frac {u _ {y} ^ {\prime} v} {c ^ {2}} {\gamma _ {v} ^ {2} \left (1 + \frac {u _ {x} ^ {\prime} v} {c2} \rite)}}\\
yaa () seey shuru honey vaali yeh prakriya wegg aur twaran kii manmaane dishaaon key samanya mamle key liye parinam detee haih
iska matlab haye, yadi saapeksh wegg sootra _ 26 key saath dou jadatviya frame sootra _ 24 aur sootra _ 25 hain, tou sootra _ 24 mein kshanik wegg sootra _ 18 key saath ack vastu kaa twaran sootra _ 9 maapa jaataa haye, jabaki sootra _ 25 mein usee vastu kaa twaran sootra _ 23 hotaa haye aur iska kshanik wegg sootra _ 19 hotaa haye. wegg jodd sootron key saath, yeye twaran parivartan bhee guaranti dete hain kii twarit vastu kii parinami gati kabhi bhee prakaash kii gati taka naheen pahunch sakati haye yaa usase adhik naheen hoe sakati haye.
chaar-twaran.
yadi teen-vactors key bajaay chaar-vactors kaa upayog kiya jaataa haye, arthaat sootra _ 33 koo chaar-sthiti key roop mein aur sootra _ 34 koo chaar-wegg key roop mein, tou kissy vastu kaa chaar-twaran sootra _ 35 koo samay key samanvaya key bajaay uchit samay sootra _ 36 key sambandh mein vibhedan dwara praapt kiya jaataa haih
 & = \left (\gamma ^ {4} \frac {\mathbf {u} \cdot\mathbf {a}} {c},\\gamma ^ {4} \frac {(\mathbf {a} \cdot\mathbf {u}) \mathbf {u}} {c ^ {2}} + \gamma ^ {2} \mathbf {a} \rite)
jahan sootra _ 9 vastu kaa teen-twaran haye aur sootra _ 18 iska kshanik parimaan sootra _ 39 kaa teen-wegg haye, jisamen sambandhit lorentz kaarak sootra _ 40 haye. yadi kewal sthaanik bhaag parr vichaar kiya jaataa haye, aur jaba wegg koo sootra _ 41 dwara x-disha mein nirdeshit kiya jaataa haye aur kewal twaran koo samanantar (x-disha) yaa wegg key lambavat (y-, z-disha) maanaa jaataa haye, tou abhivyakti koo ghataakarah
pahale charcha kiye gaye teen-twaran key vipareet, chaar-twaran key liye ack nayaa parivartan praapt karna aavashyak naheen haye, kyonki sabhi chaar-vactors key saath, sootra _ 43 aur sootra _ 44 key ghatak saapeksh gati sootra _ 45 key saath dou jadatviya frames mein ack lorentz parivartan (,) key anuroop dwara judey hue hain. chaar-vektaron kaa ack anya guna aantarik utpaad sootra _ 46 yaa isake parimaan sootra _ 47 kaa aparivartaniyata haye, joe iss mamle mein detaa haih
uchit twaran.
aseem chhoti avadhi mein hamesha ack jadatviya frame hotaa haye, jiska wegg kshanik roop seey twarit shareer key samaan hotaa haye, aur jisamen lorentz parivartan hotaa haye. inn frames mein sambandhit teen-twaran sootra _ 48 koo seedhe ack twaranmapi dwara maapa jaa sakata haye, aur ise uchit twaran yaa vishram twaran kahaa jaataa haye. ack kshanik jadatviya frame sootra mein sootra 49 kaa sambandh aur ack baahari jadatviya frame sootra mein maapa gayaa sootra 9 sootra _ 24 sootra _ 53, sootra _ 54, sootra _ 55 aur sootra _ 56 key saath (,) seey anusaran kartaa haye. isaliye () key sandarbh mein, jaba wegg sootra _ 57 dwara x-disha mein nirdeshit kiya jaataa haye aur jaba kewal wegg key samanantar (x-disha) yaa lambavat (y-, z-disha) koo maanaa jaataa haye, tou yeh nimnalikhit haih
a _ {y} ^ {0} & = \frac {a _ {y}} {1-\frac {u ^ {2}} {c ^ {2}}}\\
 & \start {sanrekhit} a _ {x} & = a _ {x} ^ {0} \left (1-\frac {u ^ {2}} {c ^ {2}} \rite) ^ {3/2}\\
a _ {y} & = a _ {y} ^ {0} \left (1-\frac {u ^ {2}} {c ^ {2}} \rite)\\
a _ {z} & = a _ {z} ^ {0} \left (1-\frac {u ^ {2}} {c ^ {2}} \rite)
 & \tect {or} & \start {aline} \mathbf {a} ^ {0} & = \mathbf {a} \left (\gamma ^ {3},\\gamma ^ {2},\\gamma ^ {2} \rite)\\
\ganit bf {a} & = \mathbf {\mathbf {a}} ^ {0} \left (\frac {1} {\gamma ^ {3}},\\frac {1} {\gamma ^ {2}},\\frac {1} {\gamma ^ {2}} \rite)
sootra _ 18 key parimaan sootra _ 39 kii manmaane dishaaon key liye () dwara saamaanyikritah
chaar-twaran key parimaan key saath bhee ack ghanishth sambandh haih kyonki yeh aparivartaniya haye, ise kshanik jadatviya frame sootra _ 25 mein nirdhaarit kiya jaa sakata haye, jisamen sootra _ 62 aur sootra _ 63 dwara yeh sootra _ 64 kaa paalan kartaa haih
iss prakaar chaar-twaran kaa parimaan uchit twaran key parimaan seey mell khaata haye. ise () key saath jodkar, sootra _ 25 mein sootra _ 49 aur sootra _ 24 mein sootra _ 9 key beach sambandh key nirdhaaran key liye ack vaikalpik vidhi dee gaee haye, arthaat
jaba wegg koo sootra _ 41 dwara x-disha mein nirdeshit kiya jaataa haye aur kewal wegg key samanantar (x-disha) yaa lambavat (y-, z-disha) koo maanaa jaataa haye.
twaran aur bull.
sthir dravyamaan sootra _ 71 maanate hue, teen-bull sootra _ 73 key falan key roop mein chaar-bull sootra _ 72 sootra _ 74 dwara chaar-twaran () seey sambandhit haye, iss prakarah
iss prakaar wegg kii manmaane dishaaon key liye teen-bull aur teen-twaran key beach sambandh haye
\mathbf {a} & = \frac {1} {m\gamma} \left (\mathbf {f}-\frac {(\mathbf {f} \cdot\mathbf {u}) \mathbf {u}} {c ^ {2}} \rite)
jaba wegg koo sootra _ 41 dwara x-disha mein nirdeshit kiya jaataa haye aur kewal wegg key samanantar (x-disha) yaa lambavat (y-, z-disha) twaran koo maanaa jaataa haye.
f _ {y} & = \frac {m _ {y}} {\sqrt {1-\frac {u ^ {2}} {c ^ {2}}}}\\
 & \start {sanrekhit} a _ {x} & = \frac {f _ {x}} {m} \left (1-\frac {u ^ {2}} {c ^ {2}} \rite) ^ {3/2}\\
a _ {y} & = \frac {f _ {y}} {m} \sqrt {1-\frac {u ^ {2}} {c ^ {2}}}\\
 & \tect {or} & \start {aline} \mathbf {f} & = m\mathbf {a} \left (\gamma ^ {3},\\gamma,\\gamma\rite)\\
\ganit bf {a} & = \frac {\mathbf {f}} {m} \left (\frac {1} {\gamma ^ {3}},\\frac {1} {\gamma},\\frac {1} {\gamma} \rite)
isaliye, dravyamaan kii neutonian paribhasha teen-bull aur teen-twaran key anupaat key roop mein aiss. aar. mein nuksandeh haye, kyonki aisaa dravyamaan wegg aur disha donon parr nirbhar karega. natijatan, purani pathyapustakon mein upayog kii jaane vaali nimnalikhit saamoohik paribhashaon kaa abb upayog naheen kiya jaataa haih
teen-twaran aur teen-bull key beach sambandh () gati key sameekaran seey bhee praapt kiya jaa sakata haye.
jahan sootra _ 78 teen-momentum haye. sootra _ 24 mein sootra _ 73 aur sootra _ 25 mein sootra _ 81 key beach teen-bull kaa sambandhit parivartan (jaba frame key beach saapeksh wegg sootra _ 10 dwara x-disha mein nirdeshit kiya jaataa haye aur kewal twaran samanantar (x-disha) yaa wegg key liye lambavat (y-, z-disha) maanaa jaataa haye) sootra _ 18, sootra _ 9, sootra _ 86, sootra _ 87, yaa chaar-bull key parivartit ghatakon seey sambandhit parivartan sootron key pratisthaapan dwara anusaran kiya jaataa haye, jisake parinam key saathah
(\mathbf {f} \cdot\mathbf {u})} {1-\frac {u _ {x} v} {c ^ {2}}}\\
f _ {y} ^ {\prime} & = \frac {f _ {y}} {\gamma _ {v} \left (1-\frac {u _ {x} v} {c ^ {2}} \rite)}\\
 & \start {sanrekhit} f _ {x} & = \frac {f _ {x} ^ {\prime} + \frac {v} {c ^ {2}} (\mathbf {f} ^ {\prime} \cdot\mathbf {u} ^ {\prime})} {1 + \frac {u _ {x} ^ {\prime} v} {c ^ {2}}}\\
f _ {y} & = \frac {f _ {y} ^ {\prime}} {\gamma _ {v} \left (1 + \frac {u _ {x} ^ {\prime} v} {c ^ {2}} \rite)}\\
yaa sootra _ 18 kii manmaane dishaaon key liye saamaanyikrit, saath hii sootra _ 26 parimaan sootra _ 13 key saathah
uchit twaran aur uchit bull.
ack kshanik jadatviya frame mein bull sootra _ 91 koo ack sahavarti spring santulan dwara maapa jaataa haye jise uchit bull kahaa jaa sakata haye. yeh sootra _ 92 aur sootra _ 54 key saath-saath sootra _ 55 aur sootra _ 56 koo nirdhaarit karke (,) seey anusaran kartaa haye. iss prakaar () dwara jahaan wegg sootra _ 57 key liye kewal samanantar (x-disha) yaa lambavat (y-, z-disha) twaran maanaa jaataa haih
 & \start {sanrekhit karein} f _ {x} & = f _ {x} ^ {0}\\
f _ {y} & = f _ {y} ^ {0} \sqrt {1-\frac {u ^ {2}} {c ^ {2}}}\\
 & \tect {or} & \start {aline} \mathbf {f} ^ {0} & = \mathbf {f} \left (1,\\gamma,\\gamma\rite)\\
\mathbf {f} & = \mathbf {f} ^ {0} \left (1,\\frac {1} {\gamma},\\frac {1} {\gamma} \rite)
sootra _ 18 key parimaan sootra _ 39 kii manmaane dishaaon key liye () dwara saamaanyikritah
choonki kshanik jadatviya frames mein ack mein chaar-bull sootra 100 aur chaar-twaran sootra 101 hotaa haye, isaliye sameekaran () neutonian sambandh sootra 102 utpann kartaa haye, isaliye (,,) koo sankshep mein prastut kiya jaa sakata haye.
isake dwara, anuprasth dravyamaan sootra _ 103 kii aitihaasik paribhashaon mein spasht virodhabhas koo samajhaaya jaa sakata haye. einstein (1905) ney teen-twaran aur uchit bull key beach sambandh kaa varnan kiya.
jabaki lorentz (1899,1904) aur plank (1906) ney teen-twaran aur teen-bull key beach sambandh kaa varnan kiya.
ghumaavdaar vishva rekhaaen.
gati key samikaranon key ekikaran seey twarit pindon kii ghumaavdaar vishva rekhaaen praapt hoti hain joe kshanik jadatviya frames key anukram key anuroop hoti hain (yahaan, abhivyakti "ghumaavdaar" minkowski aarekhon mein vishva rekhaaon key roop seey sambandhit haye, jise samanya saapekshata key "ghumaavdaar" sthaankaal key saath bhramit naheen kiya jaanaa chaahiye). isake sambandh mein, ghadee key abhidharana kii tathaakathit ghadee parikalpana parr vichaar kiya jaanaa chaahieh ghadiyon koo sthanantarit karane kaa uchit samay twaran seey swatantra haye, yaani, inn ghadiyon kaa samay failaav jaisa kii ack baahari jadatviya frame mein dekhaa jaataa haye, kewal usa frame key sambandh mein isake saapeksh wegg parr nirbhar kartaa haye. vakr vishva rekhaaon key dou saral mamle abb uchit twaran key liye sameekaran () key ekikaran dwara pradaan kiye jaate hainh
que) atee-boleek gatih sthir, anudairdhya uchit twaran sootra _ 106 dwara () vishva rekha kii or leey jaataa haye.
 & \tau (t) = \frac {c} {\alfa} \ln\left (\sqrt {1 + \left (\frac {\alfa t} {c} \rite) + \frac {\alfa t} {c} \rite),\Quadx (t) = \frac {c ^ {2}} {\alfa} \left (\sqrt {1 + \left (\frac {\alfa t} {c} \rite)-1 \rite)
vishva rekha atee-bolibolic sameekaran sootra _ 107 seey mell khaati haye, jisase naam atee-bolibolic gati praapt hoti haye. inn samikaranon kaa upayog aksar judvaan virodhabhas yaa ghanti key antariksh yaan virodhabhas key vibhinn paridrishyon kii ganana key liye yaa nirantar twaran kaa upayog karke antariksh yatra key sambandh mein kiya jaataa haye.
b) sthir, anuprasth uchit twaran sootra _ 108 koo () dwara ack kendragami twaran key roop mein dekhaa jaa sakata haye, joe ack samaan ghoornan mein ack vastu kii vishva rekha kii or leey jaataa haye.
jahan sootra 109 sparshrekha gati haye, sootra 110 kakshiya trijya haye, sootra 111 samanvaya samay key kaary key roop mein koniya wegg haye, aur sootra 112 uchit koniya wegg key roop mein haye.
ghumaavdaar vishva rekhaaon kaa vargeekaran triple vakron kii vibhedak jyaamiti kaa upayog karke praapt kiya jaa sakata haye, jise spacetime franet-serate sootron dwara vyakt kiya jaa sakata haye. vishesh roop seey, yeh dikhaaya jaa sakata haye kii atee-maukhik gati aur ack samaan golakar gati gati key vishesh mamle hain jinamen nirantar vakrata aur marod hoti haye, joe janm kii kathorta kii sthiti koo santusht karati haye. ack shareer koo janm kathor kahaa jaataa haye yadi twaran key dauran isaki aseem roop seey alag vishva rekhaaon yaa binduon key beach antariksh kaal kii doori sthir rahati haye.
twarit sandarbh frame.
jadatviya frame key bajaay, inn twarit gatiyon aur ghumaavdaar vishva rekhaaon koo twarit yaa vakr nirdeshank kaa upayog karke bhee varnit kiya jaa sakata haye. uchit sandarbh frame sthaapit kiya gayaa haye kii yeh tarika fermi nirdeshank seey nikatata seey sambandhit haye. udaaharan key liye, ack atee-maukhik roop seey twarit sandarbh frame key liye nirdeshank koo kabhi-kabhi rindler nirdeshank kahaa jaataa haye, yaa ack samaan roop seey ghoomane vaale sandarbh frame key nirdeshank koo ghoornan belnakar nirdeshank (yaa kabhi-kabhi janm nirdeshank) kahaa jaataa haye. samatulyata siddhant key sandarbh mein, inn twarit frames mein utpann honey vaale prabhaav ack sajaatiya, kaalpanik gurutvaakarshan kshetra mein prabhaavon key anuroop hain. iss tarah seey yeh dekhaa jaa sakata haye kii aiss. aar. mein twarankari frame kaa rojgar mahatvapurna ganitiya sambandh paida kartaa haye, joe (jaba aage vikasit hotaa haye) samanya saapekshata mein ghumaavdaar sthaankaal key sandarbh mein vaastavik, asamaroopi gurutvaakarshan kshetron key vivaran mein ack maulik bhoomika nibhaata haye.
itihas.
adhik jaankaari key liye wonn laa, pauli, miller, zahar, gorgaulhon aur vishesh saapekshata key itihas mein aitihaasik srot dekhein.

