uyt tng i rng hay thuyt tng i tng quỏt Xem bi vit gii thiu: Gii thiu thuyt tng i rng tng i rng cũn l c s cho cỏc mụ hỡnh v tr hc hin ti v s ang gión n khụng ngng ca v tr uyt tng i rng hay thuyt tng i tng quỏt l lý thuyt hỡnh hc ca lc hp dn nh vt lý Albert Einstein cụng b vo nm 1916[2] v hin ti c coi l lý thuyt miờu t hp dn thnh cụng ca Lch s vt lý hin i uyt tng i tng quỏt thng nht thuyt tng i hp v nh lut vt hp dn ca Newton, ng thi nú miờu t lc hp dn (trng hp dn) nh l mt tớnh cht hỡnh hc ca khụng gian v thi gian, hoc khụng thi gian c bit, cong ca khụng thi gian cú liờn h cht ch trc tip vi nng lng v ng lng ca vt cht v bc x Liờn h ny c xỏc nh bng phng trỡnh trng Einstein, mt h phng trỡnh o hm riờng phi tuyn Nhiu tiờn oỏn v h qu ca thuyt tng i rng khỏc bit hn so vi kt qu ca vt lý c in, c bit cp n s trụi i ca thi gian, hỡnh hc ca khụng gian, chuyn ng ca vt th ri t v s lan truyn ca ỏnh sỏng Nhng s khỏc bit nh vy bao gm s gión thi gian hp dn, thu kớnh hp dn, dch chuyn hp dn ca ỏnh sỏng, v s tr thi gian hp dn Mi quan sỏt v thớ nghim u xỏc nhn cỏc hiu ng ny cho ti Mc dự cú mt s lý thuyt khỏc v lc hp dn cng c nờu ra, nhng lý thuyt tng i tng quỏt l mt lý thuyt n gin nht phự hp cỏc d liu thc nghim Tuy th, cũn tn ti nhng cõu hi m, cn bn nht nh cỏc nh vt lý cha bit lm th no kt hp thuyt tng i rng vi cỏc nh lut ca vt lý lng t nhm to mt lý thuyt y v nht quỏn l thuyt hp dn lng t Lý thuyt ca Einstein cú nhiu ng dng quan trng vt lý thiờn Nú ch trc tip s tn ti ca l en nhng vựng ca khụng thi gian ú khụng gian v thi gian b un cong n mc c ỏnh sỏng cng khụng th thoỏt c mt trng thỏi cui cựng ca cỏc ngụi lng ln Cú rt nhiu ngun bc x mnh phỏt t mt vi loi thiờn th c nh da trờn s tn ti ca l en; vớ d, cỏc h ụi tia X v nhõn cỏc thiờn h hot ng th hin s cú mt ca tng ng l en lng v l en cú lng khng l S lch ca tia sỏng trng hp dn lm xut hin hiu ng thu kớnh hp dn, ú nhiu hỡnh nh ca cựng mt thiờn h hin lờn qua nh chp uyt tng i tng quỏt miờu t cỏc tớnh cht ca súng hp dn m ó c xỏc nhn mt cỏch trc tip bi nhúm Advanced LIGO Hn na, thuyt Albert Einstein, 1921 Ngay sau phỏt trin thuyt tng i c bit nm 1905, Einstein bt u suy ngh v s mõu thun gia lc hp dn Newton vi lý thuyt ny Nm 1907, ụng nhn s liờn h (hay tng ng cc b) gia lc hp dn v h quy chiu gia tc (ụng coi õy l ý tng hnh phỳc nht ca i mỡnh) v nờu mt thớ nghim suy tng n gin ú cú mt ngi quan sỏt thang mỏy ri t ễng ó phi mt tỏm nm theo ui nhm tỡm kim lý thuyt hp dn tng i tớnh Sau nhiu nhm ln v i lch hng, cui cựng ụng ó tỡm c phng trỡnh hp dn v miờu t nú cuc hp ca Vin hn lõm Khoa hc Ph vo thỏng 11 nm 1915 m ngy gi l phng trỡnh 2 T C HC C IN N THUYT TNG I RNG trng Einstein H phng trỡnh ny cho bit hỡnh hc ca khụng thi gian b nh hng bi s cú mt ca vt cht nh th no, v lc hp dn s cong ca hỡnh hc khụng thi gian Phng trỡnh trng Einstein l mnh ghộp trung tõm ca thuyt tng i tng quỏt.[3] Phng trỡnh trng Einstein l h phng trỡnh vi phõn riờng phn phi tuyn v rt khú gii Einstein ó s dng phng phỏp xp x nhm suy lun nhng h qu u tiờn ca lý thuyt Nhng u nm 1916, nh thiờn vt lý Karl Schwarzschild tỡm nghim chớnh xỏc khụng tm thng u tiờn ca phng trỡnh trng Einstein m ngy gi l mờtric Schwarzschild Nghim ny l c s lý thuyt cho mụ hỡnh vt lý v trng thỏi cui cựng ca suy sp hp dn, dn n s hỡnh thnh ca mt s thiờn th ú cú l en dng i xng cu Trong cựng nm, nghim Schwarzschild ó c tng quỏt thnh nghim chớnh xỏc cho vt th cú in tớch, hay chớnh l mờtric ReissnerNordstrửm, nghim ny mụ t l en tớch in khụng quay.[4] Nm 1917, Einstein ỏp dng lý thuyt ca ụng cho ton b v tr, khai sinh ngnh v tr hc tng i tớnh.[5] eo t tng ng thi, ụng ó gi nh v tr tnh ti vnh hng, v phi cng thờm mt tham s mi vo phng trỡnh trng ban u ca mỡnhhng s v tr hcnhm thu c kt qu nh quan sỏt t by lõu nay.[6] Tuy th, nm 1929, nhng nghiờn cu ca nh thiờn Edwin Hubble v nhng ngi khỏc li ch v tr ang gión n V kt qu quan sỏt ny li phự hp vi nghim mụ t v tr ang gión n nh vt lý ngi Nga Alexander Friedman tỡm t nm 1922 m khụng ũi hi cú hng s v tr hc Mc s v nh v tr hc ngi B Georges Lemaợtre ó s dng nghim ny nhm miờu t kch bn s khai ca mụ hỡnh V n ln, mụ hỡnh núi rng v tr ban u ó tin húa t trng thỏi cc k núng v m c.[7] Sau ny, Einstein coi hng s v tr hc l sai lm ln nht ca i ụng.[8] Trong sut thi kỡ t thp niờn 1920 n thp niờn 1950, cỏc nh vt lý coi thuyt tng i tng quỏt mt lý thuyt k l cỏc lý thuyt vt lý Nú p hn lý thuyt ca Newton, phự hp vi thuyt tng i hp v gii thớch c mt vi hiu ng m lý thuyt Newton cha thnh cụng Chớnh Einstein ó ch vo nm 1915 rng lý thuyt ca ụng ó gii thớch c chuyn ng d thng ca im cn nht ca Sao y m khụng cn ti bt kỡ mt tham s no.[9] Vo nm 1919 mt on him dn u bi Arthur Eddington ó xỏc nhn tiờn oỏn ca thuyt tng i tng quỏt v s lch ỏnh sỏng nú i gn Mt tri bng cỏch theo dừi nht thc vo thỏng 5,[10] khin Einstein lp tc tr nờn ni ting.[11] V lý thuyt tr thnh hng i chớnh ca vt lý lý thuyt v thiờn vt lý giai on phỏt trin t 1960 n 1975, hay thi k vng ca thuyt tng i rng.[12] Cỏc nh vt lý bt u nm bt c khỏi nim l en, v ng nht nhng i tng thiờn vt lý ny vi quasar thiờn quan sỏt.[13] Cú thờm nhiu kim nghim chớnh xỏc h Mt Tri ó chng t sc mnh tiờn oỏn ca lý thuyt,[14] v v tr hc tng i tớnh cng vy vi rt nhiu quan sỏt o lng nhm kim chng h qu ca lý thuyt.[15] T c hc c in n thuyt tng i rng Chỳng ta cú th hiu thuyt tng i rng thụng qua nhng im tng t v khỏc bit ca nú so vi lý thuyt Newton Bc u tiờn l ch c hc c in v nh lut vt hp dn cho phộp miờu t theo ngụn ng hỡnh hc Bng cỏch kt hp miờu t ny vi nh lut ca thuyt tng i hp s cho chỳng ta khỏm phỏ thuyt tng i rng mt cỏch t nhiờn.[16] 2.1 Mụ t bng hỡnh hc ca lc hp dn Newton Theo thuyt tng i tng quỏt, mi vt trng hp dn hnh x ging nh chỳng mt thang mỏy kớn ang gia tc Vớ d, mt ngi s thy qu o qu búng ri tờn la (trỏi) ging nh nú ri trờn mt t (phi), chng t rng gia tc ca tờn la cung cp mt lc ging nh lc hỳt ca Trỏi t C s ca vt lý c in l khỏi nim chuyn ng ca mt vt th, kt hp gia chuyn ng t (hay quỏn tớnh) v chuyn ng cú ngoi lc tỏc dng Cỏc chuyn ng ny c miờu t bng phng trỡnh khụng gian chiu Euclid v s dng khỏi nim thi gian tuyt i Nhng ngoi lc tỏc dng lờn vt th lm qu o vt lch qu o ca chuyn ng quỏn tớnh tuõn theo nh lut th hai ca Newton v chuyn ng, phỏt biu l tng lc tỏc dng lờn vt bng lng (quỏn tớnh) nhõn vi gia tc ca nú.[17] Tip theo, chuyn ng quỏn tớnh c liờn h vi hỡnh hc ca khụng gian v thi gian: h quy chiu quỏn tớnh ca c hc c in, cỏc vt chuyn ng t vi tc khụng i s cú qu o l ng thng eo ngụn ng ca vt lý hin i, qu o ca chỳng l ng trc a, nhng tuyn th gii thng (world 2.2 Chuyn sang tng i tớnh chuyn ng quỏn tớnh ny cng cho phộp xỏc nh hỡnh hc ca khụng gian v thi gian theo ngụn ng toỏn hc, qu o ca vt chớnh l chuyn ng trờn ng trc a Trong phng trỡnh ng trc a cha h s liờn thụng ph thuc vo gradien ca th nng hp dn Khụng gian ca c hc Newton theo cỏch xõy dng ny thun tỳy l hỡnh hc Euclid phng Hỡnh hc ny tỏc ng n chuyn ng ca vt cht nhng khụng b nh hng bi vt cht v tn ti mt cỏch tuyt i Tuy nhiờn ton b khụng thi gian vt lý li l mt cu trỳc phc Nh c ch bng cỏc thớ nghim tng tng n gin v qu o ri t ca cỏc ht th khỏc nhau, dch chuyn cỏc vect khụng thi gian - ký hiu cho tc ca ht (cỏc vect kiu thi gian, cú thnh phn ta ) - s cho kt qu l cỏc vect khỏc dc theo qu o ca ht; hay núi v mt toỏn hc, liờn thụng Newton khụng kh tớch c (cỏc vect tc dch chuyn trờn qu o s khụng cũn song song vi vect ban u na) T iu ny, chỳng ta cú th kt lun rng khụng thi gian l cong Mụ hỡnh hỡnh hc phng ca hp dn Newton ch Dch chuyn song song mt vect trờn cung kớn thuc mt cu t s dng cỏc khỏi nim hip bin, cú ngha l nú cụng A N B A v vect cui cựng cú hng khỏc so vi vec mụ t t ban u, gúc lch t l vi din tớch tam giỏc cu (cung kớn) nhn mt h quy chiu quỏn tớnh ton cc v hin tng hp dn ỳng mi h ta .[21] eo cỏch miờu t hỡnh hc ny, cỏc hiu ng thy triu gia tc tng i gia cỏc vt th gn ri t lines, hay ng th gii) khụng thi gian cong c liờn h vi o hm ca liờn thụng, ch hỡnh v ng trc a chớnh l ng thng hỡnh hc hc thay i nh th no bi s cú mt lng.[22] phng.[18] Ngc li, chỳng ta mong mun rng nh ỏp dng chuyn ng quỏn tớnh - mt bit c chuyn ng thc ca vt th nh hng ca ngoi lc no (nh lc in t hoc ma sỏt) - xỏc nh hỡnh hc ca khụng gian, cng nh ta thi gian Tuy nhiờn, cú mt s khú khn xut hin hp dn eo nh lut vt hp dn Newton, v nhng thớ nghim c lp ca Eửtvửs v cỏc thớ nghim sau ú (xem thớ nghim Eửtvửs), vt ri t (cũn gi l nguyờn lý tng ng yu, hay s bng gia lng quỏn tớnh v lng hp dn th ng): qu o ca vt th ri t ch ph thuc vo v trớ v tc ban u ca nú, ch khụng ph thuc vo nú cu to bng vt cht gỡ (nh lc in t cũn ph thuc vo in tớch ht th).[19] Cú mt minh n gin iu ny th hin thớ nghim tng tng ca Einstein, hỡnh bờn cnh: i vi mt quan sỏt viờn thang mỏy kớn, khụng th bit c, bng theo dừi qu o ca cỏc vt nh qu búng ri, rng ang cn phũng ng yờn trờn mt t v mt trng hp dn, hay ang tu v tr chuyn ng t khụng gian vi gia tc bng gia tc hp dn.[20] Nu ch da vo s ri t ca vt, chỳng ta khụng th phõn bit c ch bng quan sỏt gia chuyn ng quỏn tớnh v chuyn ng chu nh hng ca lc hp dn S khụng phõn bit c ny gi mt nh ngha mi cho chuyn ng quỏn tớnh: chuyn ng ca vt ri t trng hp dn nh ngha mi v 2.2 Chuyn sang tng i tớnh Nu mụ hỡnh lc hp dn Newton cú th biu din bng hỡnh hc thỡ c s vt lý ca nú, c hc c in, ch l trng hp gii hn ca c hc tng i tớnh (c bit) i vi chuyn ng cú tc nh.[23] eo ngụn ng ca i xng: b qua nh hng ca trng hp dn, cỏc phng trỡnh vt lý tuõn theo bt bin Lorentz ging nh ca thuyt tng i hp hn l tuõn theo bt bin Galileo nh c hc c in (Nhúm i xng ca thuyt tng i hp l nhúm Poincarộ bao gm c phộp tnh tin v phộp quay.) S khỏc gia c hc c in v thuyt tng i hp tr lờn rừ rt cỏc vt cú tc gn vi tc ỏnh sỏng, v xột n nhng quỏ trỡnh nng lng cao.[24] Vi i xng Lorentz, chỳng ta cú thờm nhng cu trỳc mi Chỳng c xỏc nh bng hp nún ỏnh sỏng (xem hỡnh bờn trỏi) Cỏc nún ỏnh sỏng cho phộp nh ngha cu trỳc nhõn qu: i vi mi s kin A, v nguyờn lý cú mt cỏc s kin, hoc nh hng n A hoc b nh hng bi A thụng qua tớn hiu hoc tng tỏc m khụng vt quỏ tc ỏnh sỏng (nh s kin B hỡnh), v mt cỏc s kin khụng th liờn quan c n A (nh s kin C hỡnh) Tp ny gi l nhng quan sỏt viờn c lp.[25] Khi gn vi tuyn th gii (world-lines) ca ht ri t do, chỳng ta s dng nún ỏnh sỏng nhm khụi phc li mờtric na-Riemannian ca khụng thi gian, ớt nht i vi T C HC C IN N THUYT TNG I RNG Time B A C Space Nún ỏnh sỏng s hng vụ hng dng eo thut ng toỏn hc, quỏ trỡnh ny xỏc nh lờn cu trỳc bo giỏc.[26] uyt tng i hp khụng miờu t lc hp dn, vy cỏc nh vt lý ỏp dng nú cho nhng mụ hỡnh khụng tớnh n lc hp dn Bi vỡ mụ hỡnh hp dn Newton núi rng lc hp dn gia hai vt th tỏc dng mt cỏch tc thỡ, khụng k chỳng cỏch xa bao nhiờu (hay tn ti nhng h quy chiu quỏn tớnh ton cc), vy lý thuyt Newton vi phm bt bin Lorentz Khi tớnh n trng hp dn, bng ỏp dng s ri t do, cỏch lý gii tng t nh phn trc c ỏp dng: khụng cú mt h quy chiu quỏn tớnh ton cc tn ti lý thuyt tng i tng quỏt ay vỡ vy chỳng ta ch cú th s dng nhng h quy chiu quỏn tớnh cc b xp x" di chuyn dc theo qu ao ht ri t Chuyn thnh ngụn ng ca khụng thi gian: nhng tuyn th gii thng kiu thi gian m xỏc nh h quy chiu quỏn tớnh khụng cú trng hp dn s b lch thnh nhng ng cong tng i vi trng hp dn (Ging nh th hai qu búng ri t do, tng nh chỳng ri song song vi nhng thc t qu o ca chỳng gp ti tõm Trỏi t, hay qu o hai qu búng b lch tng i vi cú mt trng hp dn.) v iu ny gi rng trng hp dn lm thay i hỡnh hc ca khụng thi gian t phng sang cong.[27] Nhng cú mt cõu hi xut hin trc tiờn l liu h quy chiu cc b mi gn lin vi vt ri t cú ging vi h quy chiu m ú cỏc nh lut ca thuyt tng i hp tha lý thuyt da trờn c s s khụng i ca tc ỏnh sỏng chõn khụng, v cng mụ t lý thuyt in t hc c in Bng s dng nhng h quy chiu tng i tớnh ca thuyt tng i hp (nh h quy chiu gn lin vi mt t-phũng thớ nghim, hay h quy chiu ri t do), chỳng ta cú th dn nhng kt qu khỏc cho hiu ng dch chuyn hp dn, hiu ng dch chuyn tn s ca ỏnh sỏng nú truyn qua trng hp dn (xem bờn di) Nhng o c th nghim ch rng ỏnh sỏng lan truyn cỏc h quy chiu ri t cú qu o v tn s ging vi ỏnh sỏng lan truyn nhng h quy chiu quỏn tớnh ca thuyt tng i hp V ỏnh sỏng lan truyn trng hp dn cú qu o v s dch chuyn tn s ging nh nú lan truyn h quy chiu ang gia tc vi gia tc bng gia tc hp dn.[28] Tng quỏt húa phỏt biu ny tng ng vi phỏt biu cỏc nh lut ca thuyt tng i hp tha mt cỏch xp x tt nhng h quy chiu ri t (v khụng quay)", cũn gi l nguyờn lý tng ng Einstein, mt nguyờn lý nn tng ca thuyt tng i tng quỏt.[29] Cỏc thớ nghim cng ch rng thi gian o bi nhng ng h trng hp dn thi gian riờng, thut ng ca vt lý hc khụng tuõn theo cỏc nh lut ca thuyt tng i hp (hm ý thi gian b cong) Trong ngụn ng ca hỡnh hc khụng thi gian, nú khụng c o bng mờtric Minkowski Nh trng hp lc hp dn Newton, iu ny gi lý thuyt tng i rng cn mt hỡnh hc tng quỏt miờu t quy mụ nh, mi h quy chiu ri t u tng ng vi v miờu t xp x bng mờtric Minkowski H qu l, chỳng ta s cn phi tng quỏt hỡnh hc Minkowski thnh hỡnh hc cỏc khụng gian cong Tenx mờtric xỏc nh lờn cu trỳc hỡnh hc c bit nú cho phộp o di v gúc khỏc vi mờtric Minkowski ca thuyt tng i hp, nú l mờtric tng quỏt ca mờtric a gi-Riemann Hn na, mi mờtric Riemann c kt hp mt cỏch t nhiờn vi mt loi liờn thụng c bit, liờn thụng Levi-Civita, v thc t liờn thụng ny tha nguyờn lý tng ng v lm cho khụng thi gian ca thuyt tng i tng quỏt trờn phng din cc b ging vi khụng thi gian Minkowski (cú ngha l chn h ta quỏn tớnh cc b phự hp, tenx mờtric ca thuyt tng i rng tr thnh tenx mờtric Minkowski, cng nh o hm riờng bc nht v cỏc h s liờn thụng trit tiờu - tng ng vi khụng cú trng hp dn h to cc b ny) Tenx mờtric th hin tớnh ng lc ca hỡnh hc khụng thi gian, nú cho thy vt cht nh hng lờn hỡnh hc nh th no cng nh s xut hin ca nú phng trỡnh chuyn ng ca ht th.[30] Trong khụng thi gian Minkowski phng, vi h ta xà (x0 , x1 , x2 , x3 ) = (ct, x, y, z) mt nhng bt bin Lorentz l khong khụng 2.3 Phng trỡnh trng Einstein thi gian gia hai s kin s2 2.3 lng-nng lng, thỡ chỳng ta cn phi la chn u tiờn mt h quy chiu quỏn tớnh v ú ũi hi 2 2 2 2 quỏn ti2 +dz mt2h chiu tớnh ton cc, iu ny = c t +x +y +z = ds = c dt +dxtn +dy = quy dx dx l khụng c phộp thuyt tng i tng quỏt Nu ds2 < thỡ hai s kin nm trờn tuyn th Nh nguyờn lý tng ng Einstein, ngoi lng, gii (world line) kiu thi gian (time-like), v mi nng lng thỡ ng sut cng tr thnh mt ngun cho s kin thc cú liờn h nhõn qu vi nhau-mt s trng hp dn V tenx ng sutnng lng kin nm nún ỏnh sỏng ca s kin kia-s lp tc tng quỏt cho khụng thi gian cong v tr thnh tenx miờu t mt ngun cho trng hp dn nm trờn ng kiu thi gian cho phộp thu v trng hp gii hn ca lc hp dn Nu ds2 > thỡ hai s kin nm trờn tuyn th Newton c in, mt cỏch t nhiờn chỳng ta gi thit gii kiu khụng gian (space-like), õy l khong rng phng trỡnh trng hp dn liờn h tenx ng khụng thi gian gia hai s kin m mt s kin sutnng lng hng hai vi mt tenx cong hng hai gi l tenx Ricci, tenx ny cú ý ngha vt lý miờu nm ngoi nún ỏnh sỏng ca s kin t mt trng hp c bit ca hiu ng thy triu: nú Nu ds2 = thỡ hai s kin nm trờn tuyn th cho bit s thay i th tớch ca mt ỏm nh ht th gii khụng (null-world line), hay chỳng nm trờn ban u ng yờn tng i vi nhau, v sau ú ri t ng i ca ỏnh sỏng trng hp dn Trong thuyt tng i hp, nh lut bo ton nng lngng lng tng ng Bt bin Lorentz l i lng khụng i vi phng trỡnh toỏn hc l phõn k ca tenx ng chỳng ta chuyn t h ta ny sang h ta sutnng lng phi bng (hay t do) Cụng thc khỏc ny cng c tng quỏt húa sang cho khụng thi gian cong bng cỏch thay th o hm riờng thụng thng theo cỏc trc ta ca a cong bng o hm hip Tenx mờtric Minkowski l bin ca cỏc ta , o hm ny c nghiờn cu hỡnh hc vi phõn Cỏc nh lut bo ton phi luụn tha 0 phm vi cc b hay l phõn k hip bin ca +1 = tenx mt ng sutnng lng bng 0, v vy +1 phõn k hip bin ca v bờn phng trỡnh trng 0 +1 - v cho bit cong cc b ca khụng thi gian - cng vi du mờtric (, +, +, +) Trong thuyt phi bng Ban u, Einstein ngh rng v hỡnh hc tng i rng, cỏc tenx mờtric gà thay ny ch cú tenx Ricci (phõn k hip bin ca tenx ny th cho tenx v m bo i lng khỏc 0), nhng sau ú ụng phỏt hin phng trỡnh ds2 = gà dxà dx l bt bin Lorentz cc b trng cn phi tuõn theo nh lý phõn k hip bin t ng thi tenx mờtric cho phộp nõng v h - v ụng ó tỡm dng phng trỡnh n gin nht ch s ca cỏc tenx khỏc Cỏc phng trỡnh tuõn theo nh lý ny, m ngy gi l Phng trỡnh vt lý vit di dng phng trỡnh tenx cú trng Einstein: mt thun li l dng phng trỡnh ca nú khụng thay i chỳng ta chuyn sang h ta khỏc bt k (th hin cho tớnh hip 8G Rà R gà = Tà bin tng quỏt v nguyờn lý tng ng c Einstein).[31] V trỏi ca phng trỡnh l tenx Einstein, phõn k hip bin ca tenx ny bng Tenx ny l t hp ca tenx Ricci Rà v tenx mờtric gà = gà c Phng trỡnh trng Einstein bit Tuy ó nhn c hỡnh hc Riemann l cụng c toỏn hc cn thit nhm mụ t cỏc hiu ng hp dn, chỳng ta cũn cn phi xỏc nh thờm nhng ngun ca trng hp dn Trong mụ hỡnh hp dn Newton, ngun hp dn l lng Trong thuyt tng i hp, lng l mt thnh phn i lng tng quỏt hn l tenx nng lngng lng, bao gm mt nng lng v mt ng lng cng nh ng sut (bao gm ỏp sut v lc ct) Tenx nng lngng lng khụng cha nng lng ca trng hp dn.[32] Nu ngun hp dn thuyt tng i rng ch l R = g R l cong vụ hng Ricci, vi g cú th coi l cỏc phn t ca ma trn nghch o ca ma trn cú phn t g Tenx Ricci liờn h vi tenx cong Riemann R thụng qua phộp thu gn ch s Rà = R Mt khỏc, h s liờn thụng (hay ký hiu Christoel, nú khụng phi l tenx) cú th c tớnh t tenx mờtric, NH NGHA V CC NG DNG C BN = g ( gà + g gà ) Rà = v tenx cong Riemann (miờu t cong ni ti cc vụ hng cong R l hm ca tenx Ricci nờn nú cng bng phng trỡnh chõn khụng b ca khụng thi gian) bng Ngoi cỏch dn phng trỡnh Einstein tuõn theo nh lut bo ton nng lng-ng lng trờn, chớnh R = + Einstein v nh toỏn hc David Hilbert cũn nờu phng phỏp bin phõn cho tỏc dng Einstein-Hilbert õy = x l o hm riờng Trong thuyt tng v cng thu c phng trỡnh trng Phng phỏp i rng, tenx xon bng 0, ú h s Christoel bin phõn cú c im l nú thun li cho vic tng cú tớnh i xng = cng nh tenx Ricci quỏt hay m rng thuyt tng i tng quỏt Rà = Rà Cỏc nh vt lý cng ó xut nhng lý thuyt khỏc Trờn v phi ca phng trỡnh trng, Tà l tenx so vi thuyt tng i tng quỏt v thu c nhng mt ng sutnng lng nh lut bo ton nng phng trỡnh trng khỏc Nhng lý thuyt ny lng-ng lng cc b tng ng vi phõn k hip cng da trờn ba iu kin m thuyt tng i tng bin (o hm hip bin) ca nú quỏt tha món: T = T ; = vi Tà = gà g T , v g ; = g ; = Tenx Einstein Gà = Rà Rgà v Gà = Gà ; = Mt gii phng trỡnh Einstein v tỡm c nghim l tenx mờtric (cho phộp xỏc nh c cu trỳc hỡnh hc ca a khụng thi gian), chỳng ta s miờu t c chuyn ng ca ht (hay k c ỏnh sỏng-photon) trng hp dn thụng qua phng trỡnh ng trc a, d2 x dx dx + =0 d2 d d vi l tham s ca ng trc a Tt c cỏc phng trỡnh trờn c vit h ta x bt k Tt c cỏc tenx v h s Christoel cú thnh phn vit theo ký hiu ch s tru tng, v tuõn theo quy tc tớnh tng Einstein.[33] cho kt qu tiờn oỏn phự hp vi kt qu lý thuyt Newton v qu o cỏc hnh tinh v trng hp dn yu, Einstein tỡm hng s t l phng trỡnh = 8G/c , vi G l hng s hp dn v c l tc ỏnh sỏng.[34] Khi khụng cú vt cht hay bc x, tenx mt ng sutnng lng bng 0, v chỳng ta thu c phng trỡnh chõn khụng Einstein, Cỏc phng trỡnh tuõn theo nguyờn lý hip bin tng quỏt (v nguyờn lý tng ng Einstein) Phng trỡnh trng tuõn theo nh lut bo ton nng lng-ng lng cc b i vi mi tenx mờtric Khi trng hp dn yu v tc cỏc vt th l nh so vi tc ỏnh sỏng, lý thuyt s thu v mụ hỡnh hp dn Newton v c hc c in Ngoi ba iu kin trờn thỡ cỏc lý thuyt ny cũn cú thờm mt s gi thit khỏc, v ú nhng lý thuyt xut ny phc hn v mt toỏn hc so vi thuyt ca Einstein Vớ d mt s lý thuyt nh thuyt BransDicke, teleparallelism, v thuyt EinsteinCartan (thuyt ny coi tenx xon khỏc 0).[35] nh ngha v cỏc ng dng c bn Mt s nột khỏi quỏt phn trc cha mi thụng tin cn thit miờu t thuyt tng i rng, cỏc tớnh cht quan trng ca nú, nhng h qu ch yu v vic ng dng lý thuyt xõy dng cỏc mụ hỡnh vt lý 3.1 nh ngha v cỏc tớnh cht c bn uyt tng i tng quỏt l lý thuyt mờtric v tng tỏc hp dn Phng trỡnh nn tng ca lý thuyt l phng trỡnh trng Einstein, ú liờn h gia hỡnh hc ca a ta Riemann bn chiu ca khụng thi gian vi nng lng v ng lng cha 3.2 C s cho mụ hỡnh vt lý khụng thi gian ú.[36] Nhng quỏ trỡnh hin tng c hc c in c gỏn cho nguyờn nhõn lc hp dn tỏc dng (nh vt ri t do, chuyn ng trờn qu o ca cỏc hnh tinh, v qu o ca cỏc v tinh nhõn to), tng ng vi chuyn ng quỏn tớnh hỡnh hc cong ca khụng thi gian thuyt tng i rng; khụng cú lc hp dn lm lch qu o chuyn ng ca vt ng thng ay vo ú, lc hp dn l s thay i tớnh cht ca khụng thi gian, dn n lm thay i qu o ca vt tr thnh ng ngn nht cú th m vt s t nhiờn chuyn ng theo (hay ng trc a hỡnh hc vi phõn).[37] Cũn ngun gc cong ca khụng thi gian l nng lng v ng lng ca vt cht Nh nh vt lý John Archibald Wheeler phỏt biu, khụng thi gian núi cho vt cht cỏch chuyn ng; vt cht núi cho khụng thi gian cong nh th no.[38] nghim l mt mụ hỡnh vt lý tha cỏc nh lut tng i tớnh tng quỏt cng nh cỏc nh lut vt lý khỏc chi phi s cú mt ca vt cht.[43] Phng trỡnh trng Einstein l h phng trỡnh vi phõn riờng phn phi tuyn cho nhng kt qu ỏng tin cy, vy rt khú tỡm c nghim chớnh xỏc.[44] Tuy vy, cỏc nh vt lý ó gii c mt s nghim chớnh xỏc, mc du ch cú vi ba nghim cú ý ngha vt lý trc tip.[45] Nhng nghim chớnh xỏc ni ting nht, v cng cú nhiu ng dng vt lý thc nghim ú l: mờtric Schwarzschild, mờtric ReissnerNordstrửm v mờtric Kerr, chỳng l cỏc nghim ca phng trỡnh chõn khụng Einstein v mi nghim tng ng vi mt kiu l en;[46] v mờtric FriedmannLemaợtre RobertsonWalker v v tr de Sier, mi loi miờu t mt v tr cú tớnh ng lc.[47] Nhng nghim chớnh xỏc hp dn v mt lý thuyt bao gm v tr Gửdel (m kh nng k l cho phộp du hnh ngc thi gian khụng thi gian cong), nghim súng-pp cho súng hp dn, khụng gian Taub-NUT (mụ hỡnh v tr ng nht, nhng phi ng hng), v khụng gian phn de Sier (m gn õy tr lờn quan trng phng oỏn Maldacena ca lý thuyt dõy).[48] Khi m thuyt tng i thay th nng hp dn vụ hng ca vt lý c in thnh tenx i xng hng hai, thỡ ng thi tenx ny s thu v trng hp gii hn c in nhng iu kin xỏc nh i vi trng hp dn yu v chuyn ng cú tc tng i chm so vi tc ỏnh sỏng, lý thuyt cho kt qu tiờn oỏn trựng vi tiờn oỏn ca nh lut vt hp Do rt khú tỡm c nghim chớnh xỏc, cỏc nh vt dn Newton.[39] lý ó tỡm cỏch gii phng trỡnh trng Einstein bng c xõy dng trờn cụng c tenx, thuyt tng i phng phỏp tớch phõn s" trờn mỏy tớnh, hoc xột tng quỏt th hin tớnh hip bin tng quỏt: mi nhng nhiu lon nh nghim chớnh xỏc Trong nh lut ca nú v hn na cỏc nh lut thit lp lnh vc mụ phng lý thuyt bng mỏy tớnh, ngi trờn khuụn kh tng i tớnh tng quỏts cú dng ta s dng cỏc siờu mỏy tớnh mụ phng hỡnh hc phng trỡnh nh mi h ta .[40] Cn bn ca khụng thi gian v gii phng trỡnh Einstein cho hn, lý thuyt khụng cha bt k mt cu trỳc hỡnh hc nhng tỡnh quan trng nh s va chm v sỏt c s bt bin no, hay thuyt tng i rng cú c nhp hai l en hay cu trỳc ca v tr trờn khong tớnh c lp vi phụng c s khụng thi gian (ng vi cỏch ln.[49] c bit, phng phỏp ny cú th ỏp dng mi s phõn b vt cht v nng lng thỡ li cú mt cho mt h bt k nu kh nng tớnh toỏn ca siờu mỏy dng hỡnh hc khụng thi gian khỏc nhau) Nú cng tớnh cho phộp, v cú th tip cn c nhng cõu hi tha iu kin cht ch ca nguyờn lý tng i cn bn nh im k d hp dn Chỳng ta cú th tỡm tng quỏt, tc l mi nh lut vt lý phi nh nhng nghim xp x bng lý thuyt nhiu lon nh i vi mi quan sỏt viờn.[41] Trờn cc b, nh ũi hi tuyn tớnh húa hp dn[50] v phng phỏp tng quỏt ca nguyờn lý tng ng, khụng thi gian cong tr húa ca nú khai trin hu Newton, c hai phng thnh khụng thi gian Minkowski, v cỏc nh lut vt phỏp ny u Einstein phỏt trin Phng phỏp sau lý tuõn theo bt bin Lorentz cc b.[42] cung cp cỏch tip cn cú h thng nhm gii hỡnh hc khụng thi gian vi s phõn b vt cht chuyn ng chm so vi tc ỏnh sỏng Phng phỏp khai 3.2 C s cho mụ hỡnh vt lý trin cha cỏc chui s hng; vi s hng th nht i din cho úng gúp ca hp dn Newton, Khỏi nim ct lừi mụ hỡnh vt lý tng i nhng s hng tip sau th hin nhng hiu chnh nh tớnh tng quỏt ú l tỡm nghim ca phng trỡnh hn ca lý thuyt Newton t thuyt tng i tng trng Einstein Khi cú phng trỡnh Einstein v nhng quỏt.[51] Phng phỏp m rng ca phng phỏp ny phng trỡnh hay iu kin gii hn c th khỏc v tớnh gi l hỡnh thc tham s húa hu Newton, cho phộp cht ca vt cht (nh phng trỡnh trng thỏi, hoc so sỏnh mt cỏch nh lng gia nhng tiờn oỏn ca gi nh v tớnh i xng ca khụng thi gian, hoc thuyt tng i rng vi nhng lý thuyt thay th phi phng trỡnh iu kin biờn, iu kin ban u) thỡ lng t khỏc.[52] nghim ca phng trỡnh s l mt a ta Riemann (thụng thng a ny c xỏc nh bi tenx Nghim Schwarzchild: miờu t khụng thi gian mờtric theo nhng h ta c bit), v trng vt tnh cú tớnh i xng cu, bờn ngoi bỏn kớnh cht c th xỏc nh trờn a ú Vt cht cng phi Schwarzchild Nú l nghim ca phng trỡnh tha bt k mt iu kin ph no ca cỏc phng chõn khụng vi Tà = trỡnh khỏc mụ t tớnh cht ca nú Hay ngn gn, mi H QU CA Lí THUYT EINSTEIN Trong h ta cu xà (ct, r, , ) 4.1 S gión thi gian hp dn v dch s dng du mờtric (-, +, +, +), mờtric chuyn tn s Schwarzchild l [53] ( ) rs ) 2 ( rs )1 2 ( ds2 = c2 d = c dt + dr +r d + sin2 d2 , r r vi l thi gian riờng (o bi ng h gn cựng vi ht th di chuyn trờn tuyn th gii kiu thi gian) t l ta thi gian (o bi mt ng h ng yờn nm rt xa so vi ngun hp dn), r l ta xuyờn tõm (o bng chu vi ng trũn chia cho 2, cỏc ng trũn nm trờn mt cu cú tõm ti ngun hp dn), l d v (tớnh t cc bc, n v radian), l kinh (radian), v Minh hiu ng dch chuyn tn s hp dn ỏnh sỏng thoỏt b mt ca thiờn th lng ln r l bỏn kớnh Schwarzschild ca ngun hp dn, nú l h s t l liờn h vi lng M ca Ban u, bng gi s nguyờn lý tng ng l tha ngun hp dn khụng cú in tớch v khụng món,[55] Einstein ó chng t trng hp dn nh quay v r = 2GM/c [54] hng ti s trụi i ca thi gian Khi ỏnh sỏng truyn vo trng hp dn mnh thỡ tn s ca nú tng lờn (hay bc súng gim i-dch chuyn xanh), hay dng ma trn ca mờtric ) ( ỏnh sỏng truyn theo hng ngc li-thoỏt 2GM 0 trng hp dn thỡ tn s ca nú gim (hay bc súng c r ( ) 2GM 0 .tng-dch chuyn ); kt hp li, hai hiu ng ny gi c2 r gà = 0 r2 chung l dch chuyn tn s hp dn Tn s ỏnh 0 r2 sin2 sỏng mt h quy chiu cc b cng chớnh l thi gian o c h quy chiu ú Do vy, tng quỏt hn, mt quỏ trỡnh s din chm chp gn thiờn Ta thy ht th nm rt xa ngun hp th lng ln so vi cựng quỏ trỡnh ú din dn r hoc khụng cú ngun hp mt ni xa hn; hiu ng ny gi l s gión thi gian dn M = thỡ mờtric Schwarzschild gà tr hp dn-hay núi v mt hỡnh hc, thi gian b cong thnh mờtric Minkowski sau chuyn s cú mt ca vt cht.[56] t ta cu sang ta (ct, x, y, z) T s r/r l rt nh, i vi Mt Tri cú bỏn kớnh Schwarzschild xp x km, nú cú bỏn kớnh gn 700.000 km T s ny s tng i ln i vi l en v neutron Ta thy ti r = r thỡ mờtric tr lờn k d (cũn gi l chõn tri s kin), thc õy l k d chỳng ta s dng h ta cu ch khụng hn l k d thc Khi la chn h ta phự hp, k d ny bin mt v ch cú r = mi l im k d vt lý H qu ca lý thuyt Einstein Hiu ng dch chuyn ó c o phũng thớ nghim[57] v nhng quan sỏt thiờn vn.[58] S gión thi gian trng hp dn ca Trỏi t cng c o nhiu ln bng cỏc ng h nguyờn t,[59] v nh hiu chnh sai lch thi gian hiu ng ny cho phộp H thng nh v ton cu (GPS) hot ng chớnh xỏc ti vi một.[60] Nhng kim nghim trng hp dn mnh thc hin trờn quan sỏt cỏc pulsar ụi.[61] Tt c cỏc kt qu thớ nghim v quan sỏt u phự hp vi thuyt tng i tng quỏt vi sai s nh.[62] Tuy vy, mc chớnh xỏc hin nay, nhng quan sỏt ny khụng th loi tr mt s lý thuyt thay th thuyt tng i rng cng da trờn nguyờn lý tng ng, v mt s lý thuyt thỡ b bỏc b.[63] uyt tng i rng cú mt s h qu vt lý Mt s xut hin trc tip t nhng tiờn ca lý thuyt, 4.2 mt s khỏc ch tr lờn rừ rng sau hn 90 nm nghiờn cu k t Einstein cụng b lý thuyt ny nh sỏng b lch v s tr thi gian hp dn 4.3 Súng hp dn lờn hỡnh hc ca khụng gian.[70] 4.3 Súng hp dn nh sỏng b b cong (phỏt t ngun im mu xanh) gn vt th nộn c (cú mu xỏm) uyt tng i tng quỏt tiờn oỏn qu o ca ỏnh sỏng b b cong trng hp dn; ỏnh sỏng lan truyn gn vt th lng ln b kộo v phớa vt ú Hiu ng ny ó c xỏc nhn t cỏc quan sỏt ỏnh sỏng phỏt t nhng ngụi sao, thiờn h hay quasar xa b lch i i gn Mt Tri.[64] Hiu ng ny v nhng tiờn oỏn liờn quan l thc t ỏnh sỏng truyn theo ng trc a kiu ỏnh sỏng hay ng trc a khụngmt ng tng quỏt húa nhng ng thng m ỏnh sỏng truyn i vt lý c in Nhng ng trc a ny cng l s tng quỏt húa tớnh bt bin ca tc ỏnh sỏng thuyt tng i hp.[65] Khi chỳng ta kho sỏt cỏc mụ hỡnh khụng thi gian mt cỏch phự hp (hoc l phớa bờn ngoi bỏn kớnh Schwarzschild, hoc cú nhiu vt th tham gia thỡ s dng phng phỏp khai trin hu Newton),[66] thỡ mt vi hiu ng ca hp dn lờn s lan truyn ca ỏnh sỏng xut hin Mc du hin tng lch ỏnh sỏng cú th suy c chỳng ta xột ỏnh sỏng truyn mt h quy chiu ang ri t do,[67] nhng kt qu tớnh thu c cho gúc lch ch bng mt na giỏ tr so vi kt qu ca thuyt tng i rng.[68] Mt hiu ng cú liờn h gn gi vi ỏnh sỏng b b cong l hiu ng tr thi gian hp dn (hay tr Shapiro), hin tng tớn hiu ỏnh sỏng truyn t im A ti im B s mt thi gian lõu hn nu cú mt trng hp dn gia hai im ú so vi khụng cú trng hp dn ó cú nhiu thớ nghim thnh cụng kim tra hiu ng ny vi chớnh xỏc cao.[69] Trong phng phỏp tham s húa hu Newton (PPN), cỏc phộp o bao gm c lch ỏnh sỏng v tr thi gian hp dn xỏc nh mt tham s , cha s nh hng ca trng hp dn Vnh cỏc ht th b nh hng cú súng hp dn i qua Cú mt vi im tng t gia trng hp dn yu v in t hc ú l, s tng t gia súng in t v súng hp dn: nhng bin i nh ca mờtric ca khụng thi gian lan truyn vi tc ỏnh sỏng.[71] Hỡnh dung n gin nht v súng hp dn cú th thy l tỏc dng ca nú lờn vnh ht th t vựng súng truyn qua Súng hỡnh sin lan truyn qua vnh ht theo hng vuụng gúc vi mt phng vnh lm búp mộo vnh theo kiu dao ng iu hũa (minh hỡnh bờn phi).[72] Do phng trỡnh trng Einstein l phi tuyn, súng hp dn cú cng bt k khụng tuõn theo nguyờn lý chng chp, khin cho vic miờu t nú rt khú khn Tuy vy, i vi trng yu chỳng ta cú th ỏp dng phng phỏp xp x tuyn tớnh Nhng súng hp dn c tuyn tớnh húa l chớnh xỏc miờu t cỏc loi súng lan truyn n Trỏi t t nhng s kin v tr t rt xa nu cỏc mỏy dũ phỏt hin chỳng Khi n Trỏi t, ngun sn sinh súng hp dn rt xa cho nờn biờn súng thu c cỏc mỏy dũ c tớnh toỏn vo khong c 1021 hay nh hn Cỏc phng phỏp phõn tớch d liu thu c t mỏy dũ s dng c im ca súng hp dn tuyn tớnh húa ú l chỳng cú th phõn tớch thnh tng cỏc chui tun hon, hay chui Fourier.[73] Mt s nghim chớnh xỏc miờu t súng hp dn m khụng cn n phng phỏp xp x, nh on súng truyn qua chõn khụng[74] cũn gi l v tr Gowdy, mt loi v tr ang gión n cha y súng hp dn.[75] Nhng i vi súng hp dn sinh t nhng s kin thiờn vt lý, nh hai l en quay trờn qu o 10 quanh v cui cựng sỏp nhp li, hoc cỏc v n siờu tõn tinh, nhng s kin ny ch cú th thc hin mụ phng trờn siờu mỏy tớnh bng cỏc mụ hỡnh phự hp.[76] H QU CA Lí THUYT EINSTEIN iu cng c cho ụng tin rng cui cựng ụng ó tỡm dng ỳng ca phng trỡnh trng hp dn.[80] Hiu ng ny cú th suy trc tip t nghim chớnh xỏc l mờtric Schwarzschild (miờu t khụng thi gian Ngy 11 thỏng nm 2016, nhúm Hp tỏc Khoa hc xung quanh vt th lng hỡnh cu)[81] hoc s LIGO v Virgo thụng bỏo ó o c trc tip súng hp dng phng phỏp khai trin hu Newton.[82] V bn dn phỏt t cp l en lng sỏp nhp vo cht hiu ng dch chuyn im cn nht l nh m mt lnh vc mi ú l thiờn súng hp hng ca hp dn lờn hỡnh hc ca khụng gian v s dn.[77][78][79] úng gúp ca nng lng t cú (self-energy) ca ngun hp dn (th hin bi tớnh phi tuyn ca phng trỡnh trng Einstein).[83] S tin ng cn im ó c 4.4 Hiu ng qu o v tớnh tng i quan sỏt cho mt s hnh tinh vi chớnh xỏc cao (Sao y, Sao Kim v Trỏi t),[84] cng nh h ụi ca phng hng pulsar, m õy hiu ng th hin rừ c vi bc ln.[85] uyt tng i tng quỏt tiờn oỏn mt s kt qu 4.4.2 Gim chu k qu o khỏc l v chuyn ng qu o ca vt th so vi c hc c in Nú tiờn oỏn s tin ng ca im cn nht ca qu o hnh tinh, cng nh s gim chu k qu o h phỏt súng hp dn v cỏc hiu ng liờn quan n tớnh tng i ca phng hng 4.4.1 S tin ng ca im cn nht Hin tng gim chu k qu o pulsar PSR1913+16: lng thi gian gim tớnh bng giõy, theo dừi trờn ba thp k.[86] Qu o Newton () v Einstein (xanh) ca hnh tinh quay quanh ngụi Trong thuyt tng i rng, cn im qu o (im ca qu o gn nht vi tõm ca h) s tin nghay qu o khụng phi l elip, m gn ging vi elip nú quay quanh tõm, m s l ng cong ging cỏnh hoa hng (xem hỡnh bờn) Einstein ln u tiờn tỡm c kt qu ny ụng s dng phng phỏp xp x mờtric v gii hn Newton v coi hnh tinh cú lng khụng ỏng k so vi Mt Tri i vi ụng, kt qu tớnh toỏn lng dch chuyn im cn nht ca Sao y bng vi giỏ tr m nh thiờn Urbain Le Verrier phỏt hin vo nm 1859, chớnh l eo thuyt tng i tng quỏt, h ụi s phỏt súng hp dn v vỡ vy h mt nng lng Vỡ s mt mỏt ny, khong cỏch qu o gia hai vt th s gim dn, v tng ng l chu k qu o Trong h Mt Tri hoc nhng h ụi, hiu ng ny rt nh v khú quan sỏt c Nhng i vi h pulsar ụi gm hai neutron quay quanh nhau, ú cú mt hoc c hai l pulsar: nhng i thiờn vụ tuyn trờn Trỏi t s nhn c nhng xung vụ tuyn rt u n t cỏc pulsar ny, chỳng c coi l nhng ng h chớnh xỏc nht t nhiờn, v cho phộp vic o cỏc tham s qu o ca h tr lờn rt chớnh xỏc Do neutron l nhng vt th nộn c v quay quanh khong cỏch nh cho nờn lng nng lng ca súng hp dn chỳng phỏt l ỏng k.[87] 23 [152] Narlikar 1993, ph 4.4.4, 4.4.5 [153] [154] [155] [156] [171] Wald 1984, tr 295 v cỏc trang tip theo; õy l kt qu quan trng cho cõu hi v s n nh ca hnu cú V chõn tri": xem Rindler 2001, ph 12.4 Hiu ng trng thỏi lng õm, thỡ khụng gian chõn khụng Unruh: Unruh 1976, v Wald 2001, ch Minkowski phng vi lng khụng cú th hỡnh thnh lờn t nhng trng thỏi lng õm ny Hawking & Ellis 1973, ph 8.1, Wald 1984, ph 9.1 [172] Townsend 1997, ch Townsend 1997, ch 2; mt miờu t k lng v nghim [173] nh ngha lngnng lng gi cc b bao gm ny Chandrasekhar 1983, ch nng lng Hawking, nng lng Geroch, hoc nng lng-ng lng gi cc b Penrose da trờn phng Townsend 1997, ch 4; i vi miờu t chi tit xem phỏp twistor; xem bi bỏo Szabados 2004 Chandrasekhar 1983, ch [157] Ellis & van Elst 1999; mt cỏi nhỡn gn hn v im k d miờu t Bửrner 1993, ph 1.2 [174] Cú rt nhiu giỏo trỡnh v c hc lng t nh Messiah 1999; giỏo trỡnh c s nh Hey & Walters 2003 [175] [158] Cú mt chỳ ý v thc t quan trng ca hin tng k d gi quang hc xut hin nhiu phng trỡnh súng, s t quang, c gii quyt cho cỏc kt qu [176] hu hn sau thay th phng phỏp xp x [177] [159] Xem Penrose 1965 [160] Hawking 1966 [161] Phng oỏn c phỏt biu Belinskii, Khalatnikov & Lifschitz 1971; bi vit ỏnh giỏ gn õy Berger 2002 Ni 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mỏy tớnh xem Berger 2002, ph 2.1 nh v sai, xem Weinberg 1996, ch 17, 18, v Goro & Sagnoi 1985 [163] Hawking & Ellis 1973, ph 7.1 [184] Giỏo trỡnh cho sinh viờn l Zwiebach 2004; bi vit vi [164] Arnowi, Deser & Misner 1962; miờu t d hiu khú hn Polchinski 1998a v Polchinski 1998b Misner, orne & Wheeler 1973, Đ21.4Đ21.7 [185] mc nng lng hin ti m cỏc my gia tc t n [165] Fourốs-Bruhat 1952 v Bruhat 1962; v gii thiu mt (14 TeV), nhng dõy ny khụng th phõn bit c vi cỏch mụ phm xem Wald 1984, ch 10; bn trc tuyn cỏc ht im, nhng cú im quan trng l, cỏc mode miờu t phng trỡnh tin húa Reula 1998 dao ng khỏc ca cựng mt dõy c bn s hin tng ng vi cỏc ht khỏc (v in tớch cng [166] Gourgoulhon 2007; bi vit miờu t c s ca mụ phng nh cỏc tớnh cht lng t khỏc), xem Ibanez 2000 Lý s thuyt tng i, bao gm xut hin cỏc thuyt ny thnh cụng ch cú mt mode dao ng c im k l ca phng trỡnh trng Einstein, xem ca dõy tng ng vi ht lng t graviton, ht gi Lehner 2001 thuyt truyn tng tỏc hp dn, xem Green, Schwarz & Wien 1987, ph 2.3, 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Tp_tin:Gravwav.gif Ngun: https://upload.wikimedia.org/wikipedia/commons/5/5c/Gravwav.gif Giy phộp: CC-BY-SA-3.0 Ngi úng gúp: self-made, using standard (TT-gauge) description of linearized sinusoidal gravitational wave Ngh s u tiờn: Mapos Tp_tin:LIGO_measurement_of_gravitational_waves.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/d/db/LIGO_ measurement_of_gravitational_waves.svg Giy phộp: CC BY 3.0 Ngi úng gúp: http://physics.aps.org/featured-article-pdf/10.1103/ PhysRevLett.116.061102 Ngh s u tiờn: B P Abbo et al (LIGO Scientic Collaboration and Virgo Collaboration) full list at the end of the article Tp_tin:Lensshoe_hubble.jpg Ngun: https://upload.wikimedia.org/wikipedia/commons/a/a9/Lensshoe_hubble.jpg Giy phộp: Public domain Ngi úng gúp: http://apod.nasa.gov/apod/image/1112/lensshoe_hubble_3235.jpg Ngh s u tiờn: ESA/Hubble & NASA Tp_tin:Light_cone.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/2/27/Light_cone.svg Giy phộp: Public domain Ngi úng gúp: Tỏc phm chớnh ngi ti lờn to Ngh s u tiờn: Sakurambo Tp_tin:Light_deflection.png Ngun: https://upload.wikimedia.org/wikipedia/commons/c/c2/Light_deflection.png Giy phộp: CC BYSA 3.0 Ngi úng gúp: self-made, using numerical integration methods to solve the geodetic equation for light near a spherical massive object (Schwarzschild metric) Ngh s u tiờn: Markus Poessel (Mapos) Tp_tin:Parallel_transport.png Ngun: https://upload.wikimedia.org/wikipedia/commons/6/6d/Parallel_transport.png Giy phộp: CC-BY-SA-3.0 Ngi úng gúp: Tỏc phm chớnh ngi ti lờn to Ngh s u tiờn: Luca Antonelli (Luke Antony) Tp_tin:Penrose.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/a/a8/Penrose.svg Giy phộp: Public domain Ngi úng gúp: Chuyn t en.wikipedia sang Commons by Andrei Stroe using CommonsHelper Ngh s u tiờn: Cronholm144 ti Wikipedia Ting Anh Tp_tin:Psr1913+16-weisberg_en.png Ngun: https://upload.wikimedia.org/wikipedia/commons/7/79/Psr1913%2B16-weisberg_en png Giy phộp: Public domain Ngi úng gúp: M Haynes et Lorimer (2001) (redrawn by Dantor as Image:Psr1913+16-weisberg.png, English labels added by mapos) Ngh s u tiờn: ? 14.3 Giy phộp ni dung 37 Tp_tin:Relativistic_precession.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/2/28/Relativistic_precession.svg Giy phộp: CC-BY-SA-3.0 Ngi úng gúp: Tỏc phm chớnh ngi ti lờn to ra, self-made using gnuplot with manual alterations Ngh s u tiờn: KSmrq Tp_tin:Spacetime_curvature.png Ngun: https://upload.wikimedia.org/wikipedia/commons/2/22/Spacetime_curvature.png Giy phộp: CC-BY-SA-3.0 Ngi úng gúp: ? Ngh s u tiờn: ? Tp_tin:Spin_network.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/5/52/Spin_network.svg Giy phộp: CC BY-SA 3.0 Ngi úng gúp: Tỏc phm chớnh ngi ti lờn to Ngh s u tiờn: Markus Poessel (Mapos) Tp_tin:Star_collapse_to_black_hole.png Ngun: https://upload.wikimedia.org/wikipedia/commons/2/20/Star_collapse_to_black_ hole.png Giy phộp: CC BY-SA 2.5 Ngi úng gúp: ? Ngh s u tiờn: ? Tp_tin:Wikisource-logo.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg Giy phộp: CC BY-SA 3.0 Ngi úng gúp: Rei-artur Ngh s u tiờn: Nicholas Moreau 14.3 Giy phộp ni dung Creative Commons Aribution-Share Alike 3.0 ... ng ca vt ng thng ay vo ú, lc hp dn l s thay i tớnh cht ca khụng thi gian, dn n lm thay i qu o ca vt tr thnh ng ngn nht cú th m vt s t nhiờn chuyn ng theo (hay ng trc a hỡnh hc vi phõn).[37] Cũn... thỡ tn s ca nú tng lờn (hay bc súng gim i-dch chuyn xanh), hay dng ma trn ca mờtric ) ( ỏnh sỏng truyn theo hng ngc li-thoỏt 2GM 0 trng hp dn thỡ tn s ca nú gim (hay bc súng c r ( ) 2GM ... ng thy triu: nú Nu ds2 = thỡ hai s kin nm trờn tuyn th cho bit s thay i th tớch ca mt ỏm nh ht th gii khụng (null-world line), hay chỳng nm trờn ban u ng yờn tng i vi nhau, v sau ú ri t ng i ca