PHYS833 Astrophysics of Compact Objects Notes Part 12

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1 1 PHYS833 Atophyic of Compact Object Note Pat 1 Black Hole If the coe of a maive ta gow too lage to fom a neuton ta then thee i nothing that can top it collaping to vey high denity The gavitational field nea the object become o tong that nothing, not even light, can ecape A egion aound the object i cut off fom the extenal univee A black hole ha been fomed The bounday of the egion that cannot communicate with the extenal univee i the event hoizon It i not poible to detemine what exactly happen to the collaping object inide the event hoizon Eintein equation of geneal elativity indicate that a ingulaity develop Howeve it may be that quantum effect pevent fomation of a ingulaity Intead condition would be finite but vey exteme The popetie of black hole depend only on thee paamete, it ma M, it angula momentum, J it electic chage, Q Becaue a chaged black hole would attact chage of oppoite ign, Q i pobably mall can be taken to be zeo To undet black hole futhe, we need ome eult fom the Geneal Theoy of Relativity 1 The peence of ma o enegy eult in cuvatue of pace time The inteval between event that ae neaby in fou-dimenional pace-time i given by β d gβdx dx =, (11) whee g β i the ymmetic metic teno Fo a flat pace-time, we have in Cateian coodinate d = dx + dy + dz c dt, (1) o that the metic teno i diagonal with diagonal component (1, 1, 1, -1) We can find the path of tet paticle moving in the cuved pace-time by uing a vaiational pinciple In the abence of gavity, tet paticle take the hotet path between two point o that δ d = 0 (13) By the pinciple of equivalence, thi vaiational pinciple alo hold in geneal elativity To deive the equation of motion, we need to conide a paamete, λ, that inceae monotonically along the path of the paticle (ie along it woldline) We can then wite β dx dx d = gβ dλ, dλ dλ (14) 1 Eintein, A (1915), Die Feldgleichungen de Gavitation, Sitzungbeichte de Peuichen Akademie de Wienchaften zu Belin: (The field equation of gavitation, Poceeding of the Puian Academy of Science in Belin)

2 whee the ign i needed becaue a paticle that ha finite ma tavel at le than the peed of light The vaiational pinciple i then δ Ldλ = 0, (15) whee The Eule-Lagange equation ae β dx dx L = γ gβ ( x ) dλ dλ (16) d L L 0, γ γ = dλ x x whee x = dx dλ Uing the expeion (16), we get the equation of motion to be (17) β d 1 dx 1 dx dx gβ gγ 0 γ = (18) dλ L dλ L dλ dλ x Fo paticle of non-zeo ma, it i poible to chooe λ o that L i contant along the path In thi cae λ i called an affine paamete One uch choice i to ue the pope time τ = / c fo λ whee i the length along the woldline The equation of motion ae then β d dx 1 dx dx gβ gγ 0 γ = dτ dτ dτ dτ x Taking the deivative thi become β β d x gγ dx dx 1 gβ dx dx gγ + = 0 β γ dτ x dτ dτ x dτ dτ Thi can be witten in tad fom by noting that o that (19) (110) g 1 γ gγ g β β βγ β x x = x x + x x, (111) xβ xβ x d x 1 β gγ gβγ g β dx dx gγ = γ dτ xβ x x dτ dτ Pemuting the indice thi become whee g β β γ d x dx dx β βγ (11) + Γ = 0, (113) dτ dτ dτ 1 gβ gγ g βγ Γ βγ = + xγ xβ x Finally, by multiplying by the invee of the metic teno, we obtain (114)

3 3 whee β γ d x dx dx βγ + Γ = 0, (115) dτ dτ dτ δ Γ = g Γ (116) βγ The Γ ymbol ae call Chitoffel ymbol Clealy if the component of the metic teno ae known then we can find the equation of motion of the tet paticle If λ i an affine paamete, we can eplace the expeion fo L in equation (16) by an equivalent Lagangian β 1 dx dx L = γ gβ ( x ) dλ dλ (117) It i then imple to deive the equation of motion fom the Eule Lagange equation (17) than uing equation (115) which equie evaluation of the Chitoffel ymbol Fo each coodinate x, the conjugate momentum i L p = (118) x (Note the poition of the index of p ) Uing the Lagangian a defined by equation (117), we have β dx p = gβ (119) dλ We alo ee, by multiplying by the invee of the metic teno, that To get ome idea of the phyical meaning of p p, p δβγ β dx = (10) dλ conide a paticle of ma m moving in a two-dimenion flat pace decibed by pola coodinate The metic i d = d + dθ (11) The metic teno it invee ae 1 0 ij 1 0 gij =, g = 0 0 (1) Taking the affine paamete to be λ = t m, the Lagangian i o that 1 d L = m + dt dθ dt, (13) d dθ p = m, pθ = m, (14) dt dt

4 4 d θ dθ p = p = m, p = pθ = m (15) dt dt Hence in thi example the pi ae the adial component of the momentum vecto, the angula momentum, wheea the p i ae ma time the genealized velocity It i the pi that ae the phyically elevant paamete Fo a ytem of coodinate x 1, x, the coodinate bai vecto ae e =, (16) x whee i the poition vecto Since d = dx = e dx, we have x Fom which we ee that β d = d d = βdx dx e e (17) g β = e e β (18) If the coodinate bai vecto ae mutually othogonal then the metic teno i diagonal If the metic teno i diagonal, we can find a et of othonomal vecto tangent to line on which all but one of the coodinate ae held contant We will ue a caet to denote a unit vecto The unit vecto ae elated to the coodinate bai vecto by 1 eˆ = e (no um) (19) g Any vecto can be witten a Hence A = A e (130) A = A e = A e e = A g (131) β β β β [The A A ae the contavaiant covaiant component of A, epectively] Retuning to the -D poblem, the phyically meaued value of a component of the momentum in a paticula diection i the pojection of the momentum p along a unit vecto in that diection The θ-component of the (linea) momentum i eθ pθ dθ p eˆ θ = p = = m, (13) dt a it hould be The Schwazchild metic The pat of pace time outide the event hoizon of an unchaged, non-otating black hole i decibed by the Schwazchild metic 1 GM GM = + + θ + θ φ d 1 c dt 1 d d in d c c (133)

5 5 Fo lage, it tend to that of flat pace time in a pheical pola coodinate ytem The lape of pope time fo a tatic obeve (ie an obeve at fixed, θ, ϕ) i GM dτ = dt 1 (134) Thi how that a clock in the gavitational field of the ma un low compaed to a clock at infinity At the Schwazchild adiu, c = GM c, we ee that the clock top Thi i the location of the event hoizon Inide the event hoizon, it i impoible fo an obeve to be tatic Intead an obeve inide the event hoizon i dawn into the cental ingulaity Uing the Lagangian of fom (117) with λ = τ, the genealized momenta ae 1 GM dt GM d dθ dφ 1, 1,, in, θ φ θ pt = p = p = p = (135) c dτ c dτ dτ dτ wheea the genealized velocitie ae t dt d θ dθ φ dφ p =, p =, p =, p = (136) dτ dτ dτ dτ The equation of motion ae d GM dt 1 = 0, τ dτ c d (137) d 1 1 GM d GM dt GM 1 GM d d θ in d φ θ, = + + dτ c dτ dτ c c dτ dτ dτ (138) d d θ in co d φ = θ θ, dτ dτ dτ (139) d dφ in θ = 0 (140) dτ dτ Since the pace - time i pheically ymmetic, we expect the obit to lie in a plane We can oient the coodinate ytem o that the plane of the obit i the equatoial plane, ie θ = π Equation (139) i then tivially atified Fom equation (140) we find that dφ pφ = l (141) dτ i a contant of the motion, which i the angula momentum pe unit ma of the tet paticle Alo fom equation (137), we have GM dt E pt 1, = c dτ c (14)

6 6 whee E i a econd contant of the motion, which, becaue t i the pace-time coodinate conjugate to the time component of the 4-momentum, we identify a the enegy pe unit ma at infinity Rathe than uing equation (138), it i eaie to conide the expeion fo L in equation (117) Since we have ued an affine paamete a independent vaiable, L i a contant Fom the definition of pope time, we have Hence fom equation (117), o that c L = (143) β c 1 dx dx gβ, = dτ dτ (144) 1 GM dt GM d dφ 1 1 c = c + + c dτ c dτ dτ 1 1 GM E GM d l = c c c dτ Re-aanging we find = c + d E GM l 1 dτ c c In the local fame nea the tet paticle, the adial component of the velocity i 1 c d GM d c d c E GM l v = = 1 = = 1 +, dt c dt E dτ E c c the tangential component of the velocity i (145) (146) (147) dφ 1 GM dφ 1 GM dτ 1 GM lc v φ = = 1 = 1 l = 1 (148) dt c dt c dt c E Hence a the event hoizon i appoached, v c, v φ 0, which how that tet paticle that co the event hoizon fall adially into the black hole in the local fame Fo tet paticle that move puely in the adial diection l = 0, equation (146) can be olved exactly Hee we only conide the cae in which the paticle i at et at infinity In thi cae E c = 1, o that d c, dτ = ± (149) whee R i the Schwazchild adiu Fo a tet paticle falling into the black hole, 3 R c τ = + contant R 3 R (150)

7 7 Subtituting thi into equation (14) give Making the ubtitution Hence 1 1 dt R c = 1 d R = R u, thi become 4 c dt u = = u 1 + R du u 1 u 1 u R t c = + ln + contant R R R R (151) (15) (153) We ee that a R, the pope time emain finite but the coodinate time tend to infinity Hence an obeve falling with the paticle will ee the paticle co the event hoizon in finite time, wheea an obeve at infinity will ee the paticle take an infinite time to co the event hoizon To analyze othe kind of obit, it i ueful to intoduce an effective potential by witing equation (146) a 1 d GM GM l 1 E + 1 = E, + (154) dτ c c whee E = E c i the enegy pe unit ma with the et ma enegy ubtacted off We ee that the Newtonian eult i obtained in the limit c, that main geneal elativitic effect i a modification of the centifugal baie tem It i eay to how that the centifugal foce i zeo at = 3R A epeentative plot of the potential function GM GM l c R l R R V ( ) = = + c c R i hown below Hee x = /R y = V/c (155)

8 8 We can ee that the potential ha a minimum a maximum The minimum in the potential coepond to a table cicula obit the maximum to an untable cicula obit Alo note that inwad moving paticle with enegie geate than the maximum potential enegy will co the event hoizon be captued by the black hole Such behavio i an effect of geneal elativity doe not happen in Newtonian gavity We can find the location of the tationay point by diffeentiating the potential, whee l = l ( cr ) Thi i zeo when Real olution exit only if The olution ae ( ) 1 3 V x = +, l (156) c x x x x l x + 3l = 0 (157) l 3 (158) 3 x = l 1± 1, l (159) whee the olution with the + ign coepond to the minimum in potential Hence thee ae no table cicula obit with adiu < 3R Fom equation (154), the enegy of a cicula obit i given by Uing equation (157) to eliminate E 1 E = l 1 (160) c c x x x l give

9 9 E E 1 x 0 + = (161) c c x x 3 We can ue thi expeion to detemine how much enegy can be eleaed when a paticle, initially at infinity, pial lowly inwad towad a black hole (pehap in an accetion dik) to the lat table obit befoe plunging into the black hole At the lat table obit, x = 3 o the enegy eleaed pe unit ma, E E ( ) Fom thi we find enegy eleae by nuclea fuion, = 3, i given by the olution of E E = c c 9 E c = Thi i much lage than the maximum poible (16) Enuc c 0009 Hence the mot luminou teady object in the Univee, uch a quaa, ae believed to be poweed by accetion onto black hole Accetion adiu of a Schwazchild black hole Conide a paticle at infinity with velocity v The impact paamete b i the cloet ditance the paticle would get to the black hole if thee wee no inteaction with the black hole The accetion adiu, RA, i the laget value of b fo which the paticle coe the event hoizon of the black hole If v c, the kinetic enegy at infinity i mall compaed to the et ma enegy of the paticle We can then take E = 0 Alo the angula momentum pe unit ma, l bv Since the mechanical enegy of the tet paticle ha been taken to be cloe to zeo, the tet paticle will be captued by the black hole if the maximum value of the potential i le than o equal to zeo The potential i zeo when x l x + l = 0, (163) the potential i tationay when Thee equation ae imultaneouly atified when x l x + 3l = 0 (164) x =, l = cr = 4 GM c (165) Hence paticle ae captued if bv < 4 GM c The co-ection fo accetion of non-inteacting paticle i theefoe σ BH 16π G M = π RA = (166) c v Fo a upemaive black hole at the cente of a galaxy, the co-ection i

10 10 σ BH 10 M v ly M 100 km (167) Null geodeic the bending of light Fo a male paticle uch a a photon, we cannot ue the pope time, τ, a the independent vaiable becaue it i a contant Howeve we can deive paametic equation of motion fo a male paticle fom thoe fo a maive paticle by uing the affine paamete λ = τ m a the independent vaiable then letting m 0 Fo the maive paticle, equation (141), (14) (146) ae dφ lm, dλ = (168) In the limit m 0, thee become GM dt Em 1, = c dλ c ( Em) ( ml) d GM = m c + 1 dλ c c (169) (170) dφ lγ, dλ (171) GM dt Eγ 1, = c dλ c (17) d Eγ GM lγ = 1, dλ c c (173) whee E γ, l γ ae the enegy angula momentum of the photon (at infinity), epectively Suppoe the photon ha momentum p The enegy angula momentum of the photon ae elated to p by E = γ pc, (174) l pb, γ = (175) whee b i the photon impact paamete Eliminating p give that l c E = γ γ b (176) Uing thi to eliminate the enegy, equation (171), (17) (173) become

11 11 d 1, d lγ φλ = (177) ( ) GM dt 1 1, = c d l γ λ bc ( ) (178) d 1 GM 1 1, d ( l γ λ = (179) ) b c which how that the tajectoy depend on the photon enegy angula momentum only though the impact paamete The ight h ide of equation (179) ha a maximum value when d GM 1 GM GM d c = = (180) c c Hence the maximum occu at = 3R, the maximum value i If the ight b 7 R h ide of equation (179) i alway poitive then an incoming photon will make it to the event hoizon will be captued by the black hole Thi will happen fo b < 3 3R, hence the captue co-ection fo photon i 7π GM γ = R = (181) σ 7 π 4 c Equation (177) (179) can alo be ued to detemine how much the Sun deflect light fom ditant ta Thi can be meaued duing total eclipe povide one of the tad tet of geneal elativity The two equation give Making the ubtitution Now let u = in θ, o that = b u give 4 R 4 d = 1 dφ b b b du = + dφ b R 3 1 u u (18) (183) 3 dθ R in 1 θ = +, (184) dφ b co θ with initial condition θ = 0at φ = 0 In the abence of gavity, R = 0 the olution would be θ = φ, which coepond to taight line motion y = in φ = b, a it hould Fo photon tajectoie that do not hit the Sun, R 6 b will alway be mall, < 510 Hence let

12 1 which give Thi ha appoximate olution R θ = φ + χ ( φ ) (185) b 3 in φ R O = + d χ dφ co φ b (186) 1 1 χ = + co φ (187) coφ Thi give R 1 R u = in θ = in φ + co in ( 1 co ) b χ φ = φ + b φ (188) Hence the photon eache infinity at φ = π + ε, whee R ( ) 1 0 in ( 1 co ( )) R = π + ε + π + ε ε + (189) b b Hence the deflection of the photon i ε = R b Fo photon that gaze the ola uface, the deflection i about 175 ac econd A team led by A S Eddington meaued a deflection of 161 ac econd in the 1919 ola eclipe, confiming the pediction of Eintein theoy The Ke metic The Ke 3 metic decibe the cuvatue of pace-time aound an unchaged black otating hole In Boye-Lindquit coodinate 4, the metic i whee R R d c dt a cdtd d d R + dθ + + a + a in θ in θdφ, = 1 in θ φ + (190) J a =, (191) Mc a co θ, = + (19) GM d = + a, (193) c Dyon, F W, Eddington, A S Davidon, C 190, Philoophical Tanaction of the Royal Society of London Seie A, Containing Pape of a Mathematical o Phyical Chaacte, 0, 91 3 Ke, R P 1963, Phyical Review Lette 11, 37 4 Boye, R H Lindquit R W 1967, J Math Phy 8, 65

13 13 RS i the Schwazchild adiu Hee M J ae the ma angula momentum of the black hole The quantitie a,, d all have dimenion of length In the limit of no otation, the metic become the Schwazchild metic The Boye-Lindquit coodinate ae elated to the Cateian coodinate by x = + a y = + a inθ co φ, inθ in φ, z = co θ A uface of contant i the uface of an oblate pheoid The metic ha two ingulaitie The puely adial element of the metic teno g i ingula when d = 0, ie at (194) RS + RS 4a = Rh (195) Thi define the event hoizon fo the Ke black hole To avoid a naked ingulaity, Rh mut be eal, which equie that a R S, o equivalently that A black hole with J = GM c i called maximally otating GM J (196) c The othe ingulaity occu when the puely tempoal component of the metic teno gtt change fom negative to poitive Thi happen when R + a co θ = 0 (197) The phyically elevant olution of the quadatic equation i R + R 4a co θ = Re, (198) with the othe olution lying inide the event hoizon Becaue of the coine dependence, R R with equality at θ = 0, ie on the otation axi The uface defined by = Re i called e h, the tatic limit o the bounday of the egophee To undet the ignificance of the tatic limit, conide a tationay obeve who tay at fixed θ but otate aound the black hole with contant angula velocity Ω We then have fo the pace-time element ( ) d c dτ = g dt + g dtdφ + g dφ = g + g Ω + g Ω dt (199) tt tφ φφ tt tφ φφ Fo the obeve to follow a time-like wold line, the expeion inide the paenthee mut be negative Since g φφ i poitive, thi i poible only if Ω lie between the oot of the equation g + g Ω + g Ω = (1100) tt tφ φφ 0 One of the oot will be zeo when 0, tt g = ie on the tatic limit Static obeve cannot exit between the event hoizon the tatic limit Stationay obeve can exit thee, with thei otational velocity being one of the two olution of equation (1100) Stationay obeve

14 14 cannot exit inide the event hoizon The figue how (in the x - z plane) the tatic limit (blue line) the event hoizon (ed line) fo a maximally otating black hole The length unit i the Schwazchild adiu 08 Event hoizon Static limit 04 z x Fame dagging The metic can alo be witten a d = g dt + g dtdφ + g d + g dθ + g dφ, (1101) tt tφ θθ φφ g tφ g tφ d = gtt dt + gd + gθθ dθ + g φφ dφ + dt g φφ g φφ Thi i equivalent to an othogonal metic in a fame co-otating with angula velocity gt φ Rac Ω = = g a a co R a in φφ + + θ + θ ( )( ) (110) (1103) Thu an inetial fame i entained to paticipate in the otation of the black hole Thi fame dagging, which caue the Lene-Thiing peceion of a gyocope, ha been obeved 5 Tet paticle motion in the Ke metic 5 Cui, W, Zhang, S N, Chen, W, 1998 Atophyical Jounal Lette 49, L53; Ciufolini, I, Pavli, E, Chieppa, F, Fene-Vieia, E, Péez-Mecade, J 1998 Science, 79, No 5359, 100

15 15 Reticting attention to the equatoial plane, the metic i R R R d = 1 c dt acdtdφ d a a dφ d Intoducing an affine paamete λ, the Lagangian i given by dt dt dφ d dφ R R R L = 1 c c a a + a dλ dλ dλ d dλ dλ (1105) Becaue L doe not depend on the coodinate t φ, thee ae two fit integal Thee lead to R dt R dφ pt = c ac = E dλ dλ 1, R dt R dφ dλ dλ pφ = ac + + a + a = l (1104) (1106) (1107) dt R R ( R + a ) c = + a + a E acl, (1108) dλ dφ R R ( R + a ) c = 1 cl + ae (1109) dλ (Note that thi equation how that, a a conequence of fame dagging, φ change even fo paticle of zeo angula momentum) The thid integal of the motion come fom L being a contant Taking λ = τ m, give L = m c, alo that E l ae the enegy angula momentum of the tet paticle The contancy of L give, afte much algeba, d 4 d R c l R a a R = c m c 1 acle 1 E (1110) dλ The ight-h ide can be conideed a an effective potential fo adial motion We can wite it a whee ( ) V ( ) U =, (1111) 3 ( ) ( ) ( ) ( ) = (111) U m c R a R c l R acle a a R E Fo a cicula obit, i contant hence it deivative with epect to λ i zeo Thi mean that fo a cicula obit, U ( ) = 0 Alo the potential mut be tationay, which equie that ( ) 0 U = Thee two condition give

16 16 ( ) ( ) ( ) m c R + a + R c l + R acle + a + a R E = 0, (1113) ( ) ( ) 4 m c 3 R + a + c l 3 + a E = 0 (1114) Thee equation can be olved fo E l in tem of, the adiu of the cicula obit Since the algeba i mey, we will only conide the pecific cae of the maximally otating black hole, fo which a = R Uing a a the length unit, mc a the unit of enegy amc a the unit of angula momentum, equation (1113) (1114) become ( ) ( ) l 4lE E = 0, (1115) ( ) + + l + E = (1116) Befoe poceeding, it i intuctive to conide the olution at the event hoizon, which in ou adopted unit i located at = 1 At thi point, the two equation above ae ( l E) l = 0, (1117) 4E = 0, (1118) which have the degeneate olution l = E Since the metic in the fom above i valid only outide the event hoizon, cicula obit olution apply only fo > 1 Eliminating l fom equation (1115) (1116) give 3 ( 3 + 1)( 1) ( 3 ) E l = (1119) E Uing thi to eliminate l fom equation (1116) give Solving fo E give ( ) 3 E 4 ( ) E ( ) = 0 (110) E = ( ) ± ( 4) (111) Equation (111) give the enegy of a cicula obit of adiu Clealy thee ae two banche to E ( ), which ae hown in the figue below The olid line i the olution with the negative ign in font of the quae oot Note that thi olution i egula at = 4, but the othe olution i ingula thee

17 17 0 Pogade Retogade 15 E The figue below how the angula momentum fo the two banche We ee that fo one banch the obit i pogade (l > 0) fo the othe banch the obit i etogade (l < 0) The etogade obit exit only fo > 4a = R 4 0 Pogade Retogade l

18 18 We can detemine whethe a cicula obit i table o untable by conideing the econd deivative of the effective potential Since the potential it fit deivative ae both zeo fo a cicula obit, the condition fo tability i U ( ) 0, which give 1 E (11) 3 Thi condition i atified fo all the pogade obit Hence fo the maximally otating black hole, an accetion dik can extend all the way to the event hoizon Compaing the enegy of the cicula obit with that at the event hoizon, we find that the efficiency of enegy eleae i 1 E ( ) E ( 1) = 1 = 043, (113) 3 which i ignificantly highe than fo the non-otating black hole