ASTR 3740 Relativity & Cosmology Spring Answers to Problem Set 4.

Transcription

1 ASTR 3740 Relativity & Comology Sping 019. Anwe to Poblem Set Tajectoie of paticle in the Schwazchild geomety The equation of motion fo a maive paticle feely falling in the Schwazchild geomety ae given a 1 ) dt dτ = E, 1.1) dφ d dτ dτ = L, 1.) ) + V eff = E 1.3) whee V eff i the effective potential given by V eff = 1 ) ) 1 + L. 1.4) a) Check Ye, the equation ae the ame, with ome light diffeence in notation John Walke ue unit c = G = 1, and my V eff i hi Ṽ ). John Walke expee anwe in tem of the ma M of the black hole athe than it Schwazchild adiu, which in unit c = 1 i of coue b) Velocity at infinity = GM. 1.5) At lage ditance, the adiu and time t coincide with the pope adial coodinate and pope time meaued by obeve in the et fame of the Schwazchild geomety. A, the effective potential educe to and the elation between time t and pope time τ educe to V eff 1 + L, 1.6) dt dτ E. 1.7) Thu in the limit of lage the adial velocity v in the Schwazchild et fame atifie ) ) d d v dτ = E V eff = 1 1 dt dτ dt E E L E. 1.8) while the tanvee velocity v i v dφ dt = dφ dτ dτ dt L E, 1.9) 1

2 a wee to be hown. It follow that the velocity v at infinity atifie which eaange to v = v + v = 1 1 E, 1.10) E = 1, 1.11) 1 v ) 1/ a wa to be hown. A poitive v equie E > 1. If intead E < 1, then the paticle cannot go to infinity: it i in a bound obit. c) Extema of the effective potential The deivative of the effective potential V eff with epect to adiu i dv eff d = L + 3L. 1.1) 3 4 The extema of the effective potential occu whee thi deivative i zeo, which happen whee L + 3L = ) The olution of thi quadatic equation ae = L L ± L 3 ). 1.14) Real olution of the quadatic exit only if the abolute value of the angula momentum exceed the citical value L > L c = ) d) Sketch i) L < L c Paticle fall taight into the black hole, neve to etun. ii) L = L c Paticle fall taight into the black hole, except fo one citical cae, whee the paticle i obiting in a cicle at = 3. Thi citical obit i the innemot table cicula obit, and the outemot untable cicula obit. iii) L > L c Hee thee i a well in the effective potential within which paticle can ocillate. Such cae coepond to bound obit, with a peiapi minimum adiu) and an apoapi maximum adiu). If the enegy i highe than the height of the inne maximum in the potential, o if

3 1.00 Effective potential V eff.95 L/ = 1.9 L/ = 3 = 1.73 L/ = Radiu / Figue 1: Effective potential a a function of adiu, fo thee value of the angula momentum. The middle cuve coepond to the citical value L = L c of the angula momentum. the paticle tat inide the adiu of the inne maximum in the potential, then the paticle once again fall into the black hole. e) Cicula obit Cicula obit occu at extema of the potential. Accoding to equation 1.13), extema occu whee ) L 3 =, 1.16) which implie L = ) 1/, 1.17) 3 a claimed. The adial velocity i zeo in cicula obit, d/dτ = 0, wheeat E = V eff = 1 ) ) 1 + L. 1.18) Plugging the angula momentum 1.17) into thi expeion yield, afte a little algeba, a claimed. f) Obital peiod E = 1/ ). 1.19) [ 3 )] 1/ 3

4 The time meaued by an obeve at et at infinity i jut the Schwazchild time t. The azimuthal angle φ evolve a dφ dt = dφ dτ dτ dt = L 1 ) = L ), 1.0) E E 3 a claimed. Fo cicula obit, plugging in the expeion 1.17) and 1.19) give dφ dt = 1/ 1/ = GM)1/. 1.1) 3/ The obital peiod t i the time taken fo φ to change by π, which i Thi i exactly Keple 3d law t = π 3/. 1.) GM) 1/ GMt π) = 3, 1.3) a claimed. Note that the pope obital peiod τ expeienced by the paticle itelf, i of coue diffeent. g) Infall time τ = dτ dt t = )t E Fo L = 0 and E = 1, the equation 1.3) fo d/dτ educe to = 1 3 ) 1/ t, 1.4) ) d + 1 ) = 1, 1.5) dτ which eaange to d dτ = ) 1/, 1.6) whee I ve taken the negative quae oot becaue the paticle i falling inwad, o the adiu deceae a the time τ inceae. Thi give an integal equation fo the pope time τ to fall fom adiu to zeo adiu 0 ) 1/ τ = d, 1.7) whoe olution i [ 3/ τ = 3 1/ ] 0 = 3/ 3 1/. 1.8) 4

5 The cae L = 0 coepond to puely adial obit, while E = 1 coepond to v = 0 accoding to equation 1.11). So L = 0 and E = 1 coepond to paticle which fall adially fom et at infinity. h) Infall time numbe Fo =, the infall time i etoing eal unit with c) Fo M = M, I get not long!. Photon in the Schwazchild geomety τ = 3c = 4GM 3c ) τ = 6.3, 1.30) The equation of motion fo male paticle uch a photon feely falling in the Schwazchild geomety ae given in tem of the affine paamete λ a 1 ) dt dλ = 1,.1) dφ d dλ dλ = J,.) ) + V eff = 1,.3) whee J = L/E i the photon angula momentum pe unit enegy, and V eff = V eff /E i the effective potential given by V eff = 1 ) J..4) a) Cicula obit Cicula obit occu whee the effective potential i a minimum table obit) o maximum untable obit). The deivative of the effective potential V eff with epect to adiu i dv eff = d + 3 ) J,.5) 3 4 which i zeo whee = 3..6) Thu photon can obit in cicle at 1.5 Schwazchild adii. The econd deivative of V eff i d V eff 6 = d 1 ) J,.7) 4 5 which i poitive fo > and negative fo <. At = 1.5 the econd deivative i poitive, which mean that the extemum mut be a maximum. Hence the obit i untable. 5