PHYS GENERAL RELATIVITY AND COSMOLOGY PROBLEM SET 7 - SOLUTIONS

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1 PHYS 54 - GENERAL RELATIVITY AND COSMOLOGY PROBLEM SET 7 - SOLUTIONS TA: Jeome Quinin Mach, 07 Noe ha houghou hee oluion, we wok in uni whee c, and we chooe he meic ignaue (,,, ) a ou convenion.. The Schwazchild meic i given by d g µ dx µ dx d d in d, () whee. The fou-velociy of a paicle i given by u µ dxµ, () bu ince i fall adially, we have d/ d / 0, hence u µ, d, 0, 0. (3) We know ha a maive paicle will follow a imelike geodeic, o i mu aify u µ u µ ) g µ dx µ dx ) g g d. (4) Noe ha o u µ u µ become " # d g g ) d d, (5) " d g #. (6) g Wih he meic given in euaion (), we find d " #. (7) The coodinae velociy i given by d, o ince he moion i inwad, we ake d " # /. () Locally, he meic of a aionay obeve i Minkowki, d µ d x µ d x d d d in d, (9) and ince we conide adial moion (o and ae kep conan), i i imply d d d. (0) Fo ueion o if you find ypo, pleae me a juinin@phyic.mcgill.ca.

2 Maching hi wih he Schwazchild meic a adiu, he local obeve meic much aify d d d d, () and hu, fom which i follow ha d d and ecalling he eul of euaion (), we find d d d d v u, () d, (3) d, (4). (5). Hee, we have a combinaion of a pecial elaiviic Dopple e ec and a geneal elaiviic Dopple e ec. The fi e ec i a anvee Dopple e ec ince i i a an angle, which can be expeed in em of he fomula 0 (ṽ)( ṽ co ), (6) whee 0 i he feuency in he fame a e, i he feuency in he fame in moion, and ṽ i he locally meaued velociy. The econd e ec follow euaion (5.03) fom Caoll, /. (7) / We ae old ha he e feuency i given by 0 (hi i he e feuency in he ocke hip fame), o 0, and calling 0 he feuency fo an obeve a e due o he pecial elaiviic Dopple e ec, we have 0 (ṽ)( ṽ co ), () and he obeved feuency a infiniy i hen (call i o a!and wih ), / o lim 0 (ṽ)( ṽ co ). (9)! / We need o deemine ṽ. Accoding o Keple hid law, which hold fo cicula obi in a Schwazchild paceime, he angula velociy i given by d v, (0) whee euaing he (Newonian) gaviaional foce o he cenifugal foce yield G N mm mv ) v GN M ) d v GN M 3. () Then, in a imila appoach o ueion, we know ha he locally meaued velociy will be ṽ d d d GN M 3 Theefoe, o 0 G NM GNM G NM ( G co NM/) G NM ( G NM/) ( /). () G NM. (3) See you favoie exbook o lecue noe on pecial elaiviy, e.g., Special Relaiviy by V. Faaoni (03). Thi i in mo exbook on claical mechanic a well, e.g., Taylo o Kleppne & Kolenkow. We give he ocke hip a ma m, buhiiielevanfoolvinghepoblem.

3 3. The Kukal coodinae T and X ae elaed o he uual Schwazchild coodinae and by T (, ) e R X(, ) e R / inh, (4) / coh, (5) valid fo >. The Schwazchild meic in Kukal coodinae fo a egion ouide he hoizon i hen given by 3 d 4R3 e /R dt dx d (), (7) which i egula a he hoizon ( ). The idea now i o exend he pace-ime o a egion inide he hoizon (0 << ), o le u find new coodinae T and X ha decibe he ineio of he black hole uch ha hey mach he uual Kukal coodinae a he hoizon, i.e., fomally, we wan T and X uch ha Le u y he coodinae lim T (, ) lim T (, ),!! T (, ) e R X(, ) e R lim X(, ) lim X(, ). ()!! / inh, (9) / coh, (30) which ae valid fo 0 <<, and which aify he condiion of euaion (). Then, one can compue d T (, d T (, X(, ) e R e R apple apple coh inh inh coh d, (3) d, (3) and o, he meic given by euaion (7) wih T eplaced by T and X eplaced by X become afe implificaion d R R d d (), (33) which i of he ame fom a he uual Schwazchild meic ouide he hoizon, bu valid inide he hoizon fo 0 <<. 4. The Reine-Nodöm meic i given by d g µ dx µ dx d d (), (34) whee The hoizon of hi black hole ae given by ± ± G N(Q P ). (35) G N M G N (Q P ). (36) When he chage i geae han he ciical chage, i.e. when Q P >, hee i no hoizon, and one find a naked ingulaiy. We now wan o how ha geodeic of maive paicle feel a epulive foce fom hi naked ingulaiy. Fo impliciy, le u eic ou aenion o adial moion, o we e and o be conan. To find he geodeic euaion, we need o compue he Chio el ymbol accoding o µ gµ (@ g ), (37) 3 Recall ha d () d in d. (6) 3

4 and o, we find ha he only non-zeo ymbol involving he and coodinae ae,,, (3) whee G N Thu, he geodeic euaion ha follow fom M Q P. (39) ae d d 0, d x µ µ dx " d dx 0 (40) # d 0. (4) Fuhemoe, a maive paicle will follow a imelike geodeic, o i mu aify u µ u µ ) g µ dx µ dx ) Thu, he adial geodeic euaion become Noe ha (ecall >0) < 0, M< Q P d ) d. (4) d. (43), < Q P M, (44) o when <(Q P )/M, we find ha d > 0, (45) meaning ha he maive paicle would feel a epulive foce below a ceain non-zeo poiive adiu, which implie ha he maive paicle would neve each he naked ingulaiy. 5. We ae given he Lagangian deniy L apple (@ µ h µ )(@ h) (@ µ h )(@ h µ ) µ (@ µ h )(@ h ) µ (@ µ h)(@ h), (46) whee h µ h µ. The acion i hen S d 4 x p g L, (47) whee g µ µ h µ, o o leading ode in mall h µ,wehave S d 4 x p L d 4 x L, (4) ince de( µ ). Thu, he acion in full i S h d 4 x (@ µ h µ )(@ h ) (@ µ h )(@ h µ ) µ (@ µ h )(@ h ) i µ! apple (@ µ h! )(@ h apple ) (49) apple I I µ I 3 µ µ! apple I!apple 4 µ. (50) Le u vay and evaluae each inegal epaaely. Fi, I d 4 x [(@ µ h µ )(@ h )(@ µ h µ )(@ h )], (5) 4

5 o inegaing by pa 4,wefind I d 4 x h µ (@ h ) d 4 x h µ h µ ), (54) o I d 4 x h µ (@ h) d 4 x h µ µ h ), (55) afe enaming ome of he dummy indice. Second, again uing inegaion by pa, we find I d 4 x [(@ µ h )(@ h µ )(@ µ h )(@ h µ )] d 4 x h (@ h µ ) d 4 x h µ µ h ), (56) o I d 4 x h (@ h µ ) d 4 x h µ µ h ) d 4 x h µ h µ) afe enaming ome of he dummy indice. Thid, uing he ame echniue, I 3 µ d 4 x [(@ µ h )(@ h )(@ µ h )(@ h )] d 4 x h (@ h ) o µ I 3 µ d 4 x h µ µ h ), (57) d 4 x h µ h ), (5) d 4 x h µ h µ ). (59) Fouh, o I!apple 4 µ d 4 x [(@ µ h! )(@ h apple )(@ µ h! )(@ h apple )] µ! apple I!apple 4 µ d 4 x h! (@ h apple ) d 4 x h apple µ h! ), (60) d 4 x h µ µ h). (6) Theefoe, combining all he em, we find S d 4 x h µ (@ h) µ h h µ h ) h µ ) µ h). we can wie S d 4 @ µ h h µ µ h. (63) Theefoe, he euaion of moion ha eul fom S µ h h h µ h 0. (64) We ecognize ha he lef-hand ide i he lineaized Einein eno, G µ, a expec. Hence, we ecove he (lineaized) Einein field euaion in vacuum, G µ (a) Uing Newonian heoy (F G N m m / and F ma), we can ay ha whee he iniial condiion ae o 4 Fo example, noe ha d 4 x (@ µ h µ )(@ h ) F ± G NM x x M d x ±, (65) lim x ±() ±, lim v ±() 0. µ[( h µ )(@ h )] (@ µ h µ )(@ h ) h µ (@ µ@ h ), (5) d 4 x@ µ[( h µ )(@ h )] d 4 x h µ (@ µ@ h )0 d 4 x h µ (@ µ@ h ), (53) ince he fi em i a oal deivaive, i.e. i i a bounday em: i vanihe on he bounday limi whee h µ 0. 5

6 wih v ± () dx±().byymmey, x x,o bu we noe ha hence G NM 4x ± ) G NM 4 ) G NM 4 ) ± G NM dv ± v ± dx ± x± apple ± d x ± x ± x ± v± 0 lim x ±!± G NM x ± M d x ±, (67) d x ± dv ± dv ± dx ± dv ± v ±, (6) dx ± dx ± dṽ ± ṽ ± [he limi of inegaion ae aken in accodance wih E. (66)] x ± v ± v± x ± ) dx ± ± G NM x ± x± p GN M ) d x ± ± x± d (uing he fac ha he colliion occu a x 0when 0) 0 0 ) 3 (±x ±) 3/ GN M ) ±x ± 3! /3 GN M /3 9 GN M ) x ± ±. (69) 6. (b) The Newonian appoximaion i eaonable a long a he pecial and geneal elaiviic e ec ae no oo ong, i.e. a long a he peed i much malle compaed o he peed of ligh, and a long a he gaviaional poenial emain mall compaed o uniy, v ±, (70) ±. (7) Hee, he fome condiion implie [aking he eul fom pa (a)] x ± ) x ±, (7) while he lae implie x ± ) x ±. (73) 6. (c) The idea i o ue he uadupole fomula, whee h ij (, ~x) G N d I ij ( ), (74), and whee he uadupole momen eno i given by I ij () d 3 ~ i j T 00 (, ~). (75) We will wan o evaluae h TT xx () a ~x 0 (0,R,0), o p ~x 0 R. Uing he eul of pa (a), we can conuc he enegy deniy by aking he wo mae M o be poin mae (dela funcion): "! /3!# /3 9 T 00 GN M 9 (, ~x) M (y) (z) x GN M x. (76) 6

7 Then, I xx () " d 3 ~x x T 00 (, ~x) d 3 ~x x M (y) (z) " M dx x x 9 GN M x 9 GN M! /3!# /3 9 GN M x! /3!# /3 9 GN M x /3 9 GN M M, (77) o aking wo ime deivaive, one find hence h xx (, 0,R,0) G N R d I xx () 3 /3 3 /3 G N M 5 /3, (7) G N M 5 /3 4(G NM) 5/3. (79) 3 /3 R( R) /3 Finally, we mu conve h ij ino h TT ij. To do o, we ue he pojecion eno P ij ij n i n j,wheen i i he uni nomal veco ha poin ino he diecion of popagaion of he gaviaional wave, and we have 5 h TT ij TT h ij P k ` i P j P ijp k` h k`. (0) Since we ae woking in Caeian coodinae, we can ake ~n ~x/, o in ou cae, we have ~n (0,, 0), o he only non-zeo componen of he pojecion eno ae P xx and P zz, and o bu I zz 0, o h zz 0, o we ae lef wih h TT xx h TT h TT xx h xx h zz, () h xx. Theefoe, xx (, 0,R,0) (G NM) 5/3. () 3 /3 R( R) /3 5 Noe ha indice can be loweed and aied wihou any change ince we ae woking in fla pace. 7

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

ASTR 3740 Relativity & Cosmology Spring Answers to Problem Set 4. ASTR 3740 Relativity & Comology Sping 019. Anwe to Poblem Set 4. 1. Tajectoie of paticle in the Schwazchild geomety The equation of motion fo a maive paticle feely falling in the Schwazchild geomety ae