ounl of Moden Phyic,,, 74-8 doi:46/jmp46 Publihed Online Apil (http://wwwscirpog/jounl/jmp) Schwzchild Geodeic in Tem of Elliptic Function nd the Relted Red Shift Abtct Günte Schf Intitut fü Theoetiche Phyik, Univeität Züich, Switzelnd E-mil: chf@phyikunizhch Received Febuy 5, ; evied Mch, ; ccepted Mch 5, Uing Weietin elliptic function the exct geodeic in the Schwzchild metic e expeed in imple nd mot tnpent fom The eult e ueful fo nlyticl nd numeicl ppliction Fo exmple we clculte the peihelion peceion nd the light deflection in the pot-einteinin ppoximtion The bounded obit e computed in the pot-newtonin ode A topicl ppliction we clculte the gvittionl ed hift fo t moving in the Schwzchild field Keywo: Schwzchild Geodeic, Red Shift Intoduction Schwzchild geodeic e elliptic function, theefoe, they hould be witten uch Fo thi pupoe the Weietin elliptic function e mot ueful becue they led to imple expeion The eon fo thi i tht the olution of qutic o cubic eqution cn be voided in thi wy In ecent ppe [] n nlytic olution fo the geodeic in the wek-field ppoximtion w given A pointed out in tht ppe the poge in the tonomicl obevtion cll fo bette nlyticl metho In thi epect it i deible to hve the exct geodeic in fom mot uited fo ppliction Fo the obit in pol coodinte (next ection) thi gol cn be chieved by uing Weiet P-function fo which mny nlyticl nd numeicl metho e known [] Conideing the motion in time (ection 4) the elted - nd -function of Weiet ppe cobin elliptic function hve been ued by Dwin [8] fo the fom of the obit Afte ome tnfomtion ou eult () gee with hi But in hi econd ppe he bndon the elliptic function becue they wee not o well dpted to tudy of the time in thoe obit Obviouly the Weiet function e bette uited fo the poblem Indeed, expeing them by Thet function one get the ntul expnion of the geodeic in powe of the Schwzchild diu, thi expnion involve elementy function only The Weiet function hve lo been ued by Hgih [] But he h choen the vible nd contnt of integtion in mnne which le to le explicit eult So it i difficult to deive the pot-newtonin coection to the geodeic given hee fom hi fomul A topicl ppliction we finlly clculte the ed hift fo t moving in the Schwzchild field The geodeic e lo needed fo the tudy of modifiction of genel eltivity ([], ection 5) The Obit in Pol Coodinte We tke the coodinte x ct, x, x, x nd wite the Schwzchild metic in the fom d c d d in d () GM whee i the Schwzchild diu We hll c ume c in the following The geodeic eqution d x with the Chitoffel diffeentil eqution () le to the following thee I m indebted to C Lämmezhl nd P Fiziev fo binging thi efeence to my ttention Copyight SciRe
G SCHARF 75 d t d () d d d e e (4) d d d (5) Hee we hve ued the tndd epeenttion e (6) nd hve choen the plin of motion The Chitoffel ymbol cn be tken fom the Appendix of [] Multiplying () by exp we find e o tht Next multiplying (5) by hence e d t cont E d t Ee we get d cont L, d L (7) Fo the contnt of integtion we ue the nottion of Chndekh [4] Finlly, ubtituting (7) nd (8) into (5) nd mul- exp d we obtin tiplying by d d e L E e (9) Conequently, the que bcket i equl to nothe contnt b Then the eulting diffeentil eqution cn be witten L E e b d () The contnt b cn be bitily djuted by ecling the ffine pmete Below we hll tke b m whee m i the et m of the tet pticle Thi will enble u to include null geodeic (light y) with m Ech geodeic i chcteized by two contnt of the motion: enegy E nd ngul momentum L Tking the que oot of () nd dividing by (8) we get d E m 4 m f () d L L Now Howeve, it i bette to conide the invee cn be witten n elliptic integl in tem of elliptic function by uing fomul of Weiet ([5], p45) Let the qutic f be witten 4 f 4 6 4, () 4 nd let be zeo i given by f, then olution of () f P g g f Hee P ; g, g invint 4 ;, 6 () i Weiet P-function with g 4 4 (4) g (5) 4 4 In ou ce we hve 4 Fo the convenience of the ede we epoduce the hot poof in the Appendix The eult () i not yet the olution of ou poblem becue it contin too mny contnt: the invint g, g nd the deivtive of f depend on E, L, but in ddition the zeo ppe Of coue one could clculte function of E, L by olving the qutic eqution f, but thi give complicted expeion It i much bette to ue nd econd zeo contnt of integtion inted of E, L Thi i even deible fom the tonome point of view becue the zeo of deivtive () e tuning point of the geodeic, fo exmple in ce of bounded obit they cn be identified with the peihelion nd phelion of the obit In ode to expe E, L by, we wite ou qutic in the fom f (6) nd compe the coefficient of,, with () Thi le to Since we cn olve fo 4 m L 6 4 E m L (7) (8) Copyight SciRe
76 G SCHARF m L E m In ddition we obtin the thid zeo, (9) m () () The eltion (9-) llow to expe eveything in tem of, Fo the invint we find g m 4L 4 () m g 6 48L 6 () 6 48 6 Hee h to be ubtituted by () Fo the deivtive f, f which ppe in ou olution () we obtin f f (4) (5) With thee ubtitution the eult () give ll poible geodeic in the fom ;, Thi will be dicued in the next ection A fit check of the olution () we conide the Newtonin limit Let the two zeo, be el nd lge comped to the Schwzchild diu in bolute vlue Then neglecting O in the invint (-) the P-function become elementy ([], p65, eqution 87): P ;,6 4in co (6) The leding ode in the deivtive of f i given by f f (7) It i convenient to intoduce the eccenticity by (8) Uing ll thi in () we find the wellknown conic co (9) Auming both zeo, poitive nd < we hve < nd the obit i n ellipe with peihelion nd phelion In the hypebolic ce > we ee fom (8) tht if i poitive mut be negtive Then thee i only one phyicl tuning point which i the point of cloet ppoch The ltte lwy coepon to The eltivitic coection to (9) e clculted in the following ection Dicuion of the Solution The olution () i n elliptic function of which implie tht it i doubly-peiodic ([], p69 o ny book on elliptic function) The vlue of the two hlf-peio, depend on the thee oot of the fundmentl cubic eqution 4e ge g () Agin it i not necey to olve thi eqution becue the olution ej, j,, cn be eily obtined fom the oot,,, of ou qutic f To ee thi we tnfom f to Weiet' noml fom follow Fit we et x o tht fom (6) we get f 4 4x 6x 4x x Next we emove the quic tem by intoducing x e () Thi give the noml fom of Weiet f 4 e g 4 e e g, () with the bove invint (4-5) Tht men oot of f e imply elted to oot of () by the tnfomtion e j 4 j j (4) The cubic eqution () with el coefficient h eithe thee el oot o one el nd two complex conjugted oot The fit ce occu if the diciminnt g 7g (5) i poitive, in the econd ce i negtive In tem of the oot i given by ([], p69, eqution 88) e e e e e e 6 (6) Copyight SciRe
The phyiclly inteeting obit coepond to the fit ce of el oot If we hve two complex conjugted * zeo then (8) implie tht the eccenticity i imginy Such obit hve been dicued by Chndekh ([4], p) Now we dicu the viou ce Bound Obit In thi ce we hve two poitive tuning point > >, conequently thee e thee el oot e >> e > e given by e (7) 4 6 4 e, e, (8) 4 4 Ou convention i choen in geement with [] The el hlf-peiod of the P-function i given by ([], p549, G SCHARF 77 whee K k i the complete elliptic integl of the fit kind with pmete k e e ee () A fit ppliction let u give the pot-einteinin coection to the obitl peceion If k () i mll we cn ue the expnion ([], p59, eqution 7) π Kk k k 4 () 4 Fom the oot e, e we find eqution 898) O e e 8 K k e 4 t g e tg e (9) Thi finlly le to the hlf-peiod 8 π π 4 8 8 O 8 () The peihelion peceion i given by π T hen the ode in () i Eintein' eult nd the O give the coection to it The ccute comput- tion of the hlf-peiod i necey to contol the obit in the lge To compute the eltivitic coection fo fom () we expe the P-function by Thet function ([ 5], p464) zq q /4 z q z q 6 z /4 6 zq, q co zq cozq co5z 8 zq, qco zq co 4zq co 6z, in in in 5 8 4 zq, q co zq co 4zq co 6 z () Hee q i the o-clled Nome ([], eq 7) k k q 8 ( 4) 6 6 Thee eie e pidly conveging ince k i mll (), they give the ntul expnion in powe of the Schwzchild diu Now the P-function i given in tem of Thet function by ([], eq 85) P e π π 4 co Oq 4 4 4q, in (5) whee π (6) Uing f thi le to 5 4 6 f (7) f in P co in O 4 4 in co (8) Copyight SciRe
78 G SCHARF Subtituting thi into () give the deied obit to O in co in co co co (9) It i impotnt to inet the peiod in (6) ccoding to () in ode to decibe the peihelion peceion coectly If the two oot coincide, it follow fom (4) tht f Accoding to () we then hve cicul motion If ll thee zeo coincide then () give which i the innemot cicul obit Unbound Obit In thi ce thee i only one phyicl point, the point of cloet ppoch The othe oot i negtive, theefoe, it i bette to ue the eccenticity (7) the econd bic quntity With given by () we then hve > > >, () becue The peiodicity of () in i now elized by jump to n unphyicl bnch with < In elity comet move on one bnch only, but it i ticky po blem to decide on which one Thi i due to the fct tht the peiod diffe little fom π in the bounded ce Conequently, neighboing phyicl bnche > e little otted gint ech othe nd the ditinction between them i not ey The quntity of phyicl inteet i the diection of the ymptote It follow fom the oiginl eqution () by integting the invee ove fom to d () f Thi i n elliptic integl which Legende noml fom cn be tnfomed to d () k in by the tnfomtion ([6], volii, p8) in, in The pmete nd k in () i given by k, () (4) The integl () i n incomplete elliptic integl of the fit kind F, k (6) which h the expnion ([6], volii, p) (5) k F 4, k in O k (7) 4 Fo mll we find k O Thi give nd O in co in (8) It i convenient to clculte co O (9) The leding ode i the Newtonin ymptote of the hypebol Null Geodeic Fo m thee i only one contnt of integtion in th e qutic () 4 f d which i the o-clled impct pmete L d E () Copyight SciRe
G SCHARF 79 Now it i necey to clculte the oot of f Thi i eily done by men of powe ei e expnion We find whee c d c c O d 8 d O 8 d, () () nd we hve odeed the zeo in the me wy in () Then in the lt ubection we cn clculte the diection of the ymptote () which now i equl to d d F, k () with given by () in O (4) 4 nd k by (4) 5 k (5) 8 nd by (5) 9 d 8 We wnt to clculte the light deflection in the pot- Einteinin ppoximtion Uing gin the expnion (7) we hve (6) 9 in 8 Fom (4) we obtin π O 4 4 Then up to O we find π π (7) 4 6 The deflection ngle i given by π π 8 (8) Inted of the impct pmete d in () we would like to ue the ditnce of cl oet ppoch (4) in the fom The two e elted by which le to The fit tem (9) O π (4) 8 i Eintein eult 4 The Motion in Time t t By dividing (7) by () we find d t L E E m d W e chooe t nd ge t E L f (4) t the point of cloet ppoch t x x E L f x x (4) Thi i um of elliptic integl of fit, econd nd thid kind The coodinte time i given function of by clculting thee Howeve, we wnt t function of nd, theefoe, ue the ubtitution (67) of the Appendix gin f x 4P f 6 f x d, (4) whee the lt eltion follow fom () The lt integl O in (4) i mll coection nd we neglect it t the moment Then integl of the following fom emin to be clculted n du (44) n P u P v whee we hve et f 4 Such integl e known ([7], vol4, p9-) P v (45) Copyight SciRe
8 G SCHARF v log v Pv v v v P v P v P v (46) (47) Thee eult e eily veified by diffeentiting nd uing ddition fomul Of coue i jut the pol ngle Then (4) le to the deied eult fo t : whee E t O (48) L (49) f 4 f 6 (4) (4) Agin we evlute thi fo bounded obit in the pot-newtonin ppoximtion by men of the expnion in Thet function The quntity v in (48-) i given the zeo of (8) Intoducing we find π V v (4) co V (4) Since <, V i complex: V lo Uing ([], eq86) P we obtin v i g π ib (44) V V V V π 4 4 4 π cov O 4 in q V π Pv i O 4 5 i (45) Similly we clculte whee nd v v fom ([], eq87) v π V V (46) π z fom ([], eq88) z Z z Thi implie πz exp, Z π v in V log v log v in V whee i given by Then co ib v log co ib v i, ctn tn tnh b (46) i equl to O q 4 O (47) (48) (49) (4) π π To expnd thi in the pot-newtonin ode we fit clculte fom tn tn tn Intoducing we get ctn tn in 4 (4) A befoe in (9) we do not expnd in (4) Howeve, if one doe o one fin contibution O ctn tn co (4) whee the fit tem, y N, i the pmete which ppe in Newtonin mechnic (Keple eqution, ee Copyight SciRe
9)) Now the expnion of G SCHARF below (4 i given by which lo follow diectly fom the definition (44) To expnd Pv (4) P v π in P v co 7 in we need 54 g P v6p v Fo we hve the imple finite limit 8 5 in π Thi finlly le to 4 in ) co 5 in 4 7 coco 65 8 8 (45) (46) Agin we do not expnd (6) in ode to keep the peihelion peceion pecie poible The limit fo i equl to 6 6 4in in co In the finl eult (48) fo the time t t t (47) the pe-fcto EL lo give coection: E L the tem popo- which follow fom (9-) In tionl to cncel t t in co (48) Appoximting Keple eqution t by thi i in geement with in (49) The pot-newtonin coection in (47) come fom viou plce To how thi we wite the eult in the fom 5 t t 4 A in (9) the pot-newtonin coection vnihe fo cicul motion 5 Gvittionl Red Shift The tudy of Schwzchild geodeic i elevnt fo the invetigtion of the ecently dicoveed S-t ne the (4) Glctic Cente ([] nd efeence given thee) Thee t move in the tong gvittionl field of the centl blck hole o tht genel eltivitic effect e obevble nd the Schwzchild metic g i fily good deciption of the itution The meuble quntity of inteet i the ed hift of pectl line in the light emitted by the moving t Theefoe we finlly conide Copyight SciRe
8 G SCHARF thi Let be the fequency of given tomic line fom the t nd the fequency of the me line obeved in the et fme of the glxy If i the velocity of the t, the two fequencie e elted by ([9] p8, equ 56) / g x / g X (5) We ume tht the obeve t X i f wy fom the cente uch tht the denominto cn be ppoximted by Fo t moving in the p lne π we hve d d g x e e (5) Fom () nd (7) we find nd (8) give d e e L E m d Le E Subtituting ll thi into (5) we ee tht L dop out nd we end up with the imple eult m m e E E (5) By () we cn expe E by the peihelion nd phelion E m whee i the mll coection () Then we finlly get / The lowet ode O i equl to O (54) (55) Since the lt tem i lwy mlle thn the econd one we indeed hve ed hift > Of coue, it i mxi- ed hift i obtined by multiplying (55) with ml t the peihelion whee i miniml The totl obeved the Dopple fcto v whee v i the component of the eltive velocity long the diection fom the obeve to the t ([9], p) 6 Acknowledgment It i pleue to cknowledge eluciing dicuion with Penjit Sh, in pticul the intoduction into the fcinting field of Glctic-cente t I lo thnk Rymond Angélil fo howing hi imultion of the coeponding dynmic 7 Refeence [] D D Ozio nd P Sh, An Anlytic Solution fo Wek-field Schwzchild Geodeic, Monthly Notice of the Royl Atonomicl Society, Vol 46, pp 787-79 [] M Abmowitz nd I A Stegun, Hndbook of Mthe- Dove Publiction, Inc, New Yok mticl Function, [] G Schf nd Gen Reltiv Gvit, Fom Mive Gvity to Modified Genel Reltivity, Genel Reltivity nd Gvittion, Vol 4, pp 47-487 doi: 7/74-9-864- [4] S Chndekh, The Mthemticl Theoy of Blck Hole, Oxfod/New Yok, Clendon Pe/ Oxfod Univeity Pe, 98 [5] E T Whittke nd G N Wton, A Coue of Moden Anlyi, Cmbidge Univeity Pe, 95 [6] A Edelyi et l, Highe Tncendentl Function, McGw-Hill Book Co, Inc, New Yok, 95 [7] Tnney, Molk, Fonction elliptique, Chele Publihing Compny, Bonx, New Yok, 97 [8] Ch Dwin, Poc Roy S London A 49 (959) 8, A 6 (96) 9 [9] S Weinbeg, Gvittion nd Comology, ohn Wiley, New Yok, 97 [] G Schf, Quntum Guge Theoie-Spin One nd Two, Google-Book () fee cce [] Y Hgih, pnee Aton Geophy 8 (9) 68 Copyight SciRe
G SCHARF 8 Appendix: Integtion of the Diffeentil Eqution We cloely follow Whittke nd Wton ([5], p45) With the nottion of the ppe (), let whee we hve whee () f x i ny zeo, f By Tylo' theoem, 4 6 A x A x f x A x A x 4 4, A, A A, o A Intoducing the new integtion vible we hve x, 4A 6A 4A A /, () To emove the econd tem in the cubic we et nd we get A A z, z A z A 4z A 4AAz / AAA A AA () (4) The coefficient of z nd z e jut the invint g, g (4-5) of the oiginl qutic Now the inveion of the integl give Weiet P-function Fom (6) nd (6) we hve z (5) P ; g, g z A A nd hence f 4P f 6 (6) (7) Copyight SciRe