Schwarzschild Geodesics in Terms of Elliptic Functions and the Related Red Shift

Home , Schwarzschild geodesics

Journal of Modern Physics, 2011, 2, 274-283 doi:10.4236/jmp.2011.24036 Published Online April 2011 (http://www.SciRP.org/journal/jmp)

Günter Scharf Institut für Theoretische Physik, Universität Zürich, Switzerland E-mail: [email protected] Received February 5, 2010; revised March 10, 2011; accepted March 15, 2011

Abstract

Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For example we calculate the perihelion precession and the light deflection in the post-Einsteinian approximation. The bounded orbits are computed in the post-Newtonian order. As a topical application we calculate the gravitational red shift for a star moving in the Schwarzschild field.

Keywords: Schwarzschild Geodesics, Red Shift

1. Introduction have also been used by Hagihara [11]1. But he has chosen the variables and constants of integration in a manner Schwarzschild geodesics are elliptic functions, therefore, which leads to less explicit results. So it is difficult to they should be written as such. For this purpose the derive the post-Newtonian corrections to the geodesics Weierstrassian elliptic functions are most useful because given here from his formulas. As a topical application we they lead to simple expressions. The reason for this is finally calculate the red shift for a star moving in the that the solution of quartic or cubic equations can be Schwarzschild field. The geodesics are also needed for avoided in this way. the study of modifications of general relativity ([10], In a recent paper [1] an analytic solution for the geo- section 5.13). desic in the weak-field approximation was given. As pointed out in that paper the progress in the astronomical 2. The Orbits in Polar Coordinates rr=   observations call for better analytical methods. In this respect it is desirable to have the exact geodesics in a We take the coordinates x0 = ct , x1 = r , x2 = , form most suited for applications. For the orbits in polar  x3 =  and write the Schwarzschild metric in the form coordinates (next section) this goal can be achieved by using Weierstrass’ P-function for which many analytical rr r d=sctrr22s d2 d 2222 d sind2(2.1) and numerical methods are known [2]. Considering the rrr motion in time (section 4) the related  - and  -func- s tions of Weierstrass appear. 2GM where r = is the Schwarzschild radius. We shall Jacobian elliptic functions have been used by Darwin s c2 [8] for the form of the orbits. After some transformation assume c =1 in the following. The geodesic equation our result (2.13) agrees with his. But in his second paper he abandons the elliptic functions because they were “not ddd2 xxx  =0 (2.2) so well adapted to a study of the time in those orbits”. ds2  ddss Obviously the Weierstrass functions are better suited for  the problem. Indeed, expressing them by Theta functions with the Christoffel  leads to the following three one gets the natural expansion of the geodesics in powers differential equations of the Schwarzschild radius, this expansion involves 1I am indebted to C. Lämmerzahl and P. Fiziev for bringing this refer- elementary functions only. The Weierstrass functions ence to my attention

ddd2ttr Taking the square root of (2.10) and dividing by (2.8)   =0 (2.3) ds2 ddss we get

22 2 22 2 2 drEmm 432 ddddrtr2     =.rrrrrrfr (2.11) ere=0 (2.4) 22ss 2     d ds 2dss 2d   d s  LL Now = r can be written as an elliptic integral. d2dd2r   However, it is better to consider the inverse rr=  2  =0. (2.5)   ds rssdd in terms of elliptic function by using a formula of Weier- Here we have used the standard representation strass ([5], p.452). Let the quartic f r be written as r fr=464 ar43 ar ar2  ara, (2.12) 1= s e (2.6)   01234 r and let r1 be a zero fr 1  =0, then a solution of (2.11) and have chosen  =2 as the plain of motion. The is given by Christoffel symbols can be taken from the Appendix of fr [3]. 1 rr=.1  (2.13) Multiplying (2.3) by exp we find 4;,Pggfr 23  1 6

  dt Here Pgg;, is Weierstrass’ P-function with e =0 23 ddss invariants 2 so that g20413=43aa aa a2 (2.14)

322  dt eE=const.= g302412320314=2aaa aaa a aa aa. (2.15) ds In our case we have a4 =0. For the convenience of the dt =Ee . (2.7) reader we reproduce the short proof in the Appendix. ds The result (2.13) is not yet the solution of our problem Next multiplying (2.5) by r 2 we get because it contains too many constants: the invariants g23, g and the derivatives of f depend on E, L, but in 2 d rL= const. = , addition the zero r1 appears. Of course one could cal- ds culate r1 as a function of E, L by solving the quartic hence equation fr  =0, but this gives complicated expressions. It is much better to use r and a second zero r d L 1 2 =. as constants of integration instead of E, L. This is even ds r 2 desirable from the astronomers point of view because the For the constants of integration we use the notation of zeros of derivative (2.11) are turning points of the geo- Chandrasekhar [4]. desic, for example in case of a bounded orbit they can be Finally, substituting (2.7) and (2.8) into (2.5) and mul- identified with the perihelion and aphelion of the orbit. In order to express E, L by rr, we write our quartic in tiplying by 2exp drs d we obtain 12 the form 2 2 ddrL2 eEe2 =0. (2.9) f rarrrrrrr = 012   3 (2.16) ddss r and compare the coefficients of rrr32,, with (2.12). Consequently, the square bracket is equal to another This leads to constant = b. Then the resulting differential equation can 2 be written as m 4=aarrr10123=2 rs 2 L drL2 =Eeb2  . (2.10)  2  6=a a rr rr rr  =1 (2.17) ds r 20121323 4=aarrrr =. The constant b can be arbitrarily adjusted by rescaling 30123s the affine parameter s. Below we shall take bm=  2 Since where m is the rest mass of the test particle. This will Em22 enable us to include null geodesics (light rays) a0 = 2 (2.18) with m2 =0. Each geodesic is characterized by two con- L stants of the motion: energy E and angular momentum L. we can solve for

2 Using all this in (2.13) we find the wellknown conic m rrr123 2 rs =, (2.19) L rr12 rr 13 rr 23 1  r r =1 . (2.29) mr2 1cos   Em22=s . (2.20) Assuming both zeros rr, positive and rr< we rrr123 12 12 have  <1 and the orbit is an ellipse with perihelion r In addition we obtain the third zero 1 and aphelion r2 . In the hyperbolic case  >1 we see rr from (2.28) that if r is positive r must be negative. rr=12 . (2.21) 1 2 3 s Then there is only one physical turning point r which rr12 rr 1s rr 2s 1 is the point of closest approach. The latter always corre- The relations (2.19-21) allow to express everything in sponds to  =0. The relativistic corrections to (2.29) terms of rr12, . For the invariants we find are calculated in the following section. 11m2 rrrr gr==2 s 123 (2.22) 2 124L2 s 12 4 rr rr rr 3. Discussion of the Solution 12 13 23 1 m2 a The solution rr=   (2.13) is an elliptic function of grr= 220 3 64832L ss16  which implies that it is doubly-periodic ([2], p.629 or (2.23) any book on elliptic functions). The values of the two 11rrrr 1 r 2 123s s half-periods , depend on the three roots of the =3 . 6 48rr12 rr 13 rr 23 16 rr 12  rr 13 rr 23 fundamental cubic equation Here r has to be substituted by (2.21). For the deriva- 3 3 4egeg23=0. (3.1) tives f r1 , f r1 which appear in our solution (2.13) we obtain Again it is not necessary to solve this equation because the solutions ejj ,=1,2,3 can be easily obtained from rr11213 r r  r the roots 0,rrr12,, 3 of our quartic fr=0. To see fr1 = (2.24) rr12 rr 13 rr 23 this we transform f r to Weierstrass' normal form as follows. First we set r=1 x so that from (2.16) we get rrrrr rrrr  121311213 fr1 =2 . (2.25) 1 32 rr12 rr 13 rr 23 f raxaxax=44 321 6 4 a0 x With these substitutions the result (2.13) gives all possi- Next we remove the quadratic term by introducing ble geodesics in the form rr=;, rr12. This will be discussed in the next section. 11a  ==xe 2 . (3.2) As a first check of the solution (2.13) we consider the  ra3 2 Newtonian limit. Let the two zeros r , r be real and 1 2 This gives the normal form of Weierstrass large compared to the Schwarzschild radius rs in abso- lute value. Then neglecting Or in the invariants a2 s f re=43 3 geg, (3.3) (2.22-23) the P-function becomes elementary ([2], p.652,  4 23 ea 12 equation 18.12.27): 2 with the above invariants (2.14-15). That means roots of 111113 P;,6=2 =  . f r are simply related to roots of (3.1) by the trans- 12 124sin 2 12 2 1 cos  formation (2.26) ar3 a2 s 1 The leading order in the derivatives of f is given by e j ==. (3.4) rrjj24 12 r1 f rr11= r2 The cubic equation (3.1) with real coefficients has ei- r 2 ther three real roots or one real and two complex conju- r  gated roots. The first case occurs if the discriminant  1 fr1 =23 2. (2.27) 32 r2 =27g2g3 (3.5) It is convenient to introduce the eccentricity  by is positive, in the second case  is negative. In terms of the roots  is given by ([2], p.629, equation 18.1.8) r1 1 =. (2.28) 2 22 r2 1  =16ee12 ee 23 ee 31. (3.6)

The physically interesting orbits correspond to the first where K k 2  is the complete elliptic integral of the case of real roots. If we have two complex conjugated first kind with parameter zeros rr= * then (2.28) implies that the eccentricity 21 ee rr is imaginary. Such orbits have been discussed by 2 23 21  kr==s Chandrasekh ar ([4], p.111). Now we discuss the various ee13 rrr 12s 2 rr 12 (3.10) cases. 1 rs 23 =1 rs . 3.1. Bound Orbits r1 11

As a first application let us give the post-Einsteinian In this case we have two positive turning points correction to the orbital precession. If k 2 (3.10) is rr>>0, consequently there are three real roots 21 small we can use the expansion ([2], p.591, equation eee>0> > given by 123 17.3.11) rr11 rr e ==ss12 (3.7) 1  22 412r3164rr2 22π 113   4 Kk=1  k   k  (3.11) 11rr 22  24  ee=,=ss  , (3.8)   2312 4rr 12 4 12 From the roots ee, we find Our convention is chosen in agreement with [2]. The real 13 half-period  of the P-function is given by ([2], p.549,  2  13rs 22rr1223 rr 12 equation 18.9.8) =2 1rOss r . 2 ee  28rr12  rr12   dt K k 13    == (3.9)  3 ee This finally leads to the half-period e1 4tg23tg 13

2 2 2  33rrrr 3313182 rr    = π 122=ss12 rr  rr rr π 1 ss Or3 (3.12) 2 12 21 21 22s  482118rr r   r 12 81rr12 1 1   

The perihelion precession is given by =2 π . (3.10), they give the natural expansion in powers of the

Then the order rs in (3.12) is Einstein's result and the Schwarzschild radius rs . Now the P-function is given in 2 Ors gives the correction to it. The accurate computa- terms of Theta functions by ([2], eq. 18.10.5) tion of the half-per iod is necessary to control the orbit in 2 π2 0  the large. Pe =  13 2 4 2 0  To compute the relativistic corrections for r   31 from (2.13) we express the P-function by Theta functions 1 r π2 1 =1s  4qO2 1q2 , ([5], p.464) 2 2 cos  12 4r1 4 sin  1/4 2 6 1 zq, = 2 q sin z q sin 3 z q sin 5 z  (3.15) 1/4 2 6 where 2 zq, = 2 q cos z q cos3 z q cos5 z π 38 =. (3.16) 3 zq, = 1 2 q cos 2 z q cos 4 z q cos 6 z  2 38 Using 4 zq, = 1 2 q cos 2 z q cos 4 z q cos 6 z  . (3.13)  r 3 2 fr=2 r s 111  r 2 Here q is the so-called Nome ([2], eq. 17.3.21) 1 1  222 kk 15  r 142 q =8 (3.14) fr =2 6s (3.17) 16 16 1 2  1  r1 1  These series are rapidly converging since k 2 is small this leads to

2 2 fr 1 12sin rs 1 32 rs  P=12 31cos2sin.O (3.18) 24 41  sin rr1 1 cos 2 2 1  1 

Substituting this into (2.13) gives the desired orbit to Or s  2 r  12rs sin1 332 =31cos2sin. (3.19) rr111cos2 1cos21c   os22  1 1

It is important to insert the period  in  (3.16) 2 according to (3.12) in order to describe the perihelion  =. (3.25) rr r precession correctly. 13 2

If the two roots rr12= coincide, it follows from The integral (3.22) is an incomplete elliptic integral of

(2.24) that fr1 =0. According to (2.13) we then have the first kind circular motion r= r1 . If all three zeros coincide  2 r123==rr then (2.21) giv es rr3 =3s which is the  =,F 2 k (3.26) innermost circular orbi t. a0 which has the expansion ([6], vol.II, p.313) 3.2. Unbound Orbits 2 24k 1 F 222,=kOsin2 2k .(3.27) In this case there is only one physical point, the point of 42 closest approach r . The other root r is negative, 1 2 For small rr we find therefore, it is better to use the eccentricity  (2.27) as s 1 2 2 r 2 the second basic quantity. With r3 given by (2.21) we s kOrr=  s 1 then have  1 r1 1  3 rs 2 rr13>>0>= r 2 r 1 , (3.20) =2Orr 1 s 1 a0 1  r1 because   1 . The periodicity of (2.13) in  is now realized by a jump to an unphysical branch with r <0. 2  1r3 sin2 = 1 . In reality a comet moves on one branch only, but it is a 2 r1 tricky problem to decide on which one. This is due to the fact that the period differs a little from 2π as in the This gives bounded case. Consequently, neighboring physical 11  r3 cos 22 = branches r >0 are a little rotated against each other r1 and the distinction between them is not easy. The quan- and tity of physical interest is the direction  of the asymptote. It foll ows from the original equation (2.11) by rs 3   =222  sin2 2 . (3.28) r 122 integrating the inverse over r from r1 to  1   dr It is convenient to calculate  =. (3.21) 2 r1 f r  11rs  1 2 cos=  3  2 Or . s (3.29) r 21 This is an elliptic integral which can be transformed to 1  Legendre’s normal form The leading order is the Newtonian asymptote of the   2 d hyperbola.  = (3.22)  22 a 0 1s k in 0 3.3. Null Geodesics by the transformation ([6], vol.II, p.308) For m2 =0 there is only one constant of integration in rrrr rr 22321 32 the quartic (2.11) sin = , sin2 = . (3.23) rr12rr 3 rr 12 4 r 2 2 f rrrr=  The parameter k in (3.22) is given by d 2 s r rr k 2 =,3 12 (3.24) which is the so-called impact parameter rr r 13 2 L d =. (3.30) and E

Now it is necessary to calculate the roots of fr  =0. (2.24) in the form This is easily done by means of a power series expansion rs 2 1 =. (3.39) rcdcrcr= 01ss 2  r1 We find The two are related by

 3 23 1 rd=1O 23 1   = 11O 1 28 2 which leads to  3 23 rd2 =1 O (3.31) 28 13 2  =21π 1. (3.40) 3 28 rd3 =,  The first term 21 is Einstein’s result. where r  = s (3.32) 4. The Motion in Time tt=  d and we have ordered the zeros in the same way as in By dividing (2.7) by (2.10) we find (3.20). Then as in the last subsection we can calculate the 12 direction of the asymptote (3.21) whic h now is equal to dtr r  L2  =1EE22s m 2  2  =dF  ,k (3.33) drrr s   rr    1    (4.1) Er3 with 1 given by (3.23) =. L rr s fr 2313 sin 1 =  O (3.34) 24 We choose t =0 at the point of closest approach 2 and k by (3.24) rr= 1 and get r 3 2225 Exd 22rs k =2  1 (3.35) txrxr=.ss (4.2) 8  Lx  r r1 fxs and  by (3.25) This is a sum of elliptic integrals of first, second and

29 2  third kind. The coordinate time is given as a function of r =1. (3.36) d 28 by calculating these. However, we want t as a function of  and, therefore, use the substitution (6.7) of the Ap- We want to calculate the light deflection in the post- pendi x again Einsteinian approximation. Using again the expansion  (3.27) we have fr1 xr= 1  (4.3) 4Pfr   1 6 92  1   =21 111    sin2 1  . 28 2  2  dx  =d , From (3.34) we obtain fx

π 3 3 where the last relation follows from (2.11). 1 =. O 3 44 The last integral Or s  in (4.2) is a small correction Then up to O  2 we find and we neglect it at the m oment. Then integrals of the  following form remain to be calculated π 33 2   =   π . (3.37) du 2416 J  = (4.4) n   n 0 Pu Pv The deflection angle is given by   where we have set π 33 2  =2 =2 π . (3.38) 228 f  r1 Pv=. (4.5) Instead of the i mpact parameter d in  (3.32) we 24 would like to use the distance of cl osest approach r1 Such integrals are known ([7], vol.4, p.109-110)

1  v  Similarly we calculate  v from ([2], eq.18.10.7) Jv1 =2  log (4.6) Pv  v v π 1V   v =  (4.16) 2 V 1 1 J =  vv 2  2   P v (4.7) where 2  2Pv P vJ . π 1 0  1   ==  (4.17) 12 0 These results are easily verified by differentiating and 1 using addition formulas. Of course J0  is just the and   z from ([2], eq.18.10.8) polar angle  . Then (4.2) leads to the desired result for 2 2zz1 Z π t  :  zZ=exp , =. (4.18) π 20 1 2 E 3 tJJ=.01122  Ors (4.8) L This implies where vV  sin 2 log = 2vO log q 22 vV  sin   01= rrrrs 1s (4.9)  cosib  r1 rs =2v log (4.19) 11=  f r (4.10)  cosib  24  2 22,vi   f  r1    2 =. (4.11) 16 where  is given by Again we evaluate this for bounded orbits in the  = arctan tan tanhb . (4.20) post-Newtonian approximation by means of the expansion in Theta functions. Then J1   (4.6) is equal to The quantity v in (4.8-11) is given as the zero of 2 2 23 1 r (3.18). Introducing 24 s JO1 =1  . ππ232 r π  1 1 Vv= (4.12) 2 To expand this in the post-Newtonian order we first we find calculate    from

131r   11rs 31 cos 2V =s  . (4.13) tan = tan  tan . 11r 1  2 r1 2 1 Since  <1, V is complex: Introducin g  2=Vi log2 1 π 2ib (4.14) 1    = 2arctan tan   (4.21) 1   Using ([2], eq.18.10.6) we get 3 3 π 2341VVV   0 r 13 Pv=  s 33  =s 2 in . 40002341      V  24r 1 1 π3 cosV 2 As before in (3.19) we do not expand  in (4.21). = 33 Oq 4sin V However, if o ne does so one finds a cont ribution we obtain Or s r1 

2 12 131r  π3  1 = 2arctan tan 2  s  2  Pv= i 32 Ors 12 r 11cos 2  1 1 (4.15) (4.22) 22 1345rs  =1i 22 . where the first term, say  N , is the parameter which 2 1  r1 1 appears in Newtonian mechanics (Kepler’s equation, see

below (4.29)). Now the expansion of J1  is given by which also follows directly from the definition (4.4). To expand 01rs JJ11=  J1 1   r1 JP2 =22  v (4.23) Pv   11 J 0  =2  π sin 2  1   1 PvJ1   cos 2  r 1 11 31 01s J  =73 sin  = JJ22  1   2 r  1 1 1 we need 15432   2.  2 g2 11 Pv =6 Pv   2

For   0 we have the simple finite limit 2 1r 11 18 5 2 =1 s . 222112 r  2 r 1  1  J  =13s 2sin2 1  =0  This finally leads to π r1

2 004211 sin2 )1 JJ21 =  (4.25) 11cos221   

2 222 1 21   15 12 11 sin 2  1  4 3 17  13 12 J2 =   11cos21cos21 2   1  (4.26) 615823 12   JJ01 . 112 1  11 

Again w e do not expand  (3.16) in order to keep which follows from (2.19-21). In t0 the terms propor- the perihelion precession as precise as possible. The limit tional to  cancel for   0 is equal to 3 2r1 1si n2  r t0  =. (4.28) J  = 2 1 6s 6 4sin 2 sin 2 cos 2 . r 11 2 1cos2   2  =0 s  1  r1 Approximating 2 by  this is in agreement with In the final result (4.8) for the time Kepler’s equation r tt =  s t (4.27) 3 01   2r1 r1 t0  =s in . (4.29) r 1  3 the pre-factor EL also gives a correction: s The post-Newtonian corrections in (4.27) come from E 2 r =1s various places. To show this we write the result in the  Lrrs 1111  r  form

2r3  115 3 t = t1 JJJJ101  0 . (4.30) 10   1 23 1 2 2 2 11r 1 2 41 2  s  11    As in (3.19) the post-Newtonian correction vanishes Galactic Center ([1] and references given there). These for circular motion  =0. stars move in the strong gravitational field of the central black hole so that general relativistic effects are observ-

5. Gravitational Red Shift able and the Schwarzschild metric g is a fairly good description of the situation. The measurabl e quantity of The study of Schw arzschild geodesics is relevant for the interest is the red shift of spectral lines in the light emit- investigation of the recently discovered S-stars near the ted by the moving star. Therefore we finally consider

Let 1 be the frequency of a given atomic line from we indeed have red shift 10>  . Of course, it is maxi- the star an d  0 the frequency of the same line observed mal at the perihelion where rr= 1 is minimal. The total in the rest frame of the galaxy. If ddx t is the velocity observed red shift is obtained by multiplying (5.5) with 1 of the star, the two frequencies are related by ([9] p.83, the Doppler factor 1 vr where vr is the compo- equ. 3.5.6) nent of the relative velocity along the direction from the observer to the star ([9], p.30). 1/2 ddxx gx   ddtt 6. Acknowledgments 1 =. (5.1) 1/2  0 gX 00 It is a pleasure to acknowledge elucidating discussions We assume that the observer at X is far away from the with Prasenjit Saha, in particular the introduction into the center such that the denominator can be approximated by fascinating field of Galactic-center stars. I also thank 1. For a star moving in the p lane  = π 2 we have Raymond Angélil for showing his simulations of the  22 corresponding dynamics. ddxx  dr2 d  gx  = eer . (5.2) ddtt  dtd t  7. References From (2.10) and (2.7) we find [1] D. J. D’Orazio and P. Saha, “An Analytic Solution for  2 dreL 2 Weak-field Schwarzschild Geodesics,” Monthly Notices =1em dt Er22 of the Royal Astronomical Society, Vol. 406, pp.  2787-2792. and (2.8) gives [2] M. Abramowitz and I. A. Stegun, “Handbook of Mathe- matical Functions,” Dover Publications, Inc., New York. d Le =.2 [3] G. Scharf and Gen. Relativ. Gravit, “From Massive Grav- dtEr ity to Modified General Relativity,” General Relativity Substituting all this into (5.1) we see that L drops out and and Gravitation, Vol. 42, pp. 471-487. we end u p with the simple result doi: 10.1007/s10714-00 9-0864-0 [4] S. Chandrasekhar, “The Mathematical Theory of Black

1 mm rs Holes,” Oxford/New York, Clarendon Press/Oxford Uni- ==1e  . (5.3) versity Press, 1983. 0 EEr [5] E. T. Whittaker and G. N. Watson, “A Course of Modern

By (2.20) we can express E by the perihelion r1 and Analysis,” Cambridge University Press, 1950. aphelion r2 [6] A. Erdelyi et al., “Higher Tr anscendental Functions,” McGraw-Hill Book Co., Inc., New York, 1953. 22r Em=1 s  [7] J. Tannery, J. Molk, “Fonctions elli ptiques,” Chelsea rrr123 Publishing Company, Bronx, New York , 1972. [8] Ch. Darwin, Proc. Roy. S. London A 249 (1959) 180, A where r3 is the small correction (2.21). Then we finally get 263 (1961) 39.

1/2 [9] S. Weinberg, “Gravitation and Cosmology,” John Wiley,  rr New York, 1972. 1 =1ss 1 .  (5.4) 01rrrr23 [10] G. Scharf, “Quantum Gauge Theories-Spin One and Two,” Google-Books (2010) free access. The lowest order Ors is equal to [11] Y. Hagihara, Japanese J. Astron. Geophys. 8 (1930) 68.

1 rrss 2 =1 Ors . (5.5) 012rr 2 r

Appendix: Integration of the Differential To remove the second term in the cubic we set Equation 11A22 A  sz=,= s1  z1  (3) We closely follow Whittaker and Watson ([5], p.452). A322 A3  With the notation of the paper (2.11), let and we get r dx   = (1)  =4 zAAAz32 3 4    213 r1 f x z1 (4) 1/2 where r is any zero, fr=0. By Taylor's theorem, 32 1 1 2dA AA A AA x we have 123 2 03 0 2 The coefficients of z and z are just the invariants fx=4 Axr31   6 Axr 21    g23, g (2.14-15) of the ori ginal qua rtic. 34 4,A11xr Axr01 Now the inversion of the inte gral gives Weierstrass’ P-function where

zPgg12=;, 3. (5) A00=,aAara 101 =  1

2 From (6.2) and (6.3) we have A2111=2aro  ar a2, A3 32 rr= 1  (6) A30111213=33ar ar ar a. zA12 2 Introducing the new integration variable and hence 11 sxr=,=11 s  rr 1, (2) fr1 rr=.1  (7) we have 46Pfr   1

 1/2 32  =4 A3210sAsAsA 6 4 ds. s1