Journal of Modern Physics, 2011, 2, 274-283
doi:10.4236/jmp.2011.24036 Published Online April 2011 (http://www.SciRP.org/journal/jmp)
Schwarzschild Geodesics in Terms of Elliptic Functions
and the Related Red Shift
Günter Scharf
Institut für Theoretische Physik, Universität Zürich, Switzerland
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 ex-
ample 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
Schwarzschild geodesics are elliptic functions, therefore,
they should be written as such. For this purpose the
Weierstrassian elliptic functions are most useful because
they lead to simple expressions. The reason for this is
that the solution of quartic or cubic equations can be
avoided in this way.
In a recent paper [1] an analytic solution for the geo-
desic in the weak-field approximation was given. As
pointed out in that paper the progress in the astronomical
observations call for better analytical methods. In this
respect it is desirable to have the exact geodesics in a
form most suited for applications. For the orbits in polar
coordinates (next section) this goal can be achieved by
using Weierstrass’ P-function for which many analytical
and numerical methods are known [2]. Considering the
motion in time (section 4) the related

- and

-func-
tions of Weierstrass appear.
Jacobian elliptic functions have been used by Darwin
[8] for the form of the orbits. After some transformation
our result (2.13) agrees with his. But in his second paper
he abandons the elliptic functions because they were “not
so well adapted to a study of the time in those orbits”.
Obviously the Weierstrass functions are better suited for
the problem. Indeed, expressing them by Theta functions
one gets the natural expansion of the geodesics in powers
of the Schwarzschild radius, this expansion involves
elementary functions only. The Weierstrass functions
have also been used by Hagihara [11]1. But he has cho-
sen the variables and constants of integration in a manner
which leads to less explicit results. So it is difficult to
derive the post-Newtonian corrections to the geodesics
given here from his formulas. As a topical application we
finally calculate the red shift for a star moving in the
Schwarzschild field. The geodesics are also needed for
the study of modifications of general relativity ([10],
section 5.13).
2. The Orbits in Polar Coordinates


=rr

We take the coordinates 0=
x
ct , 1=
x
r, 2=x

,
3=x

and write the Schwarzschild metric in the form

222 2222
d=ddd sind
s
s
rr r
sct rr
rrr
2



(2.1)
where 2
2
=
s
GM
rc is the Schwarzschild radius. We shall
assume in the following. The geodesic equation =1c
2
2
ddd
=0
dd d
xxx
ss s



 (2.2)
with the Christoffel




leads to the following three
differential equations
1I am indebted to C. Lämmerzahl and P. Fiziev for bringing this refer-
ence to my attention

G. SCHARF
275
2
2
ddd
=0
dd d
ttr
ss s

  (2.3)
22 2
2
2
2
dddd
=0
2d 2dd d
rtr
ere
ss s s

 

 

 
  (2.4)
2
2
d2dd
=0.
dd d
r
rss s

 (2.5)
Here we have used the standard representation
1=
s
re
r

 (2.6)
and have chosen =2

 as the plain of motion. The
Christoffel symbols can be taken from the Appendix of
[3].
Multiplying (2.3) by exp

we find
d=0
dd
t
e
ss




so that
d=const.=
d
t
eE
s

d=
d
tEe
s.

 (2.7)
Next multiplying (2.5) by we get
2
r
2d=const. =,
d
rL
s

hence
2
d=.
d
L
sr

For the constants of integration we use the notation of
Chandrasekhar [4].
Finally, substituting (2.7) and (2.8) into (2.5) and mul-
tiplying by

2expd drs   we obtain
22
2
2
dd =0.
dd
rL
eEe
ss r
 








(2.9)
Consequently, the square bracket is equal to another
constant = b. Then the resulting differential equation can
be written as
22
2
2
d=
d
rL
Eeb
sr


 


 
.

(2.10)
The constant b can be arbitrarily adjusted by rescaling
the affine parameter s. Below we shall take 2
=bm

where m is the rest mass of the test particle. This will
enable us to include null geodesics (light rays)
with . Each geodesic is characterized by two con-
stants of the motion: energy E and angular momentum L.
2=0m
Taking the square root of (2.10) and dividing by (2.8)
we get

22 2
432
22
d=.
dss
rEm m
rrrrrrf
LL 
r
(2.11)
Now


=r  can be written as an elliptic integral.
However, it is better to consider the inverse


=rr

in terms of elliptic function by using a formula of Weier-
strass ([5], p.452). Let the quartic

f
r be written as


43
0123
=464frarararara
2
4
, (2.12)
and let 1 be a zero r


1=0fr , then a solution of (2.11)
is given by

 
1
1
23 1
=.
4;, 6
fr
rr Pggfr



 (2.13)
Here


23
;,Pgg  is Weierstrass’ P-function with
invariants
2
204 13
=43
2
g
aaaa a (2.14)
322
3024 12320314
=2 .
g
aaaaaa aaaaa (2.15)
In our case we have4. For the convenience of the
reader we reproduce the short proof in the Appendix.
=0a
The result (2.13) is not yet the solution of our problem
because it contains too many constants: the invariants
23
,
g
g and the derivatives of f depend on E, L, but in
addition the zero 1 appears. Of course one could cal-
culate 1 as a function of E, L by solving the quartic
equation
r
r


fr=0, but this gives complicated expres-
sions. It is much better to use 1 and a second zero 2
as constants of integration instead of E, L. This is even
desirable from the astronomers point of view because the
zeros of derivative (2.11) are turning points of the geo-
desic, for example in case of a bounded orbit they can be
identified with the perihelion and aphelion of the orbit.
In order to express E, L by we write our quartic in
the form
r
2
rr
r
1
,



  
012
=

3
f
rarrrrrrr

 (2.16)
and compare the coefficients of with (2.12).
This leads to
32
,,rrr

2
10123 2
4= =
s
m
aarrrL
r


20121323
6= =1aarrrrr r


.
(2.17)
30123
4= =
s
aarrrr

Since
22
02
=Em aL
 (2.18)
we can solve for

276 G. SCHARF
2
123
2
12 1323
=
s
rrr
mrrrrr rr
L

,
(2.19)
2
22
123
=. (2.20)
s
mr
Em rrr 

In addition we obtain the third zero
12
3
12 12
=
s.
s
s
rr
rr
rrrrr r (2.21)
The relations (2.19-21) allow to express everything in
terms of . For the invariants we find
12
,rr
2
2123
22
12 13 23
11
==
1212 4 4
s
s
rrrr
m
gr
rrrrr r L


 (2.22)

2
22
0
332
2
123
3
12 132312 1323
1
=16
648
11 1
=
48 16 6
ss
ss
a
m
grr
L
rrrr r
rrrr rrrr rr rr



 .
(2.23)
Here has to be substituted by (2.21). For the deriva-
tives
3
r

1
f
r
,

1
f
r
 which appear in our solution
(2.13) we obtain
 

11213
1
12 1323
=rr rr r fr rr rr rr
 
 (2.24)
 
  
12131 1213
1
12 1323
=2 .
rrrrr rrrr
fr rrrr rr

  (2.25)
With these substitutions the result (2.13) gives all possi-
ble geodesics in the form . This will be
discussed in the next section.

12
=;,rr rr 

As a first check of the solution (2.13) we consider the
Newtonian limit. Let the two zeros 1, 2 be real and
large compared to the Schwarzschild radius
r r
s
r in abso-
lute value. Then neglecting

s
Or in the invariants
(2.22-23) the P-function becomes elementary ([2], p.652,
equation 18.12.27):

3
2
11111
;,6==
1212122 1cos 4sin 2
P  .




 
 

(2.26)
The leading order in the derivatives of f is given by
 
1
11
2
=r
2
f
rr
r
r

1
1
2
=232.
r
fr r

 




(2.27)
It is convenient to introduce the eccentricity

by
1
2
1
=
1
r
r
Using all this in (2.13) we find the wellknown conic


1
1
=
1cos
r
r

.



 (2.29)
Assuming both zeros 12
positive and 12 we
have
,rr <rr
<1

and the orbit is an ellipse with perihelion 1
and aphelion 2. In the hyperbolic case
r
r>1

we see
from (2.28) that if 1 is positive 2 must be negative.
Then there is only one physical turning point 1 which
is the point of closest approach. The latter always corre-
sponds to
r r
r
=0

. The relativistic corrections to (2.29)
are calculated in the following section.
3. Discussion of the Solution
The solution


=rr

(2.13) is an elliptic function of

which implies that it is doubly-periodic ([2], p.629 or
any book on elliptic functions). The values of the two
half-periods ,



depend on the three roots of the
fundamental cubic equation
3
23
4egeg=0. (3.1)
Again it is not necessary to solve this equation because
the solutions can be easily obtained from
the roots 12 0, of our quartic . To see
this we transform
,=1,2,3
j
ej
3
, ,rrr

=0fr


f
r to Weierstrass' normal form as
follows. First we set =1rx so that from (2.16) we get


32
321
4
1
=4 640
f
raxaxax
xa
Next we remove the quadratic term by introducing
2
3
11
== .
2
a
xe
ra






(3.2)
This gives the normal form of Weierstrass
 

2
3
3
23
4
12 2
=4
a

,
f
rege
ea

g (3.3)
with the above invariants (2.14-15). That means roots of


f
r are simply related to roots of (3.1) by the trans-
formation
321
==
24 12
s
j
jj
ar a
err  .
2
3
(3.4)
The cubic equation (3.1) with real coefficients has ei-
ther three real roots or one real and two complex conju-
gated roots. The first case occurs if the discriminant
3
2
=27
g
g (3.5)
is positive, in the second case is negative. In terms of
the roots


is given by ([2], p.629, equation 18.1.8)
.




(2.28)


22 2
1223 31
=16 .eeeeee (3.6)

G. SCHARF
277
The physically interesting orbits correspond
case of real roots. If we have two complex conjugated
ze
where


2
K
k is the complete elliptic integral of the
first kind with parameter
to the first
ros *
21
=rr
then (2.28) implies that the eccentricity

is imaginary. Such orbits have been discussed by
Chandar ([4], p.111). Now we discuss the various
ses.
3.1. Bo

223 21
13 1212
1
1
== 2
23
=1 .
11
s
s
s
s
ee rr
kr
eerrrrr
rr
r












(3.10)
rasekh
ca
und Orbits
ve two positive turning points
, consequently there are three real roots
As a first application let us give the post-Einsteinian
correction to the orbital precession. If (3.10) is
small we can use the expansion ([2], p.591, equation
17.3.11)
2
k
In this case we ha
21
23
>0> >eee given by
>>0rr
1
12
11
==
412
ss
rr
1
31 2
64
rr
er
rr

 (3.7)

22
22
π113
=1
22 24
Kkk k
4



 
 


 

 




 (3.11)
23
12
11
=,= ,
r
(3.8)
12 412 4
ss
r
ee
rr
 
Our convention is chosen in agreement with
half-period
From the roots we find
13
,ee
[2]. The real

of the P-function is given by ([2], p.549,
equation 18.9.8)

2
23
12 12
12 12
13
22
13
=2 1.
28
s
ss
rrrrr
rO
rr rr
ee r


 


 








1
3
13
23
=
4
e
2
d
=
K
k
t


ee
tg tg  (3.9) This finally leads to the half-period

 

22 2
23
12
12 2121
222
12 11
12
33 331318
=π122=π1
482118
81
ss ss
s
rr rr rr rrrrrrOr
rrr r
rr




 




 







(3.12)
The perihelion precession is given by


=2 π  .
hen the orderT
s
r in (3.12) is Einstein's result and the

2
s
Or gives the correction to it. The acc-
tion of the half-piod is necessary to control the orbit in
e.
To compute the relativistic corrections for
urate computa
er
the larg


r

from (2.13) we express the P-function by Theta fu
([
(3.13)
Here is the so-called Nome ([2], eq. 17.3.21)
nctions
5], p.464)


1/426
1,=2sinsin 3sin 5zqqzqz qz
 






1/426
2
38
3
38
4
,= 2coscos3cos5
,=12cos 2cos 4cos6
,= 12cos 2cos 4cos 6.
zqqz qz qz
zqqz qzqz
zqqz qz qz










q
2
22
=8
kk
q 
 (3.
16 16
 14)
These series are rapidly converging s
(3.10), they give the natural expansion in powers of the
Schwarzschild radius
s
r. Now the P-function is given in
terms of Theta functions by ([2], eq. 18.10.5)
 



 
31
2
2
2
22
1
0
4
1π1
=1 41,
cos
12 44sin
s
rqO
q
r


 


  

(3.15)
where
2
2
13
22
0
π
=Pe


 


π
=.
2


(3.16)
Using
 
2
11 2
1
3
=2 11
s
r
fr rr





 



 
2
12
1
15 14
=26
11
s
r
fr r





 

ince 2 k is small
(3.17)
this leads to

 
2
2
12sin 1
fr r
 
 
 

12
2
1 1
3
=1 31cos2sin.
241cos 221 41 sin
s s
r
P O
rr


  
 


 

 

(3.18)


278
Substituting this into (2.13) gives the desired orbit to
G. SCHARF


s
Or
 
2
2
11
1c os2 1cos21crr
12sin1 33
=31cos2sin.
os2 211
s
rr 









 







(3.19)
It is important to insert the period
 


in

(3.16)
according to (3.12) in order to describe the perihelion
recession correctly.
. If all three zeros coincide
p
If the two roots 12 =rr coincide, it follows from
(2.24) that

1=0fr
. According to (2.13) we then have
circular motion =r1 r
) giv
123
==rr then (2.21es 3=3r
s
rr which is the
innermost ct.
und Orbits
ircular orbi
3.2. Unbo
this case there is only one physical point, the point of
he other root is negative,
erefore, it is to use the eccentricity
In
closest approach 1 r. T2 r
th better

(2.27) as
the second basic quantity. With 3 r given by (2.21) we
then have
13 21
1
>>0>=,
1
rr rr


 (3.20)

because 1

. The periodicity of (2.13) in

is now
realized jump to an unphysical br
In moves on one branch only
by a
reality a comet
blem
anch with <0r.
, bu
.
t it is a
in the
tricky pro to decide on which one. This is due to the
fact that the period differs a little from 2π as
bounded case. Consequently, neighboring physical
branches >0r are a little rotated against each other
and the distinction between them is not easThe quan-
tity of physical interest is the direction
y

 of the as-
ymptote. Iows from the original equation (2.11) by
integrating the inverse over r from 1 r to 

t foll
1
d
=.
r
r
f
r


 (3.21)
This is an elliptic integral which
Legendre’s normal form
can be transformed to
2
0
0
d
=a


22
1s
ink




 (3.22)

by the transformation ([6], vol.II, p.308)
22
32 32 1
2
12 31
rr
rr rr
 2
sin=, sin=.
rr rr rr  


(3.23)
The parameter in (3.22) is given by
2
k
2312
13 2
rr r 

13 2
2
=
rr r


.
The integral (3.22) is an incomplete elliptic integral of
the first kind
(3.25)

2
2
0
=,
F
k
a


 (3.26)
which has the expansion ([6], vol.II, p.313)
 
2
24
222 2
1
,=sin2 .
k
42
F
kOk



(3.27)

For small 1s rr we find

2
2
1
1
1s
r


2
=s
r
kOrr



2
1
1
0
3
=2 1
s
s
rOrr
r
a





23
2
1
1
sin =1.
2
r
r







This gives
3
2
1
11
cos 2=r
r




and
22
1
3
=2sin2 .
122
s
r
r




 



2
(3.28)
It is convenient to calculate


2
2
2
1
11 1
cos =3.
21
s
s
rOr
r





  



(3.29)
The leading order is the Newtonian asymptot
hyperbola.
For there is only one constant of integrat
th .11)
e of the
3.3. Null Geodesics
2=0m
e quartic (2
ion in

4
2
2
=
s
r
f
rrrr
d
pact parameter
=,
rrr
k (3.24)
and
which is the so-called im
=.
L
dE (3.30)

G. SCHARF
279
ry to calculate the roots of Now it is necessa


=0fr
es expansion
.
This is easily done by means of a power seri
2
01 2
=ss rcdcrcr
We find

23
O  
1
3
=1
28
rd






23
2
3
=1
28
rdO


 


(3.31)

3
3=,rd
 

where
=
s
r
d
 (3.32)
and we have ordered the zeros in the sam
(3.20). Then as in the last subsection we can calcu
direction of the asymptote (3.21) whih now is equal to
e way as in
late the
c


2
1
=d ,
F
k 
 (3.33)
with given by (3.23)
1


23
1
sin =24 O
13

  (3.34)
and 2 k by (3.24)
22
25
=2 18 k

 





(3.35)
and

by (3.25)
2
29
=1
28d






. 


e want to calculate the light deflection in the post-
Einsteinian approximation. Using again
(3.27) we have
(3.36)
W
the expansion

2
9
1





11 1
1
=21sin2 .
28 22



 

 

 

From (3.34) we obtain

3
1
π3
=.
44 O

 
Then up to

O

2

we find
2
π33
=π.
2416




 


(3.37
ven by
)
The deflection angle is gi
2
π33
=2 =2π.
228







 (3.38)
f the ieter in Instead ompact paramd

(3.32) we
would like to use the distance osest approach
(2.24) in the form
of cl 1
r
1
1
=.
s
r
 (3.39)
The two are related by
r

23
11
21
1
=O

  
which leads to
2
11
28
13
=2 π.





(3.40)
The first term

1
2

is Einstein’s result.
4. The Motion in Time

=tt

by (2.10) we find By dividing (2.7)
 
12
2
22
2
d=1
d
=.
s
s
r
tr L
EE m
rrrrr


3
Er
s
Lrr fr



 









 (4.1)
e choose at the point of closest approach

W=0t
1
=rr and get

3
22
1
d
=.
r
s
ss
s r
r
Ex
txrxr
Lx
fx






r
(4.2)
This is a sum of elliptic integrals of first, sec
third kind. The coordinate time is given as a function of r
by calculating these. However, we want t as a function of
ond and

and, therefuse the substitution (6.7) of the Ap-
px again
ore,
endi

 
1
1
1
=4
fr
xr Pfr



 6
(4.3)

d=d , x
fx

where the last relation follows from (2.11).
The last integral


3
s
Or in (4.2) is a small correction
and we neglect it oment. Then integrals of the
following form remain to be calculated
at the m
  

0
d
=
nn
u
J
Pu Pv



 (4.4)
where we have set
 
1.
24
=
f
r 
Pv (4.5)
Such integrals are known ([7], vol.4, p.109-110)

280 G. SCHARF
   

1=2logJv
Pv v
1v












(4.6)
  


2
1
=
2
2
1.
J
vv
v
PvP vJ








(4.7)
These results are easily verified by differentiating and
us
P


ing addition formulas. Of course

0
J

is just the
polar angle

. Then (4.2) leads to result for the desired

t

:
  

3
011 22
=.
s
E
tJJ
L
   Or (4.8)
where
22
011
=
s
s
rrrr   (4.9)

1
11
=24
s
r r
f
r 




 (4.10)


2
1
2=.
16
f
r

 (4.11)
Again we evaluate this for bounded orbits
tonian approximation by means of the expan-
sion in Theta functions.
The quantity in (4.8-11) is
(3.18). Introducing
in the
post-New
v given as the zero of
π
=2 Vv

(4.12)
we find
1
131
cos 2=.
2
s
r
Vr




 (4.13)
Since <1

, V is complex:

2
2= loVi  g1π2ib   (4.14)
Using ([2], eq.18.10.6)
  
 

3
3
2341
33
2341
3
2
33
0
π
=4000
πcos
=4sin
VVV
Pv V
VOq
V

  




we obtain





12
2
3
2
3
=2 Pvi




v  from ([2], eq.18.10.7)
2
22
22
1
1
π
1
1345
=1 .
21
1
s
s
Or
r
ir


 










(4.15)
Similarly we calculate
 

1
1
π
=2
V
v
vV




 (4.16)
where
 

2
1
1
0
π
==
12 0

 

 (4.17)
and


z  from ([2], eq.18.10.8)
 

2
1
1
2π
=exp, =
π20 2
Z
zz
zZ


 .





 (4.18)
This implies









2
sin
log=2logsin
cos
=2 log
cos
22,
vV
vO
vV
ib
vib
vi
 









 
 
(4.19)
e
q

wher

is given by


= arctantantanh.b  

Then
(4.20)


1
J

(4.6) is equal to
   
2 2
23
123
21
1
2  4
=1 .
ππ
1
s
r
JO
r


 


 


To expand this in the post-Newtonian order we first
calculate



 from
2
1
1131
tan=tantan .
11 1
s
r
r




 




Introdug cin
1
=2arctantan
1










(4
we get
.21)
2
1
13
=s
24 1
s
r
r

in.





As before in (3.19) we do not expand  in (4.21).
However, if ne does oso one finds aribution cont


1s
Or r

1
131
= 2arctan  tan 2
1211cos
s
r
r








(4.22)
where the first term, say



N

, is the parameter which
appears in Newtonian mics (Kepler’s equation, see echan

G. SCHARF
281
.29)). Now the expansion of

1
J

below (4is given by which also follows directly from the definition (4.4).
To expand
 
01
11
1
=s
r
JJ J
r
 1 

0
1
11
=2 1 J











(4.23)
 
1
121
2.

2
1131
=73 sin
1
1543
J
11

 







 





 

For 0

 we have the simple finite limit
  

 
22
1
01
22
1
1
=2
πsin2
cos 2
=s
JP
Pv
PvJ
r
JJ
r



v





 






 

we need
 
 
22
2
2
1
=6 2
2111 185
=1
22211
1
s
g
Pv Pv
r
r



 







.
 
1=0
1
2
=132sin  2
π
s
r
Jr






 This finally leads to
 
 

00
21
sin2
42)1
=11cos221 JJ

2
11 

 












 (4.25)

 
 
222
1
22
23
01
11
2
1
2151211sin2143171312
=11cos21cos2
1
615812 .
1
11
J
JJ

2
1



 






 







 
 

 





(4.26)
ain wAge do not expand

(3.16) in order to keep
the perihelion precesble. The limit
for
sion as precise as possi
0

 is equal to
 
2=0
1
=21664sin 2sin 2cos 2.
s
r
Jr








In the final result (4.8) for the time
 
01
1
=s
r
tt t
r

  (4.27)
the pre-factor EL also gives a correction:

11
11
s
2
=1
s
r
E
Lrr r








which follows from (2.19-21). Inthe terms propor-
tional to

0 t

cancel
 
3
1
02
21si
=.
11 1cos2
1 s
r
tr
 
 






n2





(4.28)
Approximating by

this is in agreement with 2

Kepler’s equation
  
3
1
03
2
=s
1
s
r
t
r
in.

 


 (4.29)
The post-Newtonian corrections in (4.27) come from
various places. To show this we write the result in the
form
  

 
3
101
1
101 1 2 2
23
2
2115 3
0
.
11 1 22
41
11
s
r
tJJ JJ
r

=t





 






 
 

(4.30)
As in (3.19) the post-Newtonian correction vanishes
for circular motion =0

.
5. Gravitational Red Shift
chwant for the
investigation of the recently discovered S-stars near the
Galactic Center ([1] and references given there). These
stars move in the strong gravitational field of the central
black hole so that general relativistic effects are observ-
able and the Schwarzschild metric
g


easurabl
is a fairly good
description of the situation. The me quantity of
interest is the red shift of spectral lines in the light emit-
ted by the moving star. Therefore we finally consider
The study of Sarzschild geodesics is relev

282 G. SCHARF
this.
Let 1

n
be the frequency of a given atomic line from
the star ad 0

the frequency of the same line observed
in the frame of the galaxy. If rest dd
x
t

is the v
of the two frequencies are related by ([9]
equ
elocity
p.83, star, the
. 3.5.6)


1/2



1
1/2
000
dd
dd
=.
xx
gxtt
gX





(5.1)
We assume that the observer at X is far away from the
ce
in thelane
nter such that the denominator can be approximated by
1. For a star moving p=π2

we have

22
2
dd d
=
dd d
xx
gx ee
tt
 





d.
d
rr
t t

 

 
 
(5.2)
From (2.10) and (2.7) we find
2
2
22
d=1
d
reL
em
tEr






and (2.8) gives
2
d=.
d
Le
tE
r


Substituting all this into (5.1) we see that L drops out and
we endp with the simple result u

1
0
==1 .
r mm
e
EEr








(5.3)
By (2.20)
s
we can express E by the perihelion and
aphelion
1 r
2
r
22
123
=1 s
r
Em rrr




 
w
here 3 r is the small correction (2.21). Then we finally
get

1/2
1ss
rr 



01 23
=1 1.
rr rr 






 (5.4)
The lowest order

s
Or is equal to
 

2
1
012
=1 .
2
ss
s
rr 
Or
rr r 

 (5.5)
Since the last term is always smaller than the second one
we indeed have red shift 10 >

 .
1
=rr
Of course,
mal at the perihelion where is minimal. The total
ob
to the star ([9], p.30).
6. Acknowledgments
It is a pleasure to acknowledge elucidating
with Prasenjit Saha, in particular the introduction into the
fascinating field of Galactic-center stars. I also thank
Raymond Angélil for showing his simu
corresponding dynamics.
[1] D. J. D’Orazio and P. Saha, “An Analytic Solution for
Weak-field Schwarzschild Geodesics,”ly Notices
of the Royal Astronomical Society, Vol. 406, pp.
. Stegun, “Handbook of Mathe-
s,” Dover Publications, Inc., New York.
[3] G. Scharf and Gen. Relativ. Gravit, “From Mass
ity to Modified General Relativity,” General
and Gravitation, Vol. 42, pp. 471-487.
14-00
it is maxi-
served red shift is obtained by multiplying (5.5) with
the Doppler factor

1
1r v  where r v is the compo-
nent of the relative velocity along the direction from the
observer
discussions
lations of the
7. References
Month
2787-2792.
[2] M. Abramowitz and I. A
matical Function
ive Grav-
Relativity
doi: 10.1007/s1079-0864-0
[4] S. Chandrasekhar, “The Mathematical Theory of Black
Holes,” Oxford/New York, Clarendon Press/
versity Press, 1983.
er Tr
lli
n and Cosmology,” John Wiley,
Oxford Uni-
[5] E. T. Whittaker and G. N. Watson, “A Course of Modern
Analysis,” Cambridge University Press, 1950.
[6] A. Erdelyi et al., “Highanscendental Functions,”
McGraw-Hill Book Co., Inc., New York, 1953.
[7] J. Tannery, J. Molk, “Fonctions eptiques,” Chelsea
Publishing Company, Bronx, New York , 1972.
[8] Ch. Darwin, Proc. Roy. S. London A 249 (1959) 180, A
263 (1961) 39.
[9] S. Weinberg, “Gravitatio
New York, 1972.
[10] G. Scharf, “Quantum Gauge Theories-Spin One and
Two,” Google-Books (2010) free access.
[11] Y. Hagihara, Japanese J. Astron. Geophys. 8 (1930) 68.

G. SCHARF
283
on ([5], p.452).
), let
Appendix: Integration of the Differential
Equation
We closely follow Whittaker and Wats
With the notation of the paper (2.11

1
d
=
r
r
x
f
x

 (1)
where 1
r is any zero,

1=0fr . By Taylor's theorem,
we have
 
2
31 21
=4 6fxAxrAxr
 
34
11 01
4,
A
xr Axr  

where
100 101
=, =
A
aAara
2
2111
=2
o2
,
A
arara
32
301 11213
=33 .
A
ararar a
Introducing the new integration variable
, (2)
we have
.
 
11
11 1
=,=sxrs rr


1/2
32
3210
1
=4 64d
s
A
sAsAsA 





To remove the second term in the cubic we set
22
11
3
11
=,=
3
22
A
A
szsz
A
 

 
 (3)
A
 
and we get

32
2

1
1/
1 3
=434
z
zAAAz
2
32
123 203
2d
A
AA AAAx  







(4)
The coefficients of and are just the invariants
z0 z
23
,
g
g
No
(2.14-15) of grtic.
w the inversion the igral gives Weierstrass’
P-function
3
. (5)
From (6.2) and (6.3) we have
the ori
of
inal qua
nte

12
=;,zP gg 
3
1
12
=2
A
rr zA  (6)
and hence

 
1
1
1
=.
46
fr
rr Pfr




s
(7)