Page 165 - Intro to Tensor Calculus

160

which is obtained from equation (1.5.141). Substituting equations (1.5.174) and (1.5.175) into equation
(1.5.184), rearranging terms and taking the limit we find that

c 4 c 2
= h. (1.5.185)
c 5
From equations (1.5.183) and (1.5.185) we obtain the results that

c 2 2GM 1 h
2
c = , c 2 = , c 4 = p (1.5.186)
5
1+ E c 2 1+ E/c 2 c 1+ E/c 2
c 2
These values substituted into equation (1.5.181) produce the differential equation
2
du 2 2GM E 2GM 1 3
+ u − u + = u . (1.5.187)
dφ h 2 h 2 c 2 1+ E/c 2
1
Let α = c 2 2 = 2GM and β = c 2 = 2GM 1+E/c 2 ) then the differential equation (1.5.178) can be written as
c 2 (
c 4 h 2
2
d u α 3 2
+ u − = βu . (1.5.188)
dφ 2 2 2
We know the solution to equation (1.5.143) is given by

1
u = = A(1 + cos(φ − φ 0 )) (1.5.189)
ρ
and so we assume a solution to equation (1.5.188) of this same general form. We know that A is small and so
we make the assumption that the solution of equation (1.5.188) given by equation (1.5.189) is such that φ 0 is
0
approximately constant and varies slowly as a function of Aφ. Observe that if φ 0 = φ 0 (Aφ), then dφ 0 = φ A
0
dφ
2
and d φ 0 00 2
dφ 2 = φ A , where primes denote differentiation with respect to the argument of the function. (i.e.
0
Aφ for this problem.) The derivatives of equation (1.5.189) produce
du
0
= − A sin(φ − φ 0 )(1 − φ A)
0
dφ
2
d u 3 2 0 2
0
= A sin(φ − φ 0 )φ − A cos(φ − φ 0 )(1 − 2Aφ + A (φ ) )
00
0
0
0
dφ 2
3
2 0
= − A cos(φ − φ 0 )+2 A φ cos(φ − φ 0 )+ O(A ).
0
Substituting these derivatives into the differential equation (1.5.188) produces the equations
α 3β
2 0 2 2 2 2 2 3
2 A φ cos(φ − φ 0 )+ A − = A +2 A cos(φ − φ 0 )+ A cos (φ − φ 0 ) + O(A ).
0
2 2
3
Now A is small so that terms O(A ) can be neglected. Equating the constant terms and the coefficient of
the cos(φ − φ 0 ) terms we obtain the equations
α 3β 2 2 3β 2 2
2 0
A − = A 2 A φ =3β A + A cos(φ − φ 0 ).
0
2 2 2
Treating φ 0 as essentially constant, the above system has the approximate solutions
α 3β 3β
A ≈ φ 0 ≈ Aφ + A sin(φ − φ 0 ) (1.5.190)
2 2 4