§ 1.6
which can be chosen in the plane 8 motion are
!!..-= H= RZ d¢> ac ds and
215
THE GENERAL THEORY OF RELATIVITY
A
(conservation of angular momentum)
(2)
(conservation of energy)
(3)
(1 - 2) dT R ds
=
With the substitution, u
=
= TT/2. Two further integrals of the
1/R, we get the equation of the orbit: dZu d¢>z
+u=
1 z HZ+ 3u
(4)
1 HZ
(5)
The corresponding Newtonian law is
dZu d¢>z
+u=
The solution of Eq. (5) is u
=
1 HZ (1
+ € cos ¢»
(6)
This is an ellipse if € < 1. The Einstein term 3u2 is very small compared to l/Hz in planetary motion. Therefore Eq. (4) can be solved by the methods of successive approximations. One substitutes on the right-hand side the Newtonian value of u, and obtains the differential equation
dZu d¢>z
+u =
1 + H43 (1€+z2 + 2€ cos ¢> + 21 ) €z cos24>
HZ
(7)
The solution is
u
=
[~z + ~4
(
1+
~) ] (l + € cos ¢» +~:
(¢> sin ¢> -
~
€ cos2¢> )
(8)
If it were not for the term ¢> sin¢>, u would have the period 2TT. The actual period is 2TT E, where E is a small number. If this value is substituted in Eq. (8) and higher powers of E are neglected, one obtains
+
6TT 6TTa;z c Z 6TTJ(2m Z E = HZ =~ii2- = hZ'c'i~
(9)
In the case of the planet Mercury, this formula gives for the rotation of the major axis of the orbit in 100 years approximately 43 seconds of arc. This is in good agreement with observation. As A. Sommerfeld has pointed out, each deviation from the Newtonian or Coulomb potential causes a perihelion motion of the Kepler ellipse. In the theory of the fine structure of the hydrogen spectrum the deviation