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Let us turn now to the motion of Mercury. This is essentially a one-body problem, as we can neglect the effects of the mass of Mercury on the gravitational field of the Sun. Rather than following Einstein and using the method of successive approximations, we will begin our calculation from the exact solution to Einstein's field equations found in 1916 by Karl Schwarzschild. If we assume that the field of the Sun is static and isotropic, then we obtain from Eqn (12.9), with coordinates (r, 0, which, far from the origin, are the usual spherical polar coordinates, ds2 = —dr-+ r2(d02 + sin2 0 d^2) — (1 — -) c2dt2. (12.11)

Here, a is a constant of integration that depends on the mass M of the Sun. It is possible to determine a by comparing the form of the geodesic Eqns (12.7) in the case of a weak, static field with the equivalent Newtonian equation r = V(GM/r). This shows that the coefficient of c2dt2 in the metric must approach 1 — 2GM/c2r as r ^to, and, hence, a = 2GM/c2.

A lengthy calculation shows that with this value for a, the geodesics for the metric (Eqn (12.11)) in the case 0 = 1 n are the solutions to the equation d2u _ GM 3GMu2 d^2 h2 c2'

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