(^116) Theory of Relativity
equal to the corresponding R 0. As the preceding exercise shows, a black
hole can be produced either by compression or by vast accumulation of un-
compressed matter. Once you do Exercise 2.4.11, you realize that neither
you nor the Earth nor the Sun is likely to become a black hole.
The first experimental verification of general relativity was the calcula-
tion of the precession of the perihelion (the point closest to the Sun) for the
planet Mercury. Astronomical observations of the precession had been car-
ried out since the late 18th century, but the Newtonian theory of gravitation
was unable to calculate the observed precession correctly. In 1916, Einstein
used the Schwarzschild solution, with R 0 equal to the Schwarzschild radius
of the Sun, to compute the correct value of the precession; see Problem 2.2
on page 414.
The reader can derive equation (7.2.7) on page 415, describing the tra-
jectory of Mercury (or any other planet, for that matter) by following the
steps below.
Step 1. Write three equations for the geodesic involving 0(s), <p(s), and
t(s) using (2.4.30) and (2.4.37); use g to denote dg(s)/ds, and Q', to de-
note dQ(r)/dr, for the appropriate functions g,Q:
§(s) + -^ 6(s)f(s) = 0; (2.4.40)
r{s)
<p{s) + -^- ff>{s)r(s) - sin <p(s) cos ip(s) 6^2 {s) = 0; (2.4.41)
r(s)
.... H'(r(s)) ., s , , ,
t{s)+ H(r(s))t{s)^s)=°- {2AA2)
Step 2. From (2.4.40) conclude that r^2 (s)6(s) — a, with the real number
a independent of s; from (2.4.41), that <p(s) = n/2 is a possible solution;
from (2.4.42), that i(s) = /?F(r(s)), with j3 independent of s.
Step 3. Keeping in mind that the geodesic is parameterized by the arc
length, take ds — cds to have s in time units, and then, with <p(s) = TT/2,
conclude that (2.4.35) implies{dsf = c^2 H(r(s))(dt{s))^2 - F(r(s))(dr(s))^2 - r^2 (s)(d0(s))^2 ,
1 = c^2 H(r(s)) (i(s))^2 - F(r(s))(f(s))^2 - r^2 (s)(0(s))^2 ,(r(*))^2 + 4^ (l ~ TT) - ^TT = c2^ " !)• (^2 -^4 -^43 )
rJ(s) \ r(s)J r(s)
Similar to (2.4.19) on page 104, we have to switch the sign in (2.4.35) to
ensure that the expression for (ds)^2 is positive.