Einstein's Field Equations 115(cte)^2 = (dr)^2 + r^2 sin^2 <p (d6)^2 + r^2 {dip)^2 - c^2 (dt)^2. Conclude that A = -c^2 ,
where c is the speed of light. Keep in mind that (dx)^2 + (dy)^2 + (dz)^2 =
{dr)^2 + r^2 sin^2 <p (d6)^2 + r^2 (d<p)^2.
Step 6. Using (2.4.30) and (2.4.37), verify that the geodesic equations for
the trajectory (r(s),8(s),(p(s),t(s)) are as follows:
[S) + 2F(r(s))[V {S)) F(r(s)) {° [S>> 2F(s) °'9"(s) + -r'(s)9'(s) = 0;
r j (2.4.39)
¥>"(*) + -ri:r'(s)ip'(s) -r(s) sin<p{s) cosip(s)(6'(s))^2 = 0;t"(s) + ^r'(s)t'(s) = 0.Keep in mind that, since rf- = T^, the terms in (2.4.30) often come in
pairs.
Step 7. Consider the circular trajectory of radius R in the equatorial
plane (<p = 7r/2) in the Newtonian approximation, so that r(s) = R,
ip(s) = 7r/2, t'(s) — 1; the absolute time t'(s) = 1 is the key feature
of Newtonian mechanics. Use the function H(r) = —c^2 (l + l/(Br)) we
found in Steps 4 and 5, to conclude from the first equation in (2.4.39)
(6'(s))^2 = -H'(R)/(2R) = -c^2 /(2BR^3 ). On the other hand, Newton's
laws imply that, for the circular trajectory in the gravitational field of
mass M, (6»'(s))^2 = MG/R^3. Conclude that B = -c^2 /(2MG) = -1/R 0.
This completes the derivation of (2.4.35).
It is always a major accomplishment to find an explicit non-trivial so-
lution of a nonlinear partial differential equation. The importance of the
Schwarzschild solution is further enhanced by the following facts:
- This solution leads to a direct verification of general relativity in three
important problems, namely, perihelion precession of planets, gravitational
deflection of light, and gravitational red shift; we discuss these problems
below); - The solution is a lot more general than it looks: in 1927, the American
mathematician GEORGE DAVID BIRKHOFF (1884-1944) proved that, after
a suitable change of coordinates, every spherically symmetric solution of
the vacuum field equations becomes a Schwarzschild solution. - The solution formula (2.4.35) leads to a natural, and mathematically pre-
cise, definition of a black hole as an object whose size is smaller than or