112 Theory of Relativity
(c) Integrate by parts and use y%{a) = yl(b) = 0 to conclude that
dF(x(s),x'(s)) d dF(x(s),x'(s))\ i(
W Ts W )y{s)ds-
(d) Argue that, since /'(0) = 0 and y = y{s) is an arbitrary smooth
function, the expression in the big parentheses in the last equality must
vanish for all s and so, for all i = 1,..., n,
d^2 F{x(s),x(s))dxi d^2 F{x(s),x(s))d^2 x^ dF{x(s),x{s))
dqidqi ds dfdpi ds^2 dqi ~
(2.4.33)
System (2.4.33) is an example of the Euler-Lagrange equations of vari-
ational calculus.
Step 2. Verify that, with F as in (2.4.32) and s = s, equation (2.4.33)
becomes (2.4.30).
While in most situations of practical importance the geodesic does in-
deed define the shortest path between two points, and the path is unique up
to a re-parametrization, there are two main technical problems with defin-
ing the geodesic as the distance-minimizing path: (i) The a priori possibility
that several different paths have this property, and (ii) The possibility that
the sign of (2.4.32) is different in different parts of the space (even the flat
Minkowski metric (2.4.18) can change sign.) Definition (2.4.30) avoids these
problems by generalizing a different property of the straight line, namely,
that the tangent vectors at every two points of the curve are parallel. This
property formalizes the intuitive notion of going straight, and the functions
Tljk define the parallel transport of a vector along a curve by means of the
Levi-Civita connection. For more details, an interested reader should
consult a specialized reference on Riemannian geometry, such as Chapter 10
of the book Differentiable Manifolds: A First Course by L. Conlon, 1993.
The rest of the section will be an investigation of the important spe-
cial case of (2.4.22), corresponding to Tij = 0, namely, the equation
Rij [g] — (l/2)Rgij = 0. The solutions of this equation are called the vacuum
solutions of (2.4.22), because Ty = 0 in empty space.
EXERCISE 2.4.9.c Verify that Rij[g] - (1/2)11$^ = 0 if and only ifRij[g]=0. (2.4.34)Hint: use (2.4-28) and (2.4-29); the matrices g and j?-1 are non-singular.m
-I.(