Mathematics of Physics and Engineering

Einstein's Field Equations 111

It turns out that a geodesic is always parameterized by its arc length
s, although not necessarily canonically. In other words, equations (2.4.30)
imply that ds = ads for some constant a > 0; canonical parametrization
corresponds to a = 1, see page 30.

EXERCISE 2.4.8.A (a) Verify that, if a vector function x = x(s) satisfies
(2.4.30), then

d (. , ..dxUs) dxi(s)\

Ts{

3

^

s))

-dir^r)=

0

• (

2A31

Hint: differentiate and use (2.4-27). (b) Conclude that 5 = as + f3, where s
is the arc length of the curve and a, j3 are real numbers. Hint: the left-hand
side of (2.4.31) is d^2 s(s)/ds^2.

By following the steps below, the reader can prove that the geodesic,
as defined by (2.4.30), can be interpreted as the shortest path between two
points in a curved space.
Step 1. It is easier to do the basic computations in an abstract setting in IRn.
Let A and B be two fixed points in Mn and F = F(q,p), a smooth M-valued
function of 2n variables q^1 ,..., qn,px,... ,pn; we keep the convention of this
section and write the indices as superscripts. Let x — x(s), a < s < b, be
a smooth function with values in Rn. Consider the functional

L(x)= I F(x(s),x'(s))ds,

J a

where x'(s) = dx(s)/ds; being a rule assigning a number to a function, L
is indeed a functional. Later, we will take

F{q-,P) = y9ij((l)PiPj (remember the summation convention), (2.4.32)

so that L(x) is the distance from A to B along the curve x(s).
(a) Let y = y(s), a < s < b, be a smooth function with values in ffi" so
that y(a) = y(b) = 0 and let x(s) be a function for which the value L(x)
is the smallest (assume that such a function exists.) For a real number A,
define /(A) = L(x + Xy). Argue that /'(0) = 0. (b) Show that

fb fdF(x(s),x'(s)) aF(£( 3 ),x'(s))dy'(a)\

n'~Ja V W V{S)+ d^ ds )dS-