Environmental Engineering Reference
In-Depth Information
Appendix A
Deriving the Geodesic
Equation
In this appendix we shall present a derivation of the geodesic equation. We take the
approach that a geodesic is the curve through a curved space which corresponds to an
extremum of the distance between two points. For example, geodesics on the surface
of a sphere are also the curves of shortest length. We shall not prove that these curves
correspond to the curves which are generated by sewing together straight lines on
locally flat patches, as discussed in the text, but it should not come as too great a
surprise that the two are equivalent. The method we shall use is an example of what
is called the calculus of variations and it is useful in a number of areas in theoretical
physics, perhaps the most notable being the principal of least action from which one
can derive the classical equations of motion without resorting to Newton's laws or
the notion of force. By eliminating the concept of force, the path to quantum theory
is cleared. Here we shall focus only on obtaining the geodesic equation.
Generally the length of a curve between two points
A
and
B
is
B
L
=
d
s.
(A.1)
A
This is the sum of many line elements each of length d
s
. Any curve between
A
and
B
can be expressed in terms of a set of co-ordinates, i.e.
x
i
(s)
defines a general
curve such that
(
d
s)
2
=
g
ij
d
x
i
d
x
j
.
(A.2)
Consider a general curve between
A
and
B
. We can obtain a second curve from
this by varying the line elements at each and every point along its path. In this
