Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

components of a second-order tensorTare

δTij

δt

≡T

ij

;k

duk

dt

,

δTij

δt

≡Tij;k

duk

dt

,

δTij

δt

≡Tij;k

duk

dt

.

The derivative ofTalong the curver(t) may then be written in terms of, for

example, its contravariant components as

dT

dt

=

δTij

δt

ei⊗ej=Tij;k

duk

dt

ei⊗ej.

26.22 Geodesics

As an example of the use of the absolute derivative, we conclude this chapter

with a brief discussion of geodesics. A geodesic in real three-dimensional space

is a straight line, which has two equivalent defining properties. Firstly, it is the

curve of shortest length between two points and, secondly, it is the curve whose

tangent vector always points in the same direction (along the line). Although

in this chapter we have considered explicitly only our familiar three-dimensional

space, much of the mathematical formalism developed can be generalised to more

abstract spaces of higher dimensionality in which the familiar ideas of Euclidean

geometry are no longer valid. It is often of interest to find geodesic curves in

such spaces by using the defining properties of straight lines in Euclidean space.

We shall not consider these more complicated spaces explicitly but will de-

termine the equation that a geodesic in Euclidean three-dimensional space (i.e.

a straight line) must satisfy, deriving it in a sufficiently general way that our

method may be applied with little modification to finding the equations satisfied

by geodesics in more abstract spaces.

Let us consider a curver(s), parameterised by the arc lengthsfrom some point

on the curve, and choose as our defining property for a geodesic that its tangent

vectort=dr/dsalways points in the same direction everywhere on the curve, i.e.

dt

ds

= 0. (26.100)

Alternatively, we could exploit the property that the distance between two points

is a minimum along a geodesic and use the calculus of variations (see chapter 22);

this would lead to the same final result (26.101).

If we now introduce an arbitrary coordinate systemuiwith basis vectorsei,

i=1, 2 ,3, then we may writet=tiei, and from (26.99) we find

dt

ds

=ti;k

duk

ds

ei= 0.