5.4 Geodesics as Self-parallel Curves
69
Example 5.1 The above constant angle property can give us insight on how
parallel displacement works in flat and curved spaces. Let us parallel displace
a vector around a triangle in Euclidean 2-space and also parallel displace one
around a triangle on the surface of a sphere as shown in Fig. 5.4.
In flat space (a) the angle between the vector and the base of the triangle
is chosen to be α = 90º at the lower left corner; at the lower right corner the
angle between the displaced vector and the right side of the triangle becomes
β = 30º; at the top the angle between the displaced vector and the left side of
the triangle becomes γ = 150º; finally at the lower left corner the displaced
vector returns to its original orientation and the angle between it and the base
returns to 90º. On the sphere (b) we repeat the analogous displacements around
a large triangle, an octant of the sphere between the equator and the north pole.
The figure makes it clear that the vector changes its orientation by 90º.
We say that the process of parallel displacing a vector is generally not integrable,
meaning that a parallel displaced vector at a given point has an orientation that
depends on the path taken to reach the point as illustrated in Fig. 5.4. Parallel
displacement on a curved surface is not integrable. As we will study in Chap. 8
this is a fundamental and defining characteristic of a curved space in general.
5.5 Geodesics as Extremum Curves
The self-parallel definition of a geodesic is one of several equivalent ones. In
Euclidean geometry a straight line is the shortest distance between two given points.
This property can be generalized to give the following definition of a geodesic: let
the curve C have length s between two fixed points; then C is a geodesic if the
length s is an extremum, that is it is either the shortest or longest among all nearby
curves. We will show that this definition leads to the differential equation (5.28) and
is equivalent to the self-parallel definition. The extremum calculation is a problem
in the calculus of variations, well-known in classical mechanics. If the reader is not
familiar with such problems and the Euler-Lagrange method of solution he should
first consult Appendix 2.
As before the curve C is denoted by
Curve C: x
μ
= x
μ
( p).
(5.30)
Here p is an invariant parameter, which may be the arc length of the curve but need
not be. This is shown schematically in Fig. 5.5.
The line element along the curve and the arc length s can be written as
69
Example 5.1 The above constant angle property can give us insight on how
parallel displacement works in flat and curved spaces. Let us parallel displace
a vector around a triangle in Euclidean 2-space and also parallel displace one
around a triangle on the surface of a sphere as shown in Fig. 5.4.
In flat space (a) the angle between the vector and the base of the triangle
is chosen to be α = 90º at the lower left corner; at the lower right corner the
angle between the displaced vector and the right side of the triangle becomes
β = 30º; at the top the angle between the displaced vector and the left side of
the triangle becomes γ = 150º; finally at the lower left corner the displaced
vector returns to its original orientation and the angle between it and the base
returns to 90º. On the sphere (b) we repeat the analogous displacements around
a large triangle, an octant of the sphere between the equator and the north pole.
The figure makes it clear that the vector changes its orientation by 90º.
We say that the process of parallel displacing a vector is generally not integrable,
meaning that a parallel displaced vector at a given point has an orientation that
depends on the path taken to reach the point as illustrated in Fig. 5.4. Parallel
displacement on a curved surface is not integrable. As we will study in Chap. 8
this is a fundamental and defining characteristic of a curved space in general.
5.5 Geodesics as Extremum Curves
The self-parallel definition of a geodesic is one of several equivalent ones. In
Euclidean geometry a straight line is the shortest distance between two given points.
This property can be generalized to give the following definition of a geodesic: let
the curve C have length s between two fixed points; then C is a geodesic if the
length s is an extremum, that is it is either the shortest or longest among all nearby
curves. We will show that this definition leads to the differential equation (5.28) and
is equivalent to the self-parallel definition. The extremum calculation is a problem
in the calculus of variations, well-known in classical mechanics. If the reader is not
familiar with such problems and the Euler-Lagrange method of solution he should
first consult Appendix 2.
As before the curve C is denoted by
Curve C: x
μ
= x
μ
( p).
(5.30)
Here p is an invariant parameter, which may be the arc length of the curve but need
not be. This is shown schematically in Fig. 5.5.
The line element along the curve and the arc length s can be written as