Geodesic
In mathematics, a geodesic is a curve which is "straight" in some sense. It takes its name from the science of geodesy of measuring the size and shape of the earth, and was originally the shortest route between two points on the surface of the earth. For example the great circle path between points on the Earth, idealised as a sphere, is a geodesic. A small circle path is not. In intuitive terms, an elastic band stretched along a path that is not geodesic would contract its length for energy reasons to a nearby shorter path — this though only serves to explain that a geodesic is a local minimum for length.
This concept arose in differential geometry, therefore. The definitions below begin with a more abstract definition.
For metric spaces
A geodesic is a curve which is everywhere locally a distance minimizer. More precisely, if <math>M<math> is a metric space, a curve <math>\gamma:I\to M<math> is a geodesic if there is a constant <math>v\ge 0<math> such that for any <math>t\in I<math> there is a neighborhood <math>J<math> of <math>t<math> in <math>I<math> such that for any <math>t_1,t_2\in J<math> we have
- <math>d(\gamma(t_1),\gamma(t_2))=v|t_1-t_2|<math>.
This notion generalizes notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered almost always equiped with natural parametrization, i.e. in the above identity v=1 and
- <math>d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|<math>.
If the last equality is satisfied on all <math>I<math>, the geodesic is called a minimizing geodesic or shortest path.
In general, a metric space may have no geodesics, except constant curves.
Riemannian and pseudo-Riemannian manifolds
If <math>M<math> is Riemannian manifold then geodesics are always smooth curves so one can define
- <math>T=\dot\gamma(t)<math>,
and the above definition is equivalent to <math>DT/dt=0<math>,
where <math>D<math> stands for covariant derivative.
The last definition makes sense for all manifolds with connection in particular for the Levi-Civita connection on pseudo-Riemannian manifolds.
Equivalently, geodesics can be defined as extremal curves for the following energy functional
- <math>E(\gamma)=\int g(\dot\gamma(t),\dot\gamma(t))\,dt,<math>
where <math>g<math> is Riemannian (or pseudo-Riemannian) metric.
(In fact, this "energy functional" should be called action, but nobody in mathematics does so.)
The most familiar examples are the straight lines in Euclidean geometry.
On a sphere, the geodesics are the great circles.
The shortest path from point A to point B on a sphere is given by the shorter piece of the great circle passing through A and B. Note that if A and B are antipodal points (like the North pole and the South pole), then there are many shortest paths between them.
General relativity
The space-time in the theory of general relativity is a pseudo-Riemannian manifold, and geodesics can be defined exactly as before. In space-time, particles travel along geodesics. Everything in "free fall", such as the orbit of an astronaut or the orbit of a planet, follows a so-called timelike geodesic, also called a world line. Light (photons, in general) follows a path called null geodesic.
