Geodesic

For metric spaces

Geodesic is a curve which is locally distance minimizer. More precicely if is a metric space a curve is a geodesic if there is a constant such that for any there is a neighborhood of in such that for any we have . If the last equality is satisfyed on all it is called minimizing geodesic or shortest path

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. In general, metric space can have no geodesic exapt constant curves.

Riemannian and pseudo-Riemannian manifolds

If is Riemannian manifold then geodesics are allways smooth curves so one can define , and the above definition is equivalent to , where stays for covariant derivative.

The last definition has sense for all manifolds with connection in particular for Levi Civita connection on Pseudo-Riemannian manifolds.

General relativity

The space-time in the theory of general relativity is a Pseudo-Riemannian manifold, and geodesic 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) follow a path called nul geodesics.

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