Geodesic

In mathematicsspecificallydifferential geometry,geodesic ispath that furnishes shortest paths between any points onthatclose enough together. The most familiar examples arestraight linesEuclidean geometry. In more general spacesgeodesics can be more complicated, but one often still thinksthem as "straight" insense.

Onsphere,instance,geodesics aregreat circles. The shortest path from point Apoint B onspheregiven byshorter piece ofgreat circle passing through AB. Note that if ABantipodal points (likeNorth pole andSouth pole), then theremany shortest paths between them.

Intheorygeneral relativity, particles travel along geodesics through space-time. Everything"free fall" such asorbitan astronaut, ororbit ofplanet followsso called timelike geodesic, also calledworld line. Light (photonsgeneral) followpath called nul geodesics. The pathobjectslight depend onspace-time's curvature. This curvaturein turn determined byenergymass distribution; this iscontent ofEinstein field equation.

In general, geodesics can be definedany Riemannian manifold. Every shortest path from AB yieldsgeodesic, butconversenot always true, asexample ofsphere shows. Furthermore, itpossible that thereno shortest paths from AB, but theregeodesics connecting AB. An examplethis issphere withpoint between AB removed.

See also geodesic dome.