geodesic

A geodesic is a generalization of a straight line: in flat Euclidean space, it is the shortest path between two points. Within curved space or on a curved surface, the shortest path is a geodesic, but there may be additional geodesics. Regarding the curved surface of the Earth, a geodesic between two points follows the great circle through the two points, the shortest path following this great circle, but the long way around the great circle is also a geodesic, i.e., it can be thought of as a straight path between them, and an example of two points with multiple geodesics between them are the north and south poles. (I'm treating the Earth as perfectly spherical: actual geodesics between two points on the Earth's surface are not quite so simple.) The term geodesic stems from this use regarding paths along Earth's surface, e.g., for navigation.

In general relativity (GR), the path of a particle (or electromagnetic radiation) when no accelerating force other than gravity is present is a geodesic within spacetime. GR treats gravity by conceiving space and time as a spacetime such that the effects of gravity are due to geometry, i.e., an object affected by gravity and otherwise uninfluenced is following a geodesic within this spacetime. GR spacetime's more complicated curvature does have multiple geodesics between some pairs of points, an example indicated by the multiple images seen in gravitational lensing. Another example consists of those between pairs points at which a pair of orbits intersect.

Two points in spacetime (i.e., each an instantaneous event, each at some point in space) can be separated in a time-like manner, i.e., a geodesic exists that could be traveled, or in a space-like manner, i.e., too far apart to be reached in the given time difference since it would require superluminal motion (faster-than-light travel).