Geodesic

A geodesic is a path that is locally straightest in a curved space or spacetime. In flat space it is a straight line; on a sphere, it follows a great circle.

Geodesic equation: A differential equation describing paths whose tangent vectors are parallel transported along themselves. It expresses local straightness as a differential equation in curved coordinates.
Manifold: A space that locally resembles Euclidean space and supports consistent systems of coordinates. Geodesics generalize straight lines from Euclidean space to manifolds.
General relativity: Einstein's theory of gravity, in which matter and energy shape spacetime geometry. Freely falling bodies follow spacetime geodesics in general relativity.
Euclidean straight line: A line in Euclidean space whose direction remains constant and whose segments realize the distance between their endpoints. It is the flat-space case of a geodesic, not the general curved-space form.
Great circle: A circle on a sphere formed by intersecting it with a plane through its center. Its arcs are the sphere's geodesics and give its locally shortest surface paths.
Riemannian metric: A smoothly varying inner product that defines lengths and angles in each tangent space of a manifold. The metric determines which paths count as geodesics.
Tangent space: A vector space at a point that represents the manifold's possible directions of motion there. A geodesic begins with a direction chosen in its initial tangent space.
Schwarzschild geodesics: Geodesic paths in the static, spherically symmetric vacuum spacetime described by the Schwarzschild solution. They model orbital motion and light paths around an idealized spherical mass.
Minimal geodesic: A geodesic segment whose length is no greater than that of competing paths between its endpoints. Not every geodesic minimizes distance beyond a sufficiently small neighborhood.
Meridian: A curve on Earth running from the North Pole to the South Pole along a line of longitude. Each meridian is a great circle and therefore a spherical geodesic.

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