Geodesics in general relativity

Geodesics are the paths that freely falling particles and light follow through spacetime in general relativity. Their shapes are determined by spacetime geometry rather than by an ordinary force acting on them.

Connect

Geodesic equation: A differential equation describing curves whose tangent is parallel-transported along itself. It calculates free-fall paths from the spacetime metric and its derivatives.
Spacetime: The four-dimensional framework combining three dimensions of space with one dimension of time. Geodesics are curves within the spacetime on which gravity is described geometrically.
Schwarzschild geodesics: Geodesic paths in the vacuum spacetime outside a static, spherically symmetric mass. They describe orbital motion and light propagation outside an ideal nonrotating star or black hole.
Newtonian gravity: A theory in which masses attract through a gravitational force described by Newton's laws. It explains falling with force and acceleration rather than motion along spacetime geodesics.
Metric tensor: A mathematical object that defines distances, durations, and angles in spacetime. Its components determine the connection used in the geodesic equation.
Proper time: The time measured by a clock moving along a particular worldline. A massive particle's geodesic locally extremizes elapsed proper time.
Kerr geodesics: Geodesic paths in the spacetime around an ideal rotating black hole. Frame dragging alters particle orbits and light paths around rotating sources.
Minkowski spacetime: The flat spacetime of special relativity, with no gravitational curvature. Its inertial geodesics are straight, unlike paths shaped by curved spacetime.
Christoffel symbols: Coordinate-dependent quantities that encode how coordinate basis vectors change across a curved space. They supply the connection coefficients in the coordinate form of geodesic motion.
Worldline: A curve in spacetime tracing an object's position through time. A freely falling object's history is a timelike geodesic worldline.