The paths that objects follow in curved space-time, such as the trajectory of a planet orbiting a star.
Differential geometry: The branch of mathematics that deals with the study of geometrical objects such as curves, surfaces, and spaces.
Manifolds: A manifold is a mathematical object that locally looks like Euclidean space of a certain dimension, but globally can have much more complicated topology.
Curvature: Curvature measures how much a curve deviates from a straight line or a surface deviates from a plane.
Geodesics: A geodesic is a curve that locally minimizes the distance between two points on a curved space.
Riemannian geometry: The field of mathematics that studies differential geometry in the context of Riemannian manifolds, which are manifolds equipped with a metric.
General relativity: The theory of gravitation developed by Einstein that describes the interaction of matter and energy with space and time.
Schwarzschild metric: The metric that describes the geometry of spacetime outside a non-rotating, spherically symmetric mass, which is often used to derive the effects of gravity on the motion of objects in the vicinity of a massive object.
Kerr metric: The metric that describes the geometry of spacetime outside a rotating, axially symmetric black hole.
Christoffel symbols: The Christoffel symbols are a set of coefficients that appear in the equations of motion for a particle moving on a curved manifold.
Covariant derivative: A covariant derivative is a derivative that is defined in a way that is compatible with the geometry of a curved space.
Parallel transport: Parallel transport refers to the motion of a vector or tensor along a curve on a curved manifold, while keeping the object parallel to itself at each point along the curve.
Einstein's field equations: The set of equations that describe the relationship between the geometry of a curved spacetime and the matter and energy that are present in that spacetime.
Geodesic deviation: The deviation of two initially parallel geodesics provides a measure of the curvature of a manifold.
Gravitational redshift: The frequency of electromagnetic radiation is shifted when it is emitted from or absorbed by an object in a gravitational field.
Geodetic precession: The gradual change in the orientation of the axis of rotation of a gyroscope that is moving along a geodesic in a curved spacetime.
Spacelike Geodesics: These are paths that lie entirely within the spatial dimensions of spacetime and do not have a timelike component. They represent the trajectory of a particle moving at a speed less than the speed of light.
Timelike Geodesics: These are paths that have a timelike component and represent the trajectory of a massive particle. The proper time along this path is nonzero and can be measured by a clock carried by the particle.
Null Geodesics: These are paths that have zero proper time associated with them and represent the trajectory of a massless particle, such as a photon. These paths are "tangent" to the light cone at each point along the path.
Radial Geodesics: These are paths that are radial out from a central mass or black hole. They can be either timelike or null depending on the radial velocity of the particle.
Circular Geodesics: These are paths that are closed and circular around a central mass or black hole. They can be either timelike or null depending on the velocity of the particle. These paths are also known as "orbit" or "trajectory".
Parabolic Geodesics: These are paths that are not closed and unbound around a central mass or black hole, but instead have an escape velocity. They are always outgoing and have a zero total energy.
Hyperbolic Geodesics: These are paths that are unbound and asymptotic to a central mass or black hole. They can be either incoming or outgoing and have a nonzero total energy..