Coordinate singularity

A coordinate singularity is a point or region where a coordinate system makes a smooth geometric or physical description appear ill-defined. It disappears when the description is rewritten in suitable coordinates.

Coordinate chart: A coordinate chart assigns numerical coordinates to points in part of a geometric space. A singularity can mark where one chart stops describing an otherwise regular space.
Polar coordinates: A planar coordinate system that locates points by distance from an origin and angle around it. At the origin, the angle is undefined although the plane itself is smooth.
Curvature singularity: A spacetime singularity where curvature becomes unbounded or geometric evolution becomes incomplete. Unlike a coordinate singularity, it cannot be removed merely by switching coordinates.
Differentiable manifold: A space that locally resembles Euclidean space and supports smooth coordinate changes. Coordinate singularities concern the limits of local descriptions of an underlying manifold.
Coordinate transformation: A rule that converts coordinates describing the same point or geometric object between coordinate systems. Changing coordinates can remove the apparent breakdown without changing the geometry.
Spherical coordinates: A three-dimensional coordinate system using radial distance and two angles. Its angular coordinates fail at the poles, where multiple coordinate pairs describe the same point.
Event horizon: A boundary beyond which events cannot influence distant observers in a black-hole spacetime. The Schwarzschild chart becomes singular at this boundary, though the horizon itself is regular.
Atlas (mathematics): A collection of compatible coordinate charts covering a manifold. An atlas can use multiple charts where no single coordinate system works everywhere.
Jacobian matrix: A matrix of first derivatives that describes how a differentiable map changes locally. A degenerate coordinate transformation can signal that a chart has failed locally.
Schwarzschild coordinates: A coordinate system used to describe the exterior gravitational field of a spherical, nonrotating mass. Its apparent breakdown at the Schwarzschild radius is removable by changing coordinates.

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