Introduction To Mathematics of General Relativity - Curvilinear Coordinates and Curved Spacetime

Curvilinear Coordinates and Curved Spacetime

Curvilinear coordinates are coordinates in which the angles between axes can change from point-to-point. This means that rather than having a grid of straight lines, the grid instead has curvature.

A good example of this is the surface of the Earth. While maps frequently portray north, south, east and west as a simple square grid, that is not, in fact, the case. Instead, the longitude lines, running north and south, are curved, and meet at the north pole. This is because the Earth is not flat, but instead round.

In general relativity, gravity has curvature effects on the four dimensions of the universe. A common analogy is placing a heavy object on a stretched out rubber sheet, causing the sheet to bend downward. This curves the coordinate system around the object, much like an object in the universe curves the coordinate system it sits in. The mathematics here are conceptually more complex than on Earth, as it results in 4 dimensions of curved coordinates instead of 3 as used to describe a curved 2D surface.