Line Elements and Metric Tensors

Back to General Relativity

Line Elements and Metric Tensors

  • The line element ds for a curve or a space is the shortest distance between two neighboring points.  Using calculus, ds can be used to calculate distances, lengths, angles, areas, volumes, and geodesics (the shortest line between two points).
  • The line element is determined by the Metric Tensor and coordinates.
  • Thus:
    • Metric Tensor + Coordinates ➞ Line Element
    • Line Element + Calculus ➞ Distances, Lengths, Angles, Areas, Volumes, Geodesics

Examples

Metric Tensor for Euclidean Plane in Cartesian Coordinates
Metric Tensor for Euclidean Space in Spherical Coordinate
Metric Tensor for Four Dimensional Space in Cartesian Coordinates
Metric Tensor for Flat Spacetime in Cartesian Coordinates
Metric Tensor for Flat Spacetime in Spherical Coordinates