Back to General Relativity

Table of Contents

1. Line Elements and Metric Tensors
2. Examples
   1. Metric Tensor for Euclidean Plane in Cartesian Coordinates
   2. Metric Tensor for Euclidean Space in Spherical Coordinate
   3. Metric Tensor for Four Dimensional Space in Cartesian Coordinates
   4. Metric Tensor for Flat Spacetime in Cartesian Coordinates
   5. Metric Tensor for Flat Spacetime in Spherical Coordinates

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