A Model of Spacetime

Lines will correspond to our usual notion of line.

Distances will be defined using the Minkowski dot product, so that the distance between events and is . We can then get the length of a curved line by cutting it into small, approximately straight segments, and adding the lengths of each segment as found using this formula.

First, note the qualitative difference in 'timelike' and 'spacelike' vectors, as indicated by the unit imaginary number i found in timelike vectors. Also note that distances are linear -- that . In addition, the 'spheres' of spacetime - the loci of points a given distance from another - are hyperbolae in the Euclidean sense.

We can also define the hyperbolic angle between two timelike, future-pointing vectors, again use the Minkowski dot product. Where Euclidean angles are defined by , angles in spacetime are defined by the analogous formula , where * represents the Minkowski product, and the last two norms are the Minkowski norms.