Examples
A very important pseudo-Euclidean space is Minkowski space, which is the mathematical setting in which Albert Einstein's theory of special relativity is conveniently formulated. For Minkowski space, n = 4 and k = 3 so that
The geometry associated with this pseudo-metric was investigated by Poincaré. Its rotation group is the Lorentz group. The Poincaré group includes also translations and plays the same role as Euclidean groups of ordinary Euclidean spaces.
Another pseudo-Euclidean space is the plane z = x + y j consisting of split-complex numbers, equipped with the quadratic form
This is the simplest case of a pseudo-Euclidean space (n = 2, k = 1) and the only one where the null cone dissects the space to four open sets. The group SO+(1, 1) consists of so named hyperbolic rotations.
Read more about this topic: Pseudo-Euclidean Space
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