Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real $n$ -space together with a non-degenerate quadratic form $q$ . Such a quadratic form can, given a suitable choice of basis $(e 1, ..., e n)$ , be applied to a vector $x = x 1 e 1 + ... + x n e n$ , giving

For Euclidean spaces, $k = n$ , implying that the quadratic form is positive-definite. When $0 \neq k \neq n$ , $q$ is an isotropic quadratic form. Note that if $i \leq k$ and $j > k$ , then $q (e i + e j) = 0$ , so that $e i + e j$ is a null vector. In a pseudo-Euclidean space with $k \neq n$ , unlike in a Euclidean space, there exist vectors with negative magnitude.

As with the term Euclidean space, pseudo-Euclidean space may refer to either anaffine space or a vector space, though the latter may also be referred to as a pseudo-Euclidean vector space (see point–vector distinction).

The geometry of a pseudo-Euclidean space is consistent in spite of a breakdown of the some properties of Euclidean space; most notably that it is not a metric space as explained below. The affine structure is unchanged, and thus also the concepts line, plane and, generally, of an affine subspace (flat), as well as line segments.

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