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 $(e 1, \dots, e n)$ , be applied to a vector $x = x 1 e 1 + \dots + x n e n$ , giving

q(x)=\left(x_{1}^{2}+\dots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\dots +x_{n}^{2}\right)

which is called the scalar square of the vector $x$ .^: 3

For Euclidean spaces, $k = n$ , implying that the quadratic form is positive-definite. When $0 < k < n$ , $q$ is an isotropic quadratic form, otherwise it is anisotropic. Note that if $1 \leq i \leq k < j \leq n$ , then $q (e i + e j) = 0$ , so that $e i + e j$ is a null vector. In a pseudo-Euclidean space with $k < n$ , unlike in a Euclidean space, there exist vectors with negative scalar square.

As with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction).

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