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 (e1, ..., en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vector x. 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 ≤ i ≤ k < j ≤ n, then q(ei + ej) = 0, so that ei + ej 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). The geometry of a pseudo-Euclidean space is consistent despite some properties of Euclidean space not applying, 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. A null vector is a vector for which the quadratic form is zero. Unlike in a Euclidean space, such a vector can be non-zero, in which case it is self-orthogonal. If the quadratic form is indefinite, a pseudo-Euclidean space has a linear cone of null vectors given by {x : q(x) = 0}. When the pseudo-Euclidean space provides a model for spacetime (see below), the null cone is called the light cone of the origin. The null cone separates two open sets, respectively for which q(x) > 0 and q(x) < 0. If k ≥ 2, then the set of vectors for which q(x) > 0 is connected. If k = 1, then it consists of two disjoint parts, one with x1 > 0 and another with x1 < 0. Similarly, if n − k ≥ 2, then the set of vectors for which q(x) < 0 is connected.