page_title: Pseudo-Euclidean space
section_title: Positive, zero, and negative scalar squares
hierarchical_section_title: Pseudo-Euclidean space / Geometry / Positive, zero, and negative scalar squares
caption_attribution_description: Deutsch: Doppelkegel English: double cone
context_page_description: 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, 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. Note that if 1 ≤ i ≤ k and k < j ≤ n, then q = 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.
context_section_description: 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. Similar statements can be made for vectors for which q(x) < 0 if k is replaced with n − k.
caption: n = 3, k is either 1 or 2 depending on the choice of sign of q
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