Isotropic vector

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A non-zero vector that is orthogonal to itself. Let $ E $ be a vector space over the field of real or complex numbers and let $ \Phi $ be a non-degenerate bilinear form of signature $ ( p , q ) $, $ p \neq 0 $, $ q \neq 0 $, on $ E \times E $. Then an isotropic vector is a non-zero vector $ x \in E $ for which $ \Phi ( x , x ) = 0 $. One sometimes says that an isotropic vector has zero length (or norm). The set of all isotropic vectors is called the isotropic cone. A subspace $ V \subset E $ is called isotropic if there exists a non-zero vector $ z \in V $ orthogonal to $ V $( that is, the restriction of $ \Phi $ to $ V \times V $ is degenerate: $ V \cap V ^ \perp \neq \{ 0 \} $). A vector subspace $ V $ is said to be totally isotropic if all its vectors are isotropic vectors.

In the relativistic interpretation of the Universe, space-time is locally regarded as a four-dimensional vector space with a form of signature $ ( 3 , 1 ) $, the trajectories of photons are isotropic lines, while the isotropic cone is called the light cone.

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Isotropic vector. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article