Isotropic vector

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A non-zero vector that is orthogonal to itself. Let be a vector space over the field of real or complex numbers and let be a non-degenerate bilinear form of signature , , , on . Then an isotropic vector is a non-zero vector for which . 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 is called isotropic if there exists a non-zero vector orthogonal to (that is, the restriction of to is degenerate: ). A vector subspace 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 , the trajectories of photons are isotropic lines, while the isotropic cone is called the light cone.

How to Cite This Entry:
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