# Isotropic vector

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.

**How to Cite This Entry:**

Isotropic vector.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Isotropic_vector&oldid=47446