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Difference between revisions of "Isotropic vector"

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A non-zero vector that is orthogonal to itself. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529501.png" /> be a [[Vector space|vector space]] over the field of real or complex numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529502.png" /> be a non-degenerate [[Bilinear form|bilinear form]] of [[Signature|signature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529505.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529506.png" />. Then an isotropic vector is a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529507.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529508.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i0529509.png" /> is called isotropic if there exists a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295010.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295011.png" /> (that is, the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295013.png" /> is degenerate: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295014.png" />). A vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295015.png" /> is said to be totally isotropic if all its vectors are isotropic vectors.
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In the relativistic interpretation of the Universe, [[Space-time|space-time]] is locally regarded as a four-dimensional vector space with a form of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052950/i05295016.png" />, the trajectories of photons are isotropic lines, while the isotropic cone is called the light cone.
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A non-zero vector that is orthogonal to itself. Let  $  E $
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be a [[Vector space|vector space]] over the field of real or complex numbers and let  $  \Phi $
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be a non-degenerate [[Bilinear form|bilinear form]] of [[Signature|signature]]  $  ( p , q ) $,
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$  p \neq 0 $,
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$  q \neq 0 $,
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on  $  E \times E $.
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Then an isotropic vector is a non-zero vector  $  x \in E $
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for which  $  \Phi ( x , x ) = 0 $.
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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 $
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is called isotropic if there exists a non-zero vector  $  z \in V $
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orthogonal to  $  V $(
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that is, the restriction of  $  \Phi $
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to  $  V \times V $
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is degenerate:  $  V \cap V  ^  \perp  \neq \{ 0 \} $).
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A vector subspace  $  V $
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is said to be totally isotropic if all its vectors are isotropic vectors.
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In the relativistic interpretation of the Universe, [[Space-time|space-time]] is locally regarded as a four-dimensional vector space with a form of signature $  ( 3 , 1 ) $,  
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the trajectories of photons are isotropic lines, while the isotropic cone is called the light cone.

Latest revision as of 22:13, 5 June 2020


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