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[[File:Killing-vector-s2.png|thumb|450 px|A Killing vector field (red) with integral curves (blue) on a sphere.]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In [[mathematics]], a '''Killing vector field''' (often just '''Killing field'''), named after [[Wilhelm Killing]], is a [[vector field]] on a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) that preserves the [[metric tensor|metric]]. Killing fields are the [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generator]]s of [[isometry|isometries]]; that is, [[flow (geometry)|flow]]s generated by Killing fields are [[Isometry (Riemannian geometry)|continuous isometries]] of the [[manifold]]. More simply, the flow generates a [[symmetry]], in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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== Definition ==
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Specifically, a vector field ''X'' is a Killing field if the [[Lie derivative]] with respect to ''X'' of the metric ''g'' vanishes:
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math>\mathcal{L}_{X} g = 0 \,.</math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


In terms of the [[Levi-Civita connection]], this is
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math>g(\nabla_{Y} X, Z) + g(Y, \nabla_{Z} X) = 0 \,</math>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


for all vectors ''Y'' and ''Z''. In [[local coordinates]], this amounts to the Killing equation
==Demos==


:<math>\nabla_{\mu} X_{\nu} + \nabla_{\nu} X_{\mu} = 0 \,.</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


This condition is expressed in covariant form. Therefore it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.


== Examples ==
* accessibility:
* The vector field on a circle that points clockwise and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle.
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


* If the metric coefficients <math>g_{\mu \nu} \,</math> in some coordinate basis <math>dx^{a} \,</math> are independent of <math>x^{\kappa} \,</math>, then <math>K^{\mu} = \delta^{\mu}_{\kappa} \,</math> is automatically a Killing vector, where <math>\delta^{\mu}_{\kappa} \,</math> is the [[Kronecker delta]].<ref>{{cite book | title=Gravitation | last = Misner, Thorne, Wheeler | year=1973 | publisher = W H Freeman and Company| isbn=0-7167-0344-0}}</ref><br /> To prove this, let us assume <math> g_{\mu \nu},_0=0 \,</math> <br /> Then <math> K^\mu=\delta^{\mu}_{0} \,</math> and <math> K_{\mu}=g_{\mu \nu} K^{\nu}= g_{\mu \nu} \delta^{\nu}_{0}= g_{\mu 0} \,</math> <br /> Now let us look at the Killing condition <br /> <math> K_{\mu;\nu}+K_{\nu;\mu}=K_{\mu,\nu}+K_{\nu,\mu}-2\Gamma^{\rho}_{\mu\nu}K_{\rho} = g_{\mu 0,\nu}+g_{\nu 0,\mu}-g^{\rho\sigma}(g_{\sigma\mu,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma})g_{\rho 0} \,</math> <br /> and from <math> g_{\rho 0}g^{\rho \sigma} = \delta_{0}^{\sigma} \,</math> <br /> The Killing condition becomes <br /> <math> g_{\mu 0,\nu}+g_{\nu 0,\mu} - ( g_{0\mu,\nu}+g_{0\nu,\mu}-g_{\mu\nu,0} ) = 0 \,</math> <br /> that is <math>g_{\mu\nu,0}= 0 </math>, which is true.
==Test pages ==
*: The physical meaning is, for example, that, if none of the metric coefficients is a function of time, the manifold must automatically have a time-like Killing vector.
*: In layman's terms, if an object doesn't transform or "evolve" in time (when time passes), time passing won't change the measures of the object. Formulated like this, the result sounds like a tautology, but one has to understand that the example is very much contrived: Killing fields apply also to much more complex and interesting cases.


== Properties ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all [[covariant derivative]]s of the field at the point).
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*[[Help:Formula]]


The [[Lie bracket of vector fields|Lie bracket]] of two Killing fields is still a Killing field. The Killing fields on a manifold ''M'' thus form a [[Lie algebra|Lie subalgebra]] of vector fields on ''M''. This is the Lie algebra of the [[isometry group]] of the manifold if ''M'' is complete.
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For [[compact space|compact]] manifolds
==Bug reporting==
* Negative [[Ricci curvature]] implies there are no nontrivial (nonzero) Killing fields.
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
* Nonpositive [[Ricci curvature]] implies that any Killing field is parallel. i.e. covariant derivative along any vector j field is identically zero.
* If the [[sectional curvature]] is positive and the dimension of ''M'' is even, a Killing field must have a zero.
 
The divergence of every Killing vector field vanishes.
 
If <math>X</math> is a Killing vector field and <math>Y</math> is a [[Hodge theory|harmonic vector field]], then <math>g(X,Y)</math> is a [[harmonic function]].
 
If <math>X</math> is a Killing vector field and <math>\omega</math> is a [[Hodge_theory|harmonic p-form]], then <math>\mathcal{L}_{X} \omega = 0 \,.</math>
 
=== Geodesics ===
Each Killing vector corresponds to a quantity which is conserved along [[Geodesics as Hamiltonian flows|geodesics]]. This conserved quantity is the metric product between the Killing vector and the geodesic tangent vector. That is, along a geodesic with some affine parameter <math>\lambda
</math>, the equation <math>\frac d {d\lambda} ( K_\mu \frac{dx^\mu}{d\lambda} ) = 0</math> is satisfied. This aids in analytically studying motions in a [[spacetime]] with symmetries.<ref>{{Cite book|title = An Introduction to General Relativity Spacetime and Geometry|last = Carrol|first = Sean|publisher = Addison Wesley|year = 2004|isbn = |location = |pages = 133-139}}</ref>
 
== Generalizations ==
* Killing vector fields can be generalized to [[Conformal vector field|''conformal'' Killing vector fields]] defined by <math>\mathcal{L}_{X} g = \lambda g \,</math> for some scalar <math>\lambda \,.</math> The derivatives of one parameter families of [[conformal map]]s are conformal Killing fields.
* Killing ''tensor ''fields are symmetric [[tensor]] fields ''T'' such that the trace-free part of the symmetrization of <math>\nabla T \,</math> vanishes. Examples of manifolds with Killing tensors include the [[Kerr spacetime|rotating black hole]] and the [[FRW cosmology]].<ref>{{Cite book|title = An Introduction to General Relativity Spacetime and Geometry|last = Carrol|first = Sean|publisher = Addison Wesley|year = 2004|isbn = |location = |pages = 263,344}}</ref>
* Killing vector fields can also be defined on any (possibly nonmetric) manifold M if we take any Lie group G [[group action|acting]] on it instead of the group of isometries.<ref>{{citation
  |last = Choquet-Bruhat
  |first = Yvonne
  |authorlink = Yvonne Choquet-Bruhat
  |first2 = Cécile |last2=DeWitt-Morette|authorlink2 = Cécile DeWitt-Morette| title = Analysis, Manifolds and Physics| publisher = Elsevier| year= 1977| location = Amsterdam |isbn = 978-0-7204-0494-4}}</ref> In this broader sense, a Killing vector field is the pushforward of a right invariant vector field on G by the group action. If the group action is effective, then the space of the Killing vector fields is isomorphic to the Lie algebra <math>\mathfrak{g}</math> of G.
 
==See also==
* [[Affine vector field]]
* [[Curvature collineation]]
* [[Homothetic vector field]]
* [[Killing form]]
* [[Killing horizon]]
* [[Killing spinor]]
* [[Killing tensor]]
* [[Matter collineation]]
* [[Spacetime symmetries]]
 
==Notes==
{{Reflist}}
 
==References==
* {{cite book | author=Jost, Jurgen| title= Riemannian Geometry and Geometric Analysis| location=Berlin | publisher=Springer-Verlag | year=2002 | isbn=3-540-42627-2}}.
* {{cite book | author=Adler, Ronald; Bazin, Maurice & Schiffer, Menahem| title= Introduction to General Relativity (Second Edition)| location=New York | publisher=McGraw-Hill | year=1975 | isbn=0-07-000423-4}}. ''See chapters 3,9''
 
{{DEFAULTSORT:Killing Vector Field}}
[[Category:Riemannian geometry]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

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  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

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To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

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