en>Torbjørn T.: Improved math typesetting.

2014-10-16T11:07:18Z

Improved math typesetting.

en>Dthomsen8: /* Measurement of long wave downward radiation */clean up, typos fixed: so called → so-called using AWB

2013-08-04T22:44:36Z

Measurement of long wave downward radiation: clean up, typos fixed: so called → so-called using AWB

← Older revision		Revision as of 00:44, 5 August 2013
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	~~Her name~~ is ~~Felicidad Ahmad~~. The ~~preferred hobby~~ for ~~my children~~ and me is ~~playing crochet and now I'm attempting~~ to ~~make money with~~ it. ~~Delaware~~ is the ~~only location I~~'~~ve been residing~~ in. ~~I am~~ a ~~production~~ and ~~distribution officer~~.<br><br>~~Feel free~~ to ~~visit my weblog; extended auto warranty (~~[~~http~~://~~Www~~.~~Edusharer~~.~~com~~/~~members~~/~~profile~~/~~272415/AnGraves Check Out Www~~.~~Edusharer~~.~~com~~])		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.

			== Explanation ==

			Specifically, a vector field ''X'' is a Killing field if the [[Lie derivative]] with respect to ''X'' of the metric ''g'' vanishes:

			:<math>\mathcal{L}_{X} g = 0 \,.</math>

			In terms of the [[Levi-Civita connection]], this is

			:<math>g(\nabla_{Y} X, Z) + g(Y, \nabla_{Z} X) = 0 \,</math>

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

			:<math>\nabla_{\mu} X_{\nu} + \nabla_{\nu} X_{\mu} = 0 \,.</math>

			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===

			* 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.

			* If the metric coefficients <math>g_{\mu \nu} \,</math> in some coordinate basis <math>dx^{a} \,</math> are independent of <math>x^{K} \,</math>, then <math>x^{\mu} = \delta^{\mu}_{K} \,</math> is automatically a Killing vector, where <math>\delta^{\mu}_{K} \,</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>
			*: 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 of Killing fields===

			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).

			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.

			For [[compact space\|compact]] manifolds
			* Negative [[Ricci curvature]] implies there are no nontrivial (nonzero) Killing fields.
			* 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]].

			===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.

			Another generalization is to conformal Killing tensor fields. These are symmetric [[tensor]] fields ''T'' such that the trace-free part of the symmetrization of <math>\nabla T \,</math> vanishes.

			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 left 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]]

129.13.72.198: /* Measurement of long wave downward radiation */

2012-04-25T14:08:22Z

Measurement of long wave downward radiation

New page

Her name is Felicidad Ahmad. The preferred hobby for my children and me is playing crochet and now I'm attempting to make money with it. Delaware is the only location I've been residing in. I am a production and distribution officer.<br><br>Feel free to visit my weblog; extended auto warranty ([http://Www.Edusharer.com/members/profile/272415/AnGraves Check Out Www.Edusharer.com])

Pyrgeometer - Revision history

en>Torbjørn T.: Improved math typesetting.

en>Dthomsen8: /* Measurement of long wave downward radiation */clean up, typos fixed: so called → so-called using AWB

129.13.72.198: /* Measurement of long wave downward radiation */