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| {{Expert-subject|Physics|date=November 2008}}
| | I'm Thad and I live with my husband and our two children in Dirksland, in the ZH south area. My hobbies are Homebrewing, Bonsai and Auto audiophilia.<br><br>Feel free to visit my web blog ... fifa 15 coin generator ([http://Asadimpex.com/?p=1 http://asadimpex.com/?p=1]) |
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| In [[general relativity]], a [[spacetime]] is said to be '''static''' if it admits a global, non-vanishing, [[timelike]] [[Killing vector field]] <math>K</math> which is '''irrotational''', ''i.e.'', whose [[orthogonal distribution]] is [[involutive]]. (Note that the leaves of the associated [[foliation]] are necessarily space-like [[hypersurface]]s.) Thus, a static spacetime is a [[stationary spacetime]] satisfying this additional integrability condition. These spacetimes form one of the simplest classes of [[Lorentzian manifold]]s.
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| Locally, every static spacetime looks like a '''standard static spacetime''' which is a Lorentzian warped product ''R'' <math>\times</math> ''S'' with a metric of the form
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| <math>g[(t,x)] = -\beta(x) dt^{2} + g_{S}[x]</math>,
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| where ''R'' is the real line, <math>g_{S}</math> is a (positive definite) metric and <math>\beta</math> is a positive function on the [[Riemannian manifold]] ''S''.
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| In such a local coordinate representation the [[Killing vector field|Killing field]] <math>K</math> may be identified with <math>\partial_t</math> and ''S'', the manifold of <math>K</math>-''trajectories'', may be regarded as the instantaneous 3-space of stationary observers. If <math>\lambda</math> is the square of the norm of the Killing vector field, <math>\lambda = g(K,K)</math>, both <math>\lambda</math> and <math>g_S</math> are independent of time (in fact <math>\lambda = - \beta(x)</math>). It is from the latter fact that a static spacetime obtains its name, as the geometry of the space-like slice ''S'' does not change over time.
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| ==Examples of static spacetimes==
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| #The (exterior) [[Schwarzschild solution]]
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| #[[de Sitter space]] (the portion of it covered by the [[De Sitter space#Static coordinates|static patch]]).
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| #[[Reissner-Nordström]] space
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| #The [[Weyl solution]], a static axisymmetric solution of the Einstein vacuum field equations <math>R_{\mu\nu} = 0</math> discovered by [[Hermann Weyl]]
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| ==External links==
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| * [http://www.rqgravity.net/GeneralRelativity#StationaryObservers Concepts of General Relativity ] - A first principles demonstration of the metric in static spacetime
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| * [http://arxiv.org/abs/math/0406332v2 M. Sanchez, On the geometry of static space-times, preprint, 2004.]
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| {{relativity-stub}}
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| [[Category:Lorentzian manifolds]]
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I'm Thad and I live with my husband and our two children in Dirksland, in the ZH south area. My hobbies are Homebrewing, Bonsai and Auto audiophilia.
Feel free to visit my web blog ... fifa 15 coin generator (http://asadimpex.com/?p=1)