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{{Expert-subject|Physics|date=November 2008}} | |||
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. | |||
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 | |||
<math>g[(t,x)] = -\beta(x) dt^{2} + g_{S}[x]</math>, | |||
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''. | |||
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. | |||
==Examples of static spacetimes== | |||
#The (exterior) [[Schwarzschild solution]] | |||
#[[de Sitter space]] (the portion of it covered by the [[De Sitter space#Static coordinates|static patch]]). | |||
#[[Reissner-Nordström]] space | |||
#The [[Weyl solution]], a static axisymmetric solution of the Einstein vacuum field equations <math>R_{\mu\nu} = 0</math> discovered by [[Hermann Weyl]] | |||
==External links== | |||
* [http://www.rqgravity.net/GeneralRelativity#StationaryObservers Concepts of General Relativity ] - A first principles demonstration of the metric in static spacetime | |||
* [http://arxiv.org/abs/math/0406332v2 M. Sanchez, On the geometry of static space-times, preprint, 2004.] | |||
{{relativity-stub}} | |||
[[Category:Lorentzian manifolds]] | |||
Revision as of 04:32, 31 January 2014
In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field which is irrotational, i.e., whose orthogonal distribution is involutive. (Note that the leaves of the associated foliation are necessarily space-like hypersurfaces.) Thus, a static spacetime is a stationary spacetime satisfying this additional integrability condition. These spacetimes form one of the simplest classes of Lorentzian manifolds.
Locally, every static spacetime looks like a standard static spacetime which is a Lorentzian warped product R S with a metric of the form , where R is the real line, is a (positive definite) metric and is a positive function on the Riemannian manifold S.
In such a local coordinate representation the Killing field may be identified with and S, the manifold of -trajectories, may be regarded as the instantaneous 3-space of stationary observers. If is the square of the norm of the Killing vector field, , both and are independent of time (in fact ). 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.
Examples of static spacetimes
- The (exterior) Schwarzschild solution
- de Sitter space (the portion of it covered by the static patch).
- Reissner-Nordström space
- The Weyl solution, a static axisymmetric solution of the Einstein vacuum field equations discovered by Hermann Weyl
External links
- Concepts of General Relativity - A first principles demonstration of the metric in static spacetime
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