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Static spacetime : ウィキペディア英語版
Static spacetime

In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K 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'' \times ''S'' with a metric of the form
g() = -\beta(x) dt^ + g_(),
where ''R'' is the real line, g_ is a (positive definite) metric and \beta is a positive function on the Riemannian manifold ''S''.
In such a local coordinate representation the Killing field K may be identified with \partial_t and ''S'', the manifold of K-''trajectories'', may be regarded as the instantaneous 3-space of stationary observers. If \lambda is the square of the norm of the Killing vector field, \lambda = g(K,K), both \lambda and g_S are independent of time (in fact \lambda = - \beta(x)). 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 R_ = 0 discovered by Hermann Weyl

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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