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Concept
Stationary spacetime
Résumé
In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.
In a stationary spacetime, the metric tensor components, , may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form
where is the time coordinate, are the three spatial coordinates and is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field has the components . is a positive scalar representing the norm of the Killing vector, i.e., , and is a 3-vector, called the twist vector, which vanishes when the Killing vector is hypersurface orthogonal. The latter arises as the spatial components of the twist 4-vector (see, for example, p. 163) which is orthogonal to the Killing vector , i.e., satisfies . The twist vector measures the extent to which the Killing vector fails to be orthogonal to a family of 3-surfaces. A non-zero twist indicates the presence of rotation in the spacetime geometry.
The coordinate representation described above has an interesting geometrical interpretation. The time translation Killing vector generates a one-parameter group of motion in the spacetime . By identifying the spacetime points that lie on a particular trajectory (also called orbit) one gets a 3-dimensional space (the manifold of Killing trajectories) , the quotient space. Each point of represents a trajectory in the spacetime . This identification, called a canonical projection, is a mapping that sends each trajectory in onto a point in and induces a metric on via pullback. The quantities , and are all fields on and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case the spacetime is said to be static. By definition, every static spacetime is stationary, but the converse is not generally true, as the Kerr metric provides a counterexample.
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