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• Stationary_spacetime abstract "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.In a stationary spacetime satisfying the vacuum Einstein equations outside the sources, the twist 4-vector is curl-free, and is therefore locally the gradient of a scalar (called the twist scalar): Instead of the scalars and it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, and , defined as In general relativity the mass potential plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential arises for rotating sources due to the rotational kinetic energy which, because of mass-energy equivalence, can also act as the source of a gravitational field. The situation is analogous to a static electromagnetic field where one has two sets of potentials, electric and magnetic. In general relativity, rotating sources produce a gravitomagnetic field which has no Newtonian analog. A stationary vacuum metric is thus expressible in terms of the Hansen potentials (, ) and the 3-metric . In terms of these quantities the Einstein vacuum field equations can be put in the form where , and is the Ricci tensor of the spatial metric and the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.".
• Stationary_spacetime wikiPageID "2073702".
• Stationary_spacetime wikiPageRevisionID "544042900".
• Stationary_spacetime hasPhotoCollection Stationary_spacetime.
• Stationary_spacetime subject Category:Lorentzian_manifolds.
• Stationary_spacetime type Artifact100021939.
• Stationary_spacetime type Conduit103089014.
• Stationary_spacetime type LorentzianManifolds.
• Stationary_spacetime type Manifold103717750.
• Stationary_spacetime type Object100002684.
• Stationary_spacetime type Passage103895293.
• Stationary_spacetime type PhysicalEntity100001930.
• Stationary_spacetime type Pipe103944672.
• Stationary_spacetime type Tube104493505.
• Stationary_spacetime type Way104564698.
• Stationary_spacetime type Whole100003553.
• Stationary_spacetime type YagoGeoEntity.
• Stationary_spacetime type YagoPermanentlyLocatedEntity.
• Stationary_spacetime comment "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.".
• Stationary_spacetime label "Espaço-tempo estacionário".
• Stationary_spacetime label "Spaziotempo stazionario".
• Stationary_spacetime label "Stationary spacetime".
• Stationary_spacetime sameAs Spaziotempo_stazionario.
• Stationary_spacetime sameAs Espaço-tempo_estacionário.
• Stationary_spacetime sameAs m.06k9wc.
• Stationary_spacetime sameAs Q3966119.
• Stationary_spacetime sameAs Q3966119.
• Stationary_spacetime sameAs Stationary_spacetime.
• Stationary_spacetime wasDerivedFrom Stationary_spacetime?oldid=544042900.
• Stationary_spacetime isPrimaryTopicOf Stationary_spacetime.