Stationary spacetime
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. Description and analysis. In a stationary spacetime, the metric tensor components, formula_1, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form formula_2 where formula_4 is the time coordinate, formula_5 are the three spatial coordinates and formula_6 is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field formula_7 has the components formula_8. formula_9  is a positive scalar representing the norm of the Killing vector, i.e., formula_10, and formula_11 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 formula_12(see, for example, p. 163) which is orthogonal to the Killing vector formula_7, i.e., satisfies formula_14. 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 formula_15 in the spacetime formula_16. 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) formula_17, the quotient space. Each point of formula_18 represents a trajectory in the spacetime formula_16. This identification, called a canonical projection, formula_20 is a mapping that sends each trajectory in formula_16 onto a point in formula_18 and induces a metric formula_23 on formula_18 via pullback. The quantities formula_9, formula_11 and formula_6 are all fields  on formula_18 and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case formula_29 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. Use as starting point for vacuum field equations. In a stationary spacetime satisfying the vacuum Einstein equations formula_30 outside the sources, the twist 4-vector formula_31 is curl-free, and is therefore locally the gradient of a scalar formula_33 (called the twist scalar): Instead of the scalars  formula_9 and formula_33 it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, formula_37 and formula_38, defined as In general relativity the mass potential formula_37 plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential formula_38 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 that  has no Newtonian analog. A stationary vacuum metric is thus expressible in terms of the Hansen potentials formula_43 (formula_44, formula_45) and the 3-metric formula_6. In terms of these quantities the Einstein vacuum field equations can be put in the form where formula_49, and formula_50 is the Ricci tensor of the spatial metric and formula_51 the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.