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, g μ ν , may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form ( i , j = 1 , 2 , 3 )
d s 2 = λ ( d t − ω i d y i ) 2 − λ − 1 h i j d y i d y j , where t is the time coordinate, y i are the three spatial coordinates and h i j is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field ξ μ has the components ξ μ = ( 1 , 0 , 0 , 0 ) . λ is a positive scalar representing the norm of the Killing vector, i.e., λ = g μ ν ξ μ ξ ν , and ω i 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 ω μ = e μ ν ρ σ ξ ν ∇ ρ ξ σ (see, for example, p. 163) which is orthogonal to the Killing vector ξ μ , i.e., satisfies ω μ ξ μ = 0 . 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 G in the spacetime M . 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) V = M / G , the quotient space. Each point of V represents a trajectory in the spacetime M . This identification, called a canonical projection, π : M → V is a mapping that sends each trajectory in M onto a point in V and induces a metric h = − λ π ∗ g on V via pullback. The quantities λ , ω i and h i j are all fields on V and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case ω i = 0 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 R μ ν = 0 outside the sources, the twist 4-vector ω μ is curl-free,
∇ μ ω ν − ∇ ν ω μ = 0 , and is therefore locally the gradient of a scalar ω (called the twist scalar):
ω μ = ∇ μ ω . Instead of the scalars λ and ω it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, Φ M and Φ J , defined as
Φ M = 1 4 λ − 1 ( λ 2 + ω 2 − 1 ) , Φ J = 1 2 λ − 1 ω . In general relativity the mass potential Φ M plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential Φ J 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 Φ A ( A = M , J ) and the 3-metric h i j . In terms of these quantities the Einstein vacuum field equations can be put in the form
( h i j ∇ i ∇ j − 2 R ( 3 ) ) Φ A = 0 , R i j ( 3 ) = 2 [ ∇ i Φ A ∇ j Φ A − ( 1 + 4 Φ 2 ) − 1 ∇ i Φ 2 ∇ j Φ 2 ] , where Φ 2 = Φ A Φ A = ( Φ M 2 + Φ J 2 ) , and R i j ( 3 ) is the Ricci tensor of the spatial metric and R ( 3 ) = h i j R i j ( 3 ) the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
