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.
[1] 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
In this coordinate system the Killing vector field
is a positive scalar representing the norm of the Killing vector, i.e.,
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
μ ν ρ σ
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.
[3] The time translation Killing vector generates a one-parameter group of motion
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)
This identification, called a canonical projection,
is a mapping that sends each trajectory in
Thus, the geometry of a stationary spacetime does not change in time.
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
is curl-free, and is therefore locally the gradient of a scalar
, defined as[4] 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 that has no Newtonian analog.
A stationary vacuum metric is thus expressible in terms of the Hansen potentials
In terms of these quantities the Einstein vacuum field equations can be put in the form[4] where
is the Ricci tensor of the spatial metric and
These equations form the starting point for investigating exact stationary vacuum metrics.