In a stationary spacetime, the metric tensor components, ${\displaystyle g_{\mu \nu }}$ , may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form ${\displaystyle (i,j=1,2,3)}$ 

${\displaystyle ds^{2}=\lambda (dt-\omega _{i}\,dy^{i})^{2}-\lambda ^{-1}h_{ij}\,dy^{i}\,dy^{j},}$ 

where ${\displaystyle t}$  is the time coordinate, ${\displaystyle y^{i}}$  are the three spatial coordinates and ${\displaystyle h_{ij}}$  is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field ${\displaystyle \xi ^{\mu }}$  has the components ${\displaystyle \xi ^{\mu }=(1,0,0,0)}$ . ${\displaystyle \lambda }$  is a positive scalar representing the norm of the Killing vector, i.e., ${\displaystyle \lambda =g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }}$ , and ${\displaystyle \omega _{i}}$  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 ${\displaystyle \omega _{\mu }=e_{\mu \nu \rho \sigma }\xi ^{\nu }\nabla ^{\rho }\xi ^{\sigma }}$ (see, for example,^[2] p. 163) which is orthogonal to the Killing vector ${\displaystyle \xi ^{\mu }}$ , i.e., satisfies ${\displaystyle \omega _{\mu }\xi ^{\mu }=0}$ . 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.^[3] The time translation Killing vector generates a one-parameter group of motion ${\displaystyle G}$  in the spacetime ${\displaystyle M}$ . 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) ${\displaystyle V=M/G}$ , the quotient space. Each point of ${\displaystyle V}$  represents a trajectory in the spacetime ${\displaystyle M}$ . This identification, called a canonical projection, ${\displaystyle \pi :M\rightarrow V}$  is a mapping that sends each trajectory in ${\displaystyle M}$  onto a point in ${\displaystyle V}$  and induces a metric ${\displaystyle h=-\lambda \pi *g}$  on ${\displaystyle V}$  via pullback. The quantities ${\displaystyle \lambda }$ , ${\displaystyle \omega _{i}}$  and ${\displaystyle h_{ij}}$  are all fields on ${\displaystyle V}$  and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case ${\displaystyle \omega _{i}=0}$  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 ${\displaystyle R_{\mu \nu }=0}$  outside the sources, the twist 4-vector ${\displaystyle \omega _{\mu }}$  is curl-free,

${\displaystyle \nabla _{\mu }\omega _{\nu }-\nabla _{\nu }\omega _{\mu }=0,\,}$ 

and is therefore locally the gradient of a scalar ${\displaystyle \omega }$  (called the twist scalar):

${\displaystyle \omega _{\mu }=\nabla _{\mu }\omega .\,}$ 

Instead of the scalars ${\displaystyle \lambda }$  and ${\displaystyle \omega }$  it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, ${\displaystyle \Phi _{M}}$  and ${\displaystyle \Phi _{J}}$ , defined as^[4]

${\displaystyle \Phi _{M}={\frac {1}{4}}\lambda ^{-1}(\lambda ^{2}+\omega ^{2}-1),}$ 

${\displaystyle \Phi _{J}={\frac {1}{2}}\lambda ^{-1}\omega .}$ 

In general relativity the mass potential ${\displaystyle \Phi _{M}}$  plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential ${\displaystyle \Phi _{J}}$  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 that has no Newtonian analog.

A stationary vacuum metric is thus expressible in terms of the Hansen potentials ${\displaystyle \Phi _{A}}$  (${\displaystyle A=M}$ , ${\displaystyle J}$ ) and the 3-metric ${\displaystyle h_{ij}}$ . In terms of these quantities the Einstein vacuum field equations can be put in the form^[4]

${\displaystyle (h^{ij}\nabla _{i}\nabla _{j}-2R^{(3)})\Phi _{A}=0,\,}$ 

${\displaystyle R_{ij}^{(3)}=2[\nabla _{i}\Phi _{A}\nabla _{j}\Phi _{A}-(1+4\Phi ^{2})^{-1}\nabla _{i}\Phi ^{2}\nabla _{j}\Phi ^{2}],}$ 

where ${\displaystyle \Phi ^{2}=\Phi _{A}\Phi _{A}=(\Phi _{M}^{2}+\Phi _{J}^{2})}$ , and ${\displaystyle R_{ij}^{(3)}}$  is the Ricci tensor of the spatial metric and ${\displaystyle R^{(3)}=h^{ij}R_{ij}^{(3)}}$  the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.

• Static spacetime
• Spherically symmetric spacetime