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.[1]

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 which 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.