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


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

where $$t$$ is the time coordinate, $$y^{i}$$ are the three spatial coordinates and $$h_{ij}$$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field $$\xi^{\mu}$$ has the components $$\xi^{\mu} = (1,0,0,0)$$. $$\lambda$$ is a positive scalar representing the norm of the Killing vector, i.e., $$\lambda = g_{\mu\nu}\xi^{\mu}\xi^{\nu}$$, and $$ \omega_{i} $$ 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 $$ \omega_{\mu} = e_{\mu\nu\rho\sigma}\xi^{\nu}\nabla^{\rho}\xi^{\sigma}$$(see, for example, p. 163) which is orthogonal to the Killing vector $$\xi^{\mu}$$, i.e., satisfies $$\omega_{\mu} \xi^{\mu} = 0$$. 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. 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, $$ \pi : M \rightarrow V $$ is a mapping that sends each trajectory in $$M$$ onto a point in $$V$$ and induces a metric $$h = -\lambda \pi*g$$ on $$V$$ via pullback. The quantities $$\lambda$$, $$ \omega_{i} $$ and $$h_{ij}$$ 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 $$ \omega_{i} = 0 $$ 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.

Use as starting point for vacuum field equations
In a stationary spacetime satisfying the vacuum Einstein equations $$R_{\mu\nu} = 0$$ outside the sources, the twist 4-vector $$\omega_{\mu}$$ is curl-free,


 * $$\nabla_\mu \omega_\nu - \nabla_\nu \omega_\mu = 0,\,$$

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


 * $$\omega_\mu = \nabla_\mu \omega.\,$$

Instead of the scalars $$\lambda$$ and $$\omega$$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, $$\Phi_{M}$$ and $$\Phi_{J}$$, defined as


 * $$\Phi_{M} = \frac{1}{4}\lambda^{-1}(\lambda^{2} + \omega^{2} -1),$$
 * $$\Phi_{J} = \frac{1}{2}\lambda^{-1}\omega.$$

In general relativity the mass potential $$\Phi_{M}$$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential $$\Phi_{J}$$ 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 $$\Phi_{A}$$ ($$A=M$$, $$J$$) and the 3-metric $$h_{ij}$$. In terms of these quantities the Einstein vacuum field equations can be put in the form


 * $$(h^{ij}\nabla_i \nabla_j - 2R^{(3)})\Phi_A = 0,\,$$
 * $$R^{(3)}_{ij} = 2[\nabla_{i}\Phi_{A}\nabla_{j}\Phi_{A} - (1+ 4 \Phi^{2})^{-1}\nabla_{i}\Phi^{2}\nabla_{j}\Phi^{2}], $$

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