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]
Description and analysis
In a stationary spacetime, the metric tensor components, [math]\displaystyle{ g_{\mu\nu} }[/math], may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form [math]\displaystyle{ (i,j = 1,2,3) }[/math]
- [math]\displaystyle{ ds^{2} = \lambda (dt - \omega_{i}\, dy^i)^{2} - \lambda^{-1} h_{ij}\, dy^i\,dy^j, }[/math]
where [math]\displaystyle{ t }[/math] is the time coordinate, [math]\displaystyle{ y^{i} }[/math] are the three spatial coordinates and [math]\displaystyle{ h_{ij} }[/math] is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field [math]\displaystyle{ \xi^{\mu} }[/math] has the components [math]\displaystyle{ \xi^{\mu} = (1,0,0,0) }[/math]. [math]\displaystyle{ \lambda }[/math] is a positive scalar representing the norm of the Killing vector, i.e., [math]\displaystyle{ \lambda = g_{\mu\nu}\xi^{\mu}\xi^{\nu} }[/math], and [math]\displaystyle{ \omega_{i} }[/math] 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 [math]\displaystyle{ \omega_{\mu} = e_{\mu\nu\rho\sigma}\xi^{\nu}\nabla^{\rho}\xi^{\sigma} }[/math](see, for example,[2] p. 163) which is orthogonal to the Killing vector [math]\displaystyle{ \xi^{\mu} }[/math], i.e., satisfies [math]\displaystyle{ \omega_{\mu} \xi^{\mu} = 0 }[/math]. 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 [math]\displaystyle{ G }[/math] in the spacetime [math]\displaystyle{ M }[/math]. 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) [math]\displaystyle{ V= M/G }[/math], the quotient space. Each point of [math]\displaystyle{ V }[/math] represents a trajectory in the spacetime [math]\displaystyle{ M }[/math]. This identification, called a canonical projection, [math]\displaystyle{ \pi : M \rightarrow V }[/math] is a mapping that sends each trajectory in [math]\displaystyle{ M }[/math] onto a point in [math]\displaystyle{ V }[/math] and induces a metric [math]\displaystyle{ h = -\lambda \pi*g }[/math] on [math]\displaystyle{ V }[/math] via pullback. The quantities [math]\displaystyle{ \lambda }[/math], [math]\displaystyle{ \omega_{i} }[/math] and [math]\displaystyle{ h_{ij} }[/math] are all fields on [math]\displaystyle{ V }[/math] and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case [math]\displaystyle{ \omega_{i} = 0 }[/math] 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 [math]\displaystyle{ R_{\mu\nu} = 0 }[/math] outside the sources, the twist 4-vector [math]\displaystyle{ \omega_{\mu} }[/math] is curl-free,
- [math]\displaystyle{ \nabla_\mu \omega_\nu - \nabla_\nu \omega_\mu = 0,\, }[/math]
and is therefore locally the gradient of a scalar [math]\displaystyle{ \omega }[/math] (called the twist scalar):
- [math]\displaystyle{ \omega_\mu = \nabla_\mu \omega.\, }[/math]
Instead of the scalars [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ \omega }[/math] it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, [math]\displaystyle{ \Phi_{M} }[/math] and [math]\displaystyle{ \Phi_{J} }[/math], defined as[4]
- [math]\displaystyle{ \Phi_{M} = \frac{1}{4}\lambda^{-1}(\lambda^{2} + \omega^{2} -1), }[/math]
- [math]\displaystyle{ \Phi_{J} = \frac{1}{2}\lambda^{-1}\omega. }[/math]
In general relativity the mass potential [math]\displaystyle{ \Phi_{M} }[/math] plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential [math]\displaystyle{ \Phi_{J} }[/math] 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 [math]\displaystyle{ \Phi_{A} }[/math] ([math]\displaystyle{ A=M }[/math], [math]\displaystyle{ J }[/math]) and the 3-metric [math]\displaystyle{ h_{ij} }[/math]. In terms of these quantities the Einstein vacuum field equations can be put in the form[4]
- [math]\displaystyle{ (h^{ij}\nabla_i \nabla_j - 2R^{(3)})\Phi_A = 0,\, }[/math]
- [math]\displaystyle{ 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}], }[/math]
where [math]\displaystyle{ \Phi^{2} = \Phi_{A}\Phi_{A} = (\Phi_{M}^{2} + \Phi_{J}^{2}) }[/math], and [math]\displaystyle{ R^{(3)}_{ij} }[/math] is the Ricci tensor of the spatial metric and [math]\displaystyle{ R^{(3)} = h^{ij}R^{(3)}_{ij} }[/math] the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
See also
References
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