Stationary spacetime

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In general relativity, a spacetime is said to be stationary if it admits a global, nowhere zero timelike Killing vector field. In a stationary spacetime, the metric tensor components, $g μν$ , may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form $(i, j = 1,2,3)$

d s 2 = λ(d t - ω i d y i) 2 - λ - 1 h i j d y i d y j

where $t$ is the time coordinate, $y i$ are the three spatial coordinates and $h i j$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field $ξ μ$ has the components $ξ μ = (1,0,0,0)$ . $λ$ is a positive scalar representing the norm of the Killing vector, i.e., $λ = g μν ξ μ ξ ν$ , and $ω i$ 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, ^[1], p. 163) which is orthogonal to the Killing vector $ξ μ$ , i.e., satisfies $ω μ ξ μ = 0$ . 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^[2]. 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 $h = - λπ * g$ on $V$ via pullback. The quantities $λ$ , $ω i$ and $h i j$ 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 $ω i = 0$ 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 $R μν = 0$ outside the sources, the twist 4-vector $ω μ$ is curl-free,

$\nabla_{\mu}\omega_{\nu} - \nabla_{\nu}\omega_{\mu} = 0$ ,

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

$\omega_{\mu} = \nabla_{\mu}\omega.$

Instead of the scalars $λ$ and $ω$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, $Φ M$ and $Φ J$ , defined as^[3]

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

$\Phi_{J} = \frac{1}{2}\lambda^{-1}\omega.$

In general relativity the mass potential $Φ M$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential $Φ J$ 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 $Φ A$ ( $A = M$ , $J$ ) and the 3-metric $h i j$ . In terms of these quantities the Einstein vacuum field equations can be put in the form^[3]

$(h^{ij}\nabla_{i}\nabla_{j} - 2R^{(3)})\Phi_{A} = 0$ ,

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

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.