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, $g_{\mu\nu}$ , 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,^[2] 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.^[3] 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.

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^[4]

\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 which 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^[4]

(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.

References

↑ Ludvigsen, M., General Relativity: A Geometric Approach, Cambridge University Press, 1999, ISBN 052163976X
↑ Wald, R.M., (1984). General Relativity, (U. Chicago Press)
↑ Geroch, R., (1971). J. Math. Phys. 12, 918
↑ 4.0 4.1 Hansen, R.O. (1974). J. Math. Phys. 15, 46.

Stationary spacetime

See also

References