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In a stationary spacetime, the metric tensor components, , may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form
where is the time coordinate, are the three spatial coordinates and is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field has the components . is a positive scalar representing the norm of the Killing vector, i.e., , and 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 (see, for example, p. 163) which is orthogonal to the Killing vector , i.e., satisfies . 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 in the spacetime . 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) , the quotient space. Each point of represents a trajectory in the spacetime . This identification, called a canonical projection, is a mapping that sends each trajectory in onto a point in and induces a metric on via pullback. The quantities , and are all fields on and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case 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 outside the sources, the twist 4-vector is curl-free,
and is therefore locally the gradient of a scalar (called the twist scalar):
Instead of the scalars and it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, and , defined as
In general relativity the mass potential plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential 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 (, ) and the 3-metric . In terms of these quantities the Einstein vacuum field equations can be put in the form
where , and is the Ricci tensor of the spatial metric and the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.
- 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
- Hansen, R.O. (1974). J. Math. Phys. 15, 46.