Stationary spacetime

Stationary spacetime

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_{mu u}, 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 u}xi^{mu}xi^{ u}, 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 u hosigma}xi^{ u} abla^{ ho}xi^{sigma}(see, for example, [Wald, R.M., (1984). General Relativity, (U. Chicago Press)] , 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 [Geroch, R., (1971). J. Math. Phys. 12, 918] . 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 ightarrow 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 u} = 0 outside the sources, the twist 4-vector omega_{mu} is curl-free, :: abla_{mu}omega_{ u} - abla_{ u}omega_{mu} = 0,and is therefore locally the gradient of a scalar omega (called the twist scalar):::omega_{mu} = abla_{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 asHansen, R.O. (1974). J. Math. Phys. 15, 46.] ::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::(h^{ij} abla_{i} abla_{j} - 2R^{(3)})Phi_{A} = 0,::R^{(3)}_{ij} = 2 [ abla_{i}Phi_{A} abla_{j}Phi_{A} - (1+ 4 Phi^{2})^{-1} abla_{i}Phi^{2} abla_{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.


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