Stationary spacetime

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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.

See also[edit]

References[edit]

  1. ^ Ludvigsen, M., General Relativity: A Geometric Approach, Cambridge University Press, 1999, ISBN 052163976X
  2. ^ Wald, R.M., (1984). General Relativity, (U. Chicago Press)
  3. ^ Geroch, R., (1971). J. Math. Phys. 12, 918
  4. ^ a b Hansen, R.O. (1974). J. Math. Phys. 15, 46.