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_{{ij}}^{{(3)}}=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 $R_{{ij}}^{{(3)}}$ is the Ricci tensor of the spatial metric and $R^{{(3)}}=h^{{ij}}R_{{ij}}^{{(3)}}$ 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
1 2 Hansen, R.O. (1974). J. Math. Phys. 15, 46.

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Stationary spacetime

See also

References