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]

Description and analysis

In a stationary spacetime, the metric tensor components, ${\displaystyle g_{\mu \nu }}$, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form ${\displaystyle (i,j=1,2,3)}$

${\displaystyle ds^{2}=\lambda (dt-\omega _{i}\,dy^{i})^{2}-\lambda ^{-1}h_{ij}\,dy^{i}\,dy^{j},}$

where ${\displaystyle t}$ is the time coordinate, ${\displaystyle y^{i}}$ are the three spatial coordinates and ${\displaystyle h_{ij}}$ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field ${\displaystyle \xi ^{\mu }}$ has the components ${\displaystyle \xi ^{\mu }=(1,0,0,0)}$. ${\displaystyle \lambda }$ is a positive scalar representing the norm of the Killing vector, i.e., ${\displaystyle \lambda =g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }}$, and ${\displaystyle \omega _{i}}$ 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 ${\displaystyle \omega _{\mu }=e_{\mu \nu \rho \sigma }\xi ^{\nu }\nabla ^{\rho }\xi ^{\sigma }}$(see, for example, ^[2] p. 163) which is orthogonal to the Killing vector ${\displaystyle \xi ^{\mu }}$, i.e., satisfies ${\displaystyle \omega _{\mu }\xi ^{\mu }=0}$. 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 ${\displaystyle G}$ in the spacetime ${\displaystyle M}$. 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) ${\displaystyle V=M/G}$, the quotient space. Each point of ${\displaystyle V}$ represents a trajectory in the spacetime ${\displaystyle M}$. This identification, called a canonical projection, ${\displaystyle \pi :M\rightarrow V}$ is a mapping that sends each trajectory in ${\displaystyle M}$ onto a point in ${\displaystyle V}$ and induces a metric ${\displaystyle h=-\lambda \pi *g}$ on ${\displaystyle V}$ via pullback. The quantities ${\displaystyle \lambda }$, ${\displaystyle \omega _{i}}$ and ${\displaystyle h_{ij}}$ are all fields on ${\displaystyle V}$ and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case ${\displaystyle \omega _{i}=0}$ 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.

Use as starting point for vacuum field equations

In a stationary spacetime satisfying the vacuum Einstein equations ${\displaystyle R_{\mu \nu }=0}$ outside the sources, the twist 4-vector ${\displaystyle \omega _{\mu }}$ is curl-free,

${\displaystyle \nabla _{\mu }\omega _{\nu }-\nabla _{\nu }\omega _{\mu }=0,\,}$

and is therefore locally the gradient of a scalar ${\displaystyle \omega }$ (called the twist scalar):

${\displaystyle \omega _{\mu }=\nabla _{\mu }\omega .\,}$

Instead of the scalars ${\displaystyle \lambda }$ and ${\displaystyle \omega }$ it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, ${\displaystyle \Phi _{M}}$ and ${\displaystyle \Phi _{J}}$, defined as ^[4]

${\displaystyle \Phi _{M}={\frac {1}{4}}\lambda ^{-1}(\lambda ^{2}+\omega ^{2}-1),}$

${\displaystyle \Phi _{J}={\frac {1}{2}}\lambda ^{-1}\omega .}$

In general relativity the mass potential ${\displaystyle \Phi _{M}}$ plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential ${\displaystyle \Phi _{J}}$ 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 that has no Newtonian analog.

A stationary vacuum metric is thus expressible in terms of the Hansen potentials ${\displaystyle \Phi _{A}}$ (${\displaystyle A=M}$, ${\displaystyle J}$) and the 3-metric ${\displaystyle h_{ij}}$. In terms of these quantities the Einstein vacuum field equations can be put in the form ^[4]

${\displaystyle (h^{ij}\nabla _{i}\nabla _{j}-2R^{(3)})\Phi _{A}=0,\,}$

${\displaystyle R_{ij}^{(3)}=2[\nabla _{i}\Phi _{A}\nabla _{j}\Phi _{A}-(1+4\Phi ^{2})^{-1}\nabla _{i}\Phi ^{2}\nabla _{j}\Phi ^{2}],}$

where ${\displaystyle \Phi ^{2}=\Phi _{A}\Phi _{A}=(\Phi _{M}^{2}+\Phi _{J}^{2})}$, and ${\displaystyle R_{ij}^{(3)}}$ is the Ricci tensor of the spatial metric and ${\displaystyle R^{(3)}=h^{ij}R_{ij}^{(3)}}$ the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.

See also

• Static spacetime
• Spherically symmetric spacetime

References

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. Hansen, R.O. (1974). J. Math. Phys. 15, 46.