Stationary spacetime

Spacetime that admits a Killing vector that is asymptotically timelike

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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]

## Description and analysis

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In a stationary spacetime, the metric tensor components,  g μ ν {\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  ( i , j = 1 , 2 , 3 ) {\displaystyle (i,j=1,2,3)}

d s 2 = λ ( d t − ω i d y i ) 2 − λ − 1 h i j d y i d y j , {\displaystyle ds^{2}=\lambda (dt-\omega _{i}\,dy^{i})^{2}-\lambda ^{-1}h_{ij}\,dy^{i}\,dy^{j},}

where  t {\displaystyle t} is the time coordinate,  y i {\displaystyle y^{i}} are the three spatial coordinates and  h i j {\displaystyle h_{ij}} is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field  ξ μ {\displaystyle \xi ^{\mu }} has the components  ξ μ = ( 1 , 0 , 0 , 0 ) {\displaystyle \xi ^{\mu }=(1,0,0,0)} .  λ {\displaystyle \lambda } is a positive scalar representing the norm of the Killing vector, i.e.,  λ = g μ ν ξ μ ξ ν {\displaystyle \lambda =g_{\mu \nu }\xi ^{\mu }\xi ^{\nu }} , and  ω i {\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  ω μ = e μ ν ρ σ ξ ν ∇ ρ ξ σ {\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  ξ μ {\displaystyle \xi ^{\mu }} , i.e., satisfies  ω μ ξ μ = 0 {\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  G {\displaystyle G} in the spacetime  M {\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)  V = M / G {\displaystyle V=M/G} , the quotient space. Each point of  V {\displaystyle V} represents a trajectory in the spacetime  M {\displaystyle M} . This identification, called a canonical projection,  π : M -> V {\displaystyle \pi :M\rightarrow V} is a mapping that sends each trajectory in  M {\displaystyle M} onto a point in  V {\displaystyle V} and induces a metric  h = − λ π ∗ g {\displaystyle h=-\lambda \pi *g} on  V {\displaystyle V} via pullback. The quantities  λ {\displaystyle \lambda } ,  ω i {\displaystyle \omega _{i}} and  h i j {\displaystyle h_{ij}} are all fields on  V {\displaystyle V} and are consequently independent of time. Thus, the geometry of a stationary spacetime does not change in time. In the special case  ω i = 0 {\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

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

∇ μ ω ν − ∇ ν ω μ = 0 , {\displaystyle \nabla _{\mu }\omega _{\nu }-\nabla _{\nu }\omega _{\mu }=0,\,}

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

ω μ = ∇ μ ω . {\displaystyle \omega _{\mu }=\nabla _{\mu }\omega .\,}

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

Φ M = 1 4 λ − 1 ( λ 2 + ω 2 − 1 ) , {\displaystyle \Phi _{M}={\frac {1}{4}}\lambda ^{-1}(\lambda ^{2}+\omega ^{2}-1),}
Φ J = 1 2 λ − 1 ω . {\displaystyle \Phi _{J}={\frac {1}{2}}\lambda ^{-1}\omega .}

In general relativity the mass potential  Φ M {\displaystyle \Phi _{M}} plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential  Φ J {\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  Φ A {\displaystyle \Phi _{A}} ( A = M {\displaystyle A=M} ,  J {\displaystyle J} ) and the 3-metric  h i j {\displaystyle h_{ij}} . In terms of these quantities the Einstein vacuum field equations can be put in the form[4]

( h i j ∇ i ∇ j − 2 R ( 3 ) ) Φ A = 0 , {\displaystyle (h^{ij}\nabla _{i}\nabla _{j}-2R^{(3)})\Phi _{A}=0,\,}
R i j ( 3 ) = 2 [ ∇ i Φ A ∇ j Φ A − ( 1 + 4 Φ 2 ) − 1 ∇ i Φ 2 ∇ j Φ 2 ] , {\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  Φ 2 = Φ A Φ A = ( Φ M 2 + Φ J 2 ) {\displaystyle \Phi ^{2}=\Phi _{A}\Phi _{A}=(\Phi _{M}^{2}+\Phi _{J}^{2})} , and  R i j ( 3 ) {\displaystyle R_{ij}^{(3)}} is the Ricci tensor of the spatial metric and  R ( 3 ) = h i j R i j ( 3 ) {\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

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* Static spacetime
* Spherically symmetric spacetime

## References

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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.