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. 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. In a stationary spacetime, the metric tensor components, g μ ν {displaystyle 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 ) {displaystyle (i,j=1,2,3)} 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 u }xi ^{mu }xi ^{ u }} , 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 u ho sigma }xi ^{ u } abla ^{ ho }xi ^{sigma }} (see, for example, 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. 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 ightarrow 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. In a stationary spacetime satisfying the vacuum Einstein equations R μ ν = 0 {displaystyle R_{mu u }=0} outside the sources, the twist 4-vector ω μ {displaystyle omega _{mu }} is curl-free, and is therefore locally the gradient of a scalar ω {displaystyle omega } (called the twist scalar): 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 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 which 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