Principal curvature

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Template:Expert-subject 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, gμν, 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)

ds2=λ(dtωidyi)2λ1hijdyidyj,

where t is the time coordinate, yi are the three spatial coordinates and hij is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field ξμ has the components ξμ=(1,0,0,0). λ is a positive scalar representing the norm of the Killing vector, i.e., λ=gμνξμξν, and ωi 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 ωμ=eμνρσξνρξσ(see, for example,[2] p. 163) which is orthogonal to the Killing vector ξμ, i.e., satisfies ωμξμ=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, π:MV is a mapping that sends each trajectory in M onto a point in V and induces a metric h=λπ*g on V via pullback. The quantities λ, ωi and hij 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 ωi=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 Rμν=0 outside the sources, the twist 4-vector ωμ is curl-free,

μωννωμ=0,

and is therefore locally the gradient of a scalar ω (called the twist scalar):

ωμ=μω.

Instead of the scalars λ and ω it is more convenient to use the two Hansen potentials, the mass and angular momentum potentials, ΦM and ΦJ, defined as[4]

ΦM=14λ1(λ2+ω21),
ΦJ=12λ1ω.

In general relativity the mass potential ΦM plays the role of the Newtonian gravitational potential. A nontrivial angular momentum potential ΦJ 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 ΦA (A=M, J) and the 3-metric hij. In terms of these quantities the Einstein vacuum field equations can be put in the form[4]

(hijij2R(3))ΦA=0,
Rij(3)=2[iΦAjΦA(1+4Φ2)1iΦ2jΦ2],

where Φ2=ΦAΦA=(ΦM2+ΦJ2), and Rij(3) is the Ricci tensor of the spatial metric and R(3)=hijRij(3) the corresponding Ricci scalar. These equations form the starting point for investigating exact stationary vacuum metrics.

See also

References