Rotation in cosmology : Comments on " Imparting rotation to a Bianchi type II space - time , " by M . J . Rebouqas and J . B . S . d ' Olival

Flat Minkowski space-time or open submanifolds of it may be sliced by a family of three-dimensional spacelike orbits of three-dimensional subgroups of the Poincare group of Bianchi types I, III, V, VIIo, and VIIh;o'o and thus be made to appear as a spatially homogeneous Bianchi-type cosmological model. I Timelike congruences that are spatially homogeneous with respect to any of the non-Abelian groups of this list (all but type I) are in general rotating congruences: they have nonzero vorticity. No one would correctly call Minkowski space-time a rotating cosmology because of this fact, yet articles in the literature continue to do exactly this in similar circumstances. Rotation in cosmology can refer to one of two distinct notions that are often related. Either ( 1 ) the space-time possesses an intrinsically defined timelike congruence with nonzero vorticity,2 or (2) a natural slicing exists in terms of which an orthonormal basis of eigenvectors of the extrinsic curvature necessarily rotates as one moves along the congruence normal to the slicing. The first idea is relevant to stationary space-times where rotation is usually first met in studying relativity; unfortunately intuition about this case is often extended to other situations where it is no longer appropriate. Nonstatic stationary space-times possess a Killing vector field that is timelike on an open submanifold of the space-time and has nonzero vorticity, i.e., the corresponding one-form is not hypersurface forming.2 On the other hand, perfect fluid filled space-times whose fluid velocity vector has nonzero vorticity are often justifiably referred to as rotating cosmologies. In both cases the rotation refers to a component of the motion along the congruence of the perpendicular projections of Lie dragged "connecting vectors" associated with the congruence relative to a Fermi-propagated triad of orthonormal vectors spanning the local rest space relative to that congruence (a "nonrotating spatial frame"). 3,4 In the famous Godel solution,5 which originally challenged people's ideas about rotation in relativity, the fluid velocity vector is a timelike Killing vector field, combining both of these possibilities into a single example. The second idea is relevant to space-times where a natural slicing exists, since it refers to quantities defined not by the space-time but by a slicing of the space-time. A "Kasner frame6" could be defined as an orthogonal spatial frame consisting of eigenvectors of the extrinsic curvature relative to a particular slicing. The orthonormal frame obtained by normalizing such a frame (a unit Kasner frame) can then be compared to an orthonormal spatial frame that is Fermipropagated along the congruence of unit normals to the slicing. If the unit Kasner frame rotates relative to the nonrotating spatial frame and is unique (nondegenerate eigenvalues), the slicing might be called a rotating slicing of the space-time. When the eigenvalues of the extrinsic curvature are degenerate, one may freeze out the rotational freedom in the eigenvectors due to this degeneracy by minimizing the square of the angular velocity vector, which describes the rotation. If the rotation is still nonzero, the term rotating slicing may again be used. Like rotating congruences, all space-times have such rotating slicings; for this to be significant the rotating slicing must be intrinsically defined by the space-time. Probably the best candidate for such a slicing is one for which the trace of the extrinsic curvature (Tr K), also called the mean extrinsic curvature, is constant on each slice.714 Such a slicing is referred to as a constant mean curvature slicing or a "Tr K = const" slicing, and in the case of vanishing mean curvature, a maximal slicing, and is a choice preferred by the simplifications that occur both in the initial value problem15 and in geometric coordinate conditions. 16,17 A space-time with a synchronous spacelike singularity also has a unique slicing associated with the maximum lifetime function. 18.19 For a nonstatic, stationary, axially symmetric spacetime, an example of which is the Kerr rotating black hole,20 a unique maximal slicing21 exists consisting of the hypersurfaces orthogonal to the congruence of locally nonrotating observers.20-23 Some thought shows that this slicing is a rotating slicing, suggesting that the idea of describing rotation of a space-time by an intrinsically defined slicing rather than a congruence is not unreasonable. Spatially homogeneous space-times have a natural constant mean curvature slicing by the family of spacelike orbits of the homogeneity group. When these space-times have an initial big bang or final big crunch singularity, this slicing coincides with the maximum lifetime slicing. It therefore makes sense to classify spatially homogeneous space-times as rotating or nonrotating according to the second sense using the natural slicing. Such a space-time is rotating in this sense if one cannot diagonalize (for all time) the matrix of components of the spatial metric with respect to an invariant spatial frame that is comoving with the normal vector field to the natural slicing. This means that in orthogonal spatial