Spacelike conformal Killing vectors and spacelike congruences

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Abstract
Necessary and sufficient conditions are derived for space-time to admit a spacelike conformal motion with symmetry vector parallel to a unit spacelike vector field na. These conditions are expressed in terms of the shear and expansion of the spacelike congruence generated by na and in terms of the four-velocity of the observer employed at any given point of the congruence. It is shown that either the expansion or the rotation of this spacelike congruence must vanish if Dna/dp =0, where p denotes arc length measured along the integral curves of na, and also that there exist no proper spacelike homothetic motions with constant expansion. Propagation equations for the projection tensor and the rotation tensor are derived and it is proved that every isometric spacelike congruence is rigid. Fluid space-times are studied in detail. A relation is established between spacelike conformal motions and material curves in the fluid: if a fluid space-time admits a spacelike conformal Killing vector parallel to na and naua =0, where ua is the fluid four-velocity, then the integral curves of na are material curves in an irrotational fluid, while if the fluid vorticity is nonzero, then the integral curves of na are material curves if and only if they are vortex lines. An alternative derivation, based on the theory of spacelike congruences, of some of the results of Collins [J. Math. Phys. 25, 995 (1984)] on conformal Killing vectors parallel to the local vorticity vector in shear-free perfect fluids with zero magnetic Weyl tensor is given. The necessary and sufficient condition for vortex lines to be material lines is derived and the restriction this places on the flow of a thermodynamical perfect fluid is determined. As an application, a pure magnetic field in a rotational fluid is considered and results similar in nature to Ferraro’s law of isorotation are obtained. Throughout, corresponding results for a timelike conformal motion and for Newtonian theory are given for comparison.

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