X iv : g r - qc / 9 70 80 20 v 1 1 1 A ug 1 99 7 Lie Groups of Conformal Motions acting on Null Orbits

Space-times admitting a 3-dimensional Lie group of conformal motions C3 acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are provided (irrespectively of the matter content) and then all possible perfect fluid solutions are found, although none of these verify the weak and dominant energy conditions over the whole spacetime manifold. In this letter we shall consider space-times (M, g) admitting a maximal threeparameter conformal group C3 containing an Abelian two-parameter subgroup of isometries G2 whose orbits S2 are spacelike, diffeomorphic to IR 2 and admit orthogonal two-surfaces. Furthermore, we shall assume that the C3 acts transitively on null orbits N3, thus complementing a previous paper [1] in which the case of null conformal orbits was explicitly excluded. In particular, in this letter we shall provide the coordinate expressions for the metric and the CKV for each Lie algebra structure and give all possible perfect fluid solutions. A few remarks concerning Lie groups acting on null orbits are in order here. In most cases the study of null orbits has been restricted to isometries only. It is a well known fact that groups Gr, r ≥ 4, acting on N3 have at least one subgroup G3 acting on N3, N2 or S2 [2]. In the case in which the subgroup G3 acts on S2, the space-time is a LRS model, and the Gr admits either a different subgroup G3 acting on N3 or a null Killing vector (KV) [3]. The case G3 acting on N2 was studied by Barnes [4]; the group G3 is then of Bianchi type II and perfect fluid solutions are excluded since the metric leads to a Ricci tensor whose Segre type is not that of a perfect fluid. Another case that has been considered in the literature is that of a G3 acting on N3 in which Rabk k = 0, and this condition excludes perfect fluid sources with μ + p 6= 0. It Max-Planck-Institut für Gravitationsphysik. Albert-Einstein-Institut. D-14473 Potsdam. Germany. Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, NS. Canada B3H 3J5. Departament de F́ısica. Universitat de les Illes Balears. E-07071 Palma de Mallorca. Spain.