Formalism for space-times with a homothety

The formalism developed by Fayos and Sopuerta for isometries in vacuum spacetimes is generalised in three ways: to include the case of homotheties, to allow for a tetrad not aligned to the symmetry structure and to allow for a non-normalised tetrad. The new formalism is used to recover results by McIntosh and also applied to the particular cases of Robinson-Trautman and Kundt type N. AMS Classification: 83C15 83C20 1: Introduction In [2], Fayos and Sopuerta presented a general formalism for the study of spacetimes admitting a Killing vector. In this paper this formalism is extended in three ways. Firstly the case of a homothetic vector is covered; secondly the tetrad is not assumed to be aligned with the (homethetic) bivector and finally the tetrad is not assumed to be normalised. The key to the method in [2] is that in vacuum the Killing bivector satisfies the source-free Maxwell equations (a result due to Papapatrou [11], see [1]) That the homothetic bivector also satisfies these equations was first pointed out by McIntosh [9], and this allows us to extend the formalism to homotheties. Since homotheties are the most general transformations that are both affine and conformal, they are the most general transformations that would be amenable to this sort of study. The advantage of not assuming alignment is that one can make use of the formalism to completely describe all metrics of a given class in terms of the transformations they admit: this is explicitly done for the case of Kundt type N metrics in this paper. Having a tetrad that is not normalised requires us to use the extended spin coefficients of Penrose and Rindler [12], and is included because these coefficients proved useful in [15], require little extra effort and also because the symmetry of the equations under the prime operation becomes fully apparant. All my conventions and notation will follow Penrose and Rindler. In particular, note that the sign of the Riemann tensor is not the one most commonly employed (as in, for example, Kramer et al [5]), see [12] p.209. In the appendix I give a translation table to the more common notation as used in [2]. A homothetic vector ξ by definition satisfies the equation ξa;b = Fab + ψgab. (1) Here ψ, the divergence, is a constant, and Fab will be called the homothetic bivector. The calculations are most easily accomplished by turning to a spinors. Let {o, ι} be a spinor dyad, with oAι = χ. A complex null tetrad is related to this dyad in the standard way: ` = oo ′ ; n = ιι ′ ; m = oι ′ ; m = ιo ′ , ([12], (4.5.19)), and `an = −mam = χχ. 2: Basic Equations In the same manner as [2], we define components of the homothety: ξa = ξn`a + ξ`na − ξmma − ξmma. (2) Thus, for example, χχξ` = ξa`. Note also that ξm = ξm. For the homothetic bivector Fab we define its spinor form and anti-self dual by Fab = φAB A′B′ + φA′B′ AB ; (3) − Fab = 1 2 (Fab + iF ∗ ab) = φAB A′B′ ; (4) where φAB is symmetric. We write φAB = χ−1 ( φ00 oAoB + 2φ01 o(AιB) + φ11 ιAιB ) ,