ar X iv : g r - qc / 0 30 30 20 v 1 5 M ar 2 00 3 Causal symmetries

Based on the recent work [4] we put forward a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid which contains as its maximal subgroup the set of conformal transformations. We find the necessary and sufficient conditions for a vector field ~ ξ to be the infinitesimal generator of a one-parameter submonoid of pure causal symmetries. We speculate about possible applications to gravitation theory by means of some relevant examples. Our goal is to introduce a new type of spacetime symmetry which generalizes the conformal one while still preserving many causal properties of the Lorentzian manifolds. To that end, we will need the results on null-cone preserving maps analyzed and classified in [1]. The whole idea will be based on the new concept of causal mapping (leading to a definition of isocausal spacetimes) which was recently introduced in [4]. This letter is inspired by [1] and [4] which will be referred to as PI and PII from now on, respectively, and we use their notations. Herein, we will just give the fundamental results. A longer detailed exposition will be given elsewhere [5]. Some related ideas were used in [7]. According to PII, a causal relation between two Lorentzian manifolds is any diffeomorphism which maps non-spacelike (also called causal) future-directed vectors onto causal future-directed vectors. Here we will say that a transformation φ : (V, g) → (V, g) is a causal symmetry if it sets a causal relation of (V, g) with itself. From theorem 3.1 in PII follows that φ is a causal symmetry iff φg satisfies the dominant energy condition, or in the notation of PI and PII, iff φg is a future tensor: φg ∈ DP2 (V ). The set of causal symmetries of (V, g) will be denoted by C(V, g) (in short C(V ) if no confusion arises). This is a subset of the transformation group of V and clearly (prop. 3.3 of PII) the composition of causal symmetries is a causal symmetry. As the identity map is also a causal symmetry, C(V ) has the algebraic structure of a submonoid, see e.g. [9]. Nonetheless C(V ) will not in general be a subgroup because the inverse of a causal relation need not be a causal relation. Actually, both φ and φ are causal iff φ is a conformal transformation (theorem 4.2 of PII), and therefore the maximal subgroup C(V ) ∩ C(V ) of C(V ) [9] is just the group of conformal transformations of V : every subgroup of C(V ) is formed exclusively by conformal symmetries. We call proper causal symmetries the causal symmetries which are not conformal transformations.