Distinguishability Condition and the Future Subsemigroup

The paper deals with two simply connected solvable four-dimensional Lie groups M1 and M2 . The first group is a direct product of the nilpotent Heisenberg Lie group and the one-dimensional Lie group. The second one is a direct product of the two-dimensional non-abelian Lie group and the two-dimensional abelian Lie group. Applying Methods of [4, 6] we investigate the causal structure of left-invariant Lorentzian metrics on M1 [8] and M2 [7]. Here we focus our attention on one concrete metric on M1 and on a certain one-parameter family gq , q > 0 of metrics on M2 . We have proved in [7, 8] these Lorentzian spaces to be geodesically complete, satisfying the causality condition with a violation of uniform stable causality. In the present paper, we prove these spaces to be future distinguishing (that involves, because of their homogeneity, also the conditions of past distinguishing, strong causality, stable causality and continuity of causality). This result is of interest in causality theory since in accordance with [9], respectively, [5] the chronological (respectively, causal) structure of such spaces codes their conformal structure. It also characterizes the structure of the subsemigroup I , respectively, J which defines the chronological, respectively, causal structure of the considered Lorentzian Lie group. For all unfamiliar definitions, the reader is referred to [1, 6]. 1. General method to prove future distinguishability of a Lorentzian Lie group Assume M to be a solvable connected Lie group and fix a symmetric nondegenerated form of Lorentzian signature +, . . . ,+,− in the Lie algebra L of M . After the choice of future cone K in L the group M becomes a Lorentzian Lie group, or LLG for short. If an ideal [L,L] is lightlike, i.e., its intersection with K is a single ray lying in ∂K , then such an LLG M satisfies the causality ∗ This paper was presented by title at the Oberwolfach Conference on the analytical and topological theory of semigroups at Oberwolfach, January 30, 1989 206 Levichev and Levicheva condition [6, Theorem 4.2]. If the intersection of K and [L,L] is {0} , then M is uniformly stably causal [6, Theorem 4.1], hence distinguishing. We may, therefore, restrict our attention to the case K ∩ [L,L] = ` where ` is a light ray in L . Suppose that the hyperplane N contains ` and is a support hyperplane of K . Introduce in M a canonical coordinate system (x1, . . . , xn) of the second type associated with N . Then the Lie subgroup corresponding to N is characterized by the equation xn = 0. A Lorentzian manifold M with a prescribed time orientation is said to be future distinguishing (see [1], p. 24) if for any x, y ∈M the assumption I x = I y implies x = y , where I x , respectively, I − x , as usual, denotes the chronological future, respectively, past of x . If x = 1 then we shall simply write I instead of I 1 etc. Also the causal future, respectively, past, of x will be denoted by J x , respectively, J − x (and J + , respectively, J− if x = 1 .). We want to make use of a result due to R. Penrose: If M fails to be future distinguishing, then Condition (e) of his Theorem 4.31 from [10] is valid. The latter condition deals with a certain light geodesic γ . Suppose now that our LLG M fails to be future distinguishing. It follows from the proof of Theorem 4.2 of [6] that the γ above corresponds to ` . We recall Condition (e) itself: For any u, v ∈ γ with u ≤ v , if u x and y v , then y x . We may assume without loss of generality that 1 ∈ γ . Lemma . γ is entirely contained in I+ ∩ I− . Proof. In Condition (e) we take u = 1 , v ∈ γ ∩ J . Let 1 x , i.e., x ∈ I and y v , i.e., y ∈ I− v . Let y tend to v in I− v and let x tend to 1 in I . This choice is possible, since v ∈ I− v ⊂ I− v , 1 ∈ J ⊂ I+ . Taking into consideration the continuous dependence of I± y on the point y itself, we deduce v ∈ I− . We return to M and the canonical coordinates (x1, . . . , xn). Let x = x(t) denote a future timelike curve λ issuing from 1 = (0, . . . , 0). The subsemigroup I , the chronological future of 1 , consists of the points on all such λ . Observe that the component zn of the product z = x ·y equals xn+yn . Thus the coordinate xn of the point x(t) is increasing while we move along λ from 1 to the future. But the above lemma states that it must somehow reach the vicinity of γ− = J− ∩ γ . Let exp denote the exponential map (in the geometrical sense) defined on some neighborhood U of 1 . When t > 0 is sufficiently small, the points x(t) “concentrate near” exp K . They can return to γ− only above such a “level” xn , at or below which there are conjugate points to 1 along null geodesics issuing from 1 . Therefore the we have following result. Theorem 1. If, under the assumptions above, in a certain slice U def = {x : 0 < xn < ε} there are no points conjugate to 1 along future null geodesics issuing from 1 , and if the set of all points on all lightlike future geodesics from 1 in U divides U into two components, then the Lorentzian Lie group M is future distinguishing. Levichev and Levicheva 207 We note that similar arguments have been used in [5] in the course of proving the future distinguishability of a certain class of Lorentzian symmetric spaces. Note added in 1992. In [2] the authors introduced the notion of strict causality. For homogeneous Lorentzian spaces this concept agrees with that of distinguishability (see e.g. [4, 6]). In particular, the Lorentzian in the present article as well as the symmetric spaces in [5] are strictly causal. We also avail ourselves the opportunity of pointing out that, in the English translation [5] of our article “Prescribing the conformal geometry . . .” the formula labelled (3) was inadvertantly omitted. It should read