On Lorentzian causality with continuous metrics

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, and that C differentiability of the metric suffices for many key results of the smooth causality theory. PACS numbers: 04.20 Gz ar X iv :1 11 1. 04 00 v3 [ gr -q c] 2 7 M ay 2 01 2 On Lorentzian causality with continuous metrics 2 1. Causality for continuous metrics One of the factors that constrains the differentiability requirements of the proof of the celebrated Choquet-Bruhat–Geroch theorem [1], of existence and uniqueness of maximal globally hyperbolic vacuum developments of general relativistic initial data, is the degree of differentiability needed to carry out the Lorentzian causality arguments that arise. Here one should keep in mind that classical local existence and uniqueness of solutions of vacuum Einstein equations in dimension 3 + 1 applies to initial data (g,K) in the product of Sobolev spaces H × Hs−1 for s > 5/2, and that the recent studies of the Einstein equations [2–8] assume even less differentiability. On the other hand, the standard references on causality seem to assume smoothness of the metric [9–14], while the presentation in [15, 16] requires C metrics.‡ Hence the need to revisit the causality theory for Lorentzian metrics which are merely assumed to be continuous. Surprisingly enough, some standard facts of the C theory fail to hold for metrics with lower differentiability. For example, we will show that the following statements are wrong: (i) light-cones are topological hypersurfaces of codimension one; (ii) a piecewise differentiable causal curve which is not null everywhere can be deformed, with end points fixed, to a timelike curve. Concerning point (i) above, we exhibit metrics with light-cones which have nonempty interior. Researchers familiar with Lorentzian geometry will recognize point (ii) as an essential tool in many arguments. One needs therefore to reexamine the corresponding results, documenting their failure or finding a replacement for the proof. In the course of the analysis, one is naturally led to the notion of a causal bubble which, roughly speaking, is defined to be an open set which can be reached from, say a point p, by causal curves but not by timelike ones. The object of this paper is to present the above, reassessing that part of causality theory which has been presented in [16] from the perspective of continuous metrics. One of our main results is the proof that domains of dependence equipped with continuous metrics continue to admit Cauchy time functions.§ Another key result is the observation that the causality theory developed in [16] for C metrics remains true for C metrics. An application of our work to the general relativistic Cauchy problem can be found in [18]. In fact, the main motivation for the current work was to develop the Lorentzian-causality tools needed for that last reference. The existing treatments of continuous Lorentzian metrics known to us (see, e.g., the references in Section 2 of [19]) are, unfortunately, not well-suited to the study of this particular application. ‡ See also [15], where some of the issues involved in trying to prove the singularity theorems for metric below C regularity are discussed. § Once the first draft of this paper was completed (arXiv:1111.0400v1) we were informed of [17], where the result is proved by completely different methods, and in greater generality. Subsequent email exchanges with A. Fathi inspired the proof of Theorem 2.7 below. On Lorentzian causality with continuous metrics 3 The conventions and notations of [16] are used throughout. In particular all manifolds are assumed to be connected, Hausdorff, and paracompact. As we are interested in C metrics, the natural associated differentiability class of the manifolds is C. Now, C manifolds always possess a C∞ subatlas, and we will choose some such subatlas whenever convenient. For instance, when a smooth nearby metric is needed, we choose some smooth subatlas, obtaining thus a smooth manifold which is C-diffeomorphic to the original one. We carry out the smoothing construction on this new manifold, obtain the desired conclusions there, and return to the original atlas at the end of the argument. A space-time, throughout, will mean a time-oriented Lorentzian manifold (M , g). 1.1. Some background on manifolds Let M be an n-dimensional smooth manifold. By this, we will mean that we have a maximal atlas A = {(Vα, φα) : α ∈ A} of charts, each of which consists of an open set Vα ⊆ M and a bijection φα:Vα → R, where φα (Vα) is an open subset of R. These charts are compatible in the sense that • For all α, β ∈ A such that Vα ∩ Vβ 6= ∅, the sets φα (Vα ∩ Vβ) and φβ (Vα ∩ Vβ) are open subsets of R; • For all α, β ∈ A such that Vα ∩ Vβ 6= ∅, the maps φα ◦ φ−1 β :φβ (Vα ∩ Vβ)→ φα (Vα ∩ Vβ) (1.1) are C∞. The collection of subsets B := {Vα : α ∈ A} defines a basis for a topology on M with respect to which the maps φα: Vα → R are continuous. The manifold M , with this topology, is automatically locally compact, i.e., any point has a compact neighbourhood. Note that we impose that the transition maps (1.1) are C∞ since we will later wish to approximate tensorial objects on M by corresponding smooth objects. In reality, imposing that the transition functions are C would be sufficient for most of our considerations. We impose the additional topological conditions that the manifold M be connected, Hausdorff and paracompact. The fact that M is Hausdorff implies that, in addition to being locally compact, M has the property that any point p ∈ M has an open neighbourhood with compact closure.‖ Remark 1.1. A theorem of Geroch [20] shows that a manifold with a C Lorentzian metric is necessarily paracompact. However, Geroch’s construction requires extensive use of the exponential map, and therefore this method cannot be applied when the metric is merely continuous. As such, we impose the condition that M be paracompact by hand. ‖ Or, equivalently, that every point has a compact, closed neighbourhood. On Lorentzian causality with continuous metrics 4 Since M is Hausdorff and paracompact, it follows that M admits smooth partitions of unity. In particular, M admits a smooth Riemannian metric. Let us once and for all choose a Riemannian metric, say h, on M , as differentiable as the atlas allows. Without loss of generality [21], we will assume that this metric is complete. 1.2. Causality: basic notions Somewhat surprisingly, the causality theory for continuous metrics appears to present various difficulties. For instance, following [16], it is tempting to continue to define a timelike curve as a locally Lipschitz curve with tangent vector which is timelike almost everywhere. With this definition, for metrics which are only continuous, it is not even clear whether or not the timelike futures remain open, and in fact we do not know the answer to this question. Next, one would like to keep the definition of a causal curve as a locally Lipschitz curve with tangent vector which is causal almost everywhere. With this definition, or for that matter with any alternative, it is not clear that a pointwise limit of causal curves is causal. To settle this last question, and some others, we will extensively use families of smooth metrics which approach the given metric g in a specific way. For this, some notation will be useful. Let ǧ be a Lorentzian metric; we will say that the light-cones of ǧ are strictly smaller than those of g, or that those of g are strictly larger than those of ǧ, and we shall write