# Do (3+1)-dimensional Lorentzian manifolds admit unique smoothings?

Of course, 3-dimensional topological manifolds admit unique smoothings while 4-dimensional topological manifolds generally do not. A (3+1)-dimensional topological Lorentzian manifold (definition below) is a topological 4-manifold, but with extra structure which locally fibers it over 3-dimensional manifolds, so perhaps there's some hope that such a space can be uniquely smoothed.

Questions: Let $$(M,g)$$ be a $$C^k$$ Lorentzian manifold of dimension $$d+1$$.

1. Does there exist a refinement of the $$C^k$$ structure on $$M$$ to a $$C^\infty$$ structure?

2. If so, can this refinement be chosen such that $$g$$ is $$C^\infty$$?

3. If so, is the $$C^\infty$$ structure unique subject to requirement (2)?

Notes:

• I'm thinking that the case $$d=3$$ is the most interesting, but I'm not sure. I think the answer must be no for $$d\geq 4$$. The answer to (1) is yes for $$d \leq 2$$, but I'm not sure about (2) or (3) in this case.

• On the face of it, the question only makes sense for $$k \geq 1$$.

Thus, I will include some speculative remarks about the case $$k=0$$, i.e. the notion of a $$C^0$$ Lorentzian manifold.

Definitions: Here is a speculative definition of the notion of a chronological $$C^0$$ Lorentzian manifold ("chronological" refers to the absence of closed timelike curves, enforced by the existence of a distance function.)

• Following Noldus (Definition 1), a Lorentzian distance on a set $$X$$ is a function $$d: X \times X \to [0,\infty]$$ which is reflexive, antisymmetric, and satisfies the reverse triangle inequality.

• The Lorentzian length of a function $$\gamma: [0,1] \to X$$ is $$L(\gamma) = \limsup_{0 = t_0 < t_1 < \dots < t_n = 1 \\ \quad |t_{i+1} - t_i| \to 0} \sum_{i=1}^n d(t_i,t_{i+1})$$

• A function $$\gamma: [0,1] \to X$$ is timelike if $$s < t \Rightarrow d(f(s),f(t)) >0$$.

• $$(X,d)$$ is a Lorentzian length space if for all $$x,y \in X$$, $$d(x,y) = \sup_{\gamma: [0,1] \to X \, \text{timelike} \\ ~~ \gamma(0) = x,\, \gamma(1) = y} L(\gamma)$$

• $$(X,d)$$ is a chronological $$C^0$$ Lorentzian manifold if it is a Lorentzian length space and, in the coarsest topology such that $$d$$ is separately continuous in each variable, $$X$$ is a topological manifold and $$d$$ is continuous.

I might go on to define a notion of "(not necessarily chronological) $$C^0$$ Lorentzian manifold" by asking for the local structure of a chronological $$C^0$$ Lorentzian manifold, but perhaps what I've already written is speculative enough.

Question:

1. Do $$(3+1)$$ dimensional chronological $$C^0$$ Lorentzian manifolds admit unique smoothings?
• Not an answer, but a minor quibble with the definition. Your last requirement, about the local equivalence of a pre-causal structure to that of Minkowski space may be too strong. The pre-causal structure gives you the light cones and, at least in the smooth case, the light cones fix the conformal class of the metric. Hence only those Lorentzian manifolds that are conformally isometric to Minkowski space (i.e., conformally flat) would satisfy your definition of a "topological Lorentzian manifold". Of course, obviously, a generic smooth Lorentzian manifold is not conformally flat. Jun 16, 2019 at 18:21
• @IgorKhavkine Thanks for pointing this out! This seems like an important issue. Presumably I want to somehow ask instead that the causality structure agrees with Minkowski space "to first order" -- but it's quite unclear what this should mean in a $C^0$ setting! Perhaps the whole notion of a $C^0$ Lorentzian manifold is incoherent... Maybe I should just ask about smoothing $C^1$ Lorentzian manifolds (which I think are unproblematic) and leave it at that. Jun 16, 2019 at 18:49
• Depending on what you are looking for, you might already find it in this review article on "cone structures" on manifolds. I'm not sure about the weakest regularity/differentiability assumptions reviewed there, but certainly Lipschitz is still OK. The review is also not the end of the story. There are new results still being published on this topic. Jun 16, 2019 at 20:54

Concerning your first three question: depending on what you exactly mean by "refinement", any $$C^k$$-manifold $$M$$ with $$k\geq 1$$ possesses a unique $$C^\infty$$-structure that is $$C^k$$-compatible with the given $$C^k$$-structure on $$M$$, see Thm. 2.9 in Hirsch, Morris W., Differential topology, Graduate Texts in Mathematics. 33. New York - Heidelberg - Berlin: Springer-Verlag. x, 221 p.(1976). [ZBL0356.57001] So it is no loss of generality to work on smooth manifolds if you want at least $$C^1$$-regularity of your manifold. If you just want a topological manifold that is of course a complete different story. Moreover, the regularity of metric can in general be not improved.

The notion of a $$C^0$$ Lorentzian manifold / spacetime makes perfect sense (if the $$C^0$$ refers to the regularity of the metric) and can be quite useful:

Chruściel, Piotr T.; Grant, James D. E., On Lorentzian causality with continuous metrics, Classical Quantum Gravity 29, No. 14, Article ID 145001, 32 p. (2012). ZBL1246.83025.

Sbierski, Jan, The (C^0)-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry, J. Differ. Geom. 108, No. 2, 319-378 (2018). ZBL1401.53058.

Galloway, Gregory J.; Ling, Eric; Sbierski, Jan, Timelike completeness as an obstruction to (C^{0})-extensions, Commun. Math. Phys. 359, No. 3, 937-949 (2018). ZBL1396.53095.

Sämann, Clemens, Global hyperbolicity for spacetimes with continuous metrics, Ann. Henri Poincaré 17, No. 6, 1429-1455 (2016). ZBL1342.83014.

However, if you just have a continuous metric several pathologies in the causal structure can occur, see Chrusciel, Grant above and also https://arxiv.org/abs/1901.07996

Also, we introduced the notion of "Lorentzian length spaces", in the same spirit as you outline above:

Kunzinger, Michael; Sämann, Clemens, Lorentzian length spaces, ZBL06970105.

Grant, James D. E.; Kunzinger, Michael; Sämann, Clemens, Inextendibility of spacetimes and Lorentzian length spaces, ZBL07030953.

This framework allows you for example to define (synthetic) timelike/causal curvature bounds via triangle comparison, analogously to Alexandrov and CAT(k) spaces.

Of course, all this does not answer your original question but several of the sides questions and hopefully gives you some context...

• Thanks, I will take a look at these references! In the meantime, I'm embarrassed not to have been aware that smoothing $C^k$ manifolds is so much easier for $k\geq 1$! According to Hirsch, the uniqueness of the smoothing is up to $C^k$ diffeomorphism, leaving open the possibility that two smoothings might be $C^\infty$-non-diffeomorphic, which is what I was after (in analogy to the Riemannian case). Where can I read about an example of a $C^k$ Lorentzian manifold which does not admit a $C^\infty$ smoothing making the original metric $C^\infty$, and in what dimensions this can occur? Jun 18, 2019 at 21:15
• Ok, I understand now better what you are after. Can you give an example in the Riemannian case, where two smoothings are not $C^\infty$ diffeomorphic and/or it does not smooth the metric? (In my answer above I was referring to the situation that you have already fixed a $C^\infty$-structure and the metric has some regularity wrt to that structure.) Jun 19, 2019 at 8:12
• Sorry, it's been a long time, but I just circled back to this question and remembered that I never got back to you. In the Riemannian case, the study of distinct smooth structures on the same topological manifold $M$ occupies the whole field of differential topology. In the case $M = S^n$, these are called exotic spheres. I am not an expert, but there exist plenty of exotic spheres in most dimensions $\geq 5$, but not in dimension $\leq 3$. Nov 5, 2021 at 12:48
• Famously, the whole subject of differential topology has a sort of "phase change" exactly at dimension 4. That is, in dimensions $\leq 3$, manifolds admit unique smoothings. In dimensions $\geq 5$, they generally do not, but the different smooth structures can be efficiently studied using algebraic topology. Uniquely in dimension 4, at the boundary between "low" and "high" dimensions, things get extremely subtle. For example, the smooth Poincare conjecture in dimension 4 is wide open -- it's not known if there exist exotic smooth 4-spheres. Nov 5, 2021 at 12:51
• For another example, $\mathbb R^n$ is uniquely smoothable for all $n$ except for $n=4$, and there are uncountably many non-diffeomorphic smooth structures on the topological manifold $\mathbb R^4$! For yet another indication of the wildness present, Freedman's proof of the topological Poincare conjecture in dimension 4 uses some very serious point-set topology, fractal structures, etc, none of which is remotely relevant in any other dimension. There's a recent book expositing Freedman's work. Nov 5, 2021 at 12:52