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

Question

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 $k \geq 1$.

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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 $s < t \Rightarrow d(\gamma(s),\gamma(t)) >0$.

• $(X,d)$ 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:

4. Do $(3+1)$ dimensional chronological $C^0$ Lorentzian manifolds admit unique smoothings?

Answer 1

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...