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 $d+1$.

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

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

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:**

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