Is the set of Lorentzian metrics metrizable?

Question

Fix a differentiable non-compact manifold $M$. Denote by $\mathrm{Lor}(M) := \{\text{Lorentzian metrics on $M$}\}.$ One can define a topology on this set via: fix any open covering $\mathcal{A}$ on $M$. For each $g \in \mathrm{Lor}(M)$ and for each positive continuous function $r : M\to ]0,\infty[,$ one defines: $$\mathcal V(g,r) := \{h \in \mathrm{Lor}(M) : \forall p \in M, |\nabla^kg_{ij}(p) - \nabla^kh_{ij}(p)| < r(p)\},$$ where $g_{ij}, h_{ij}$ are the coordinates on some local chart with open domain contained on some open of $\mathcal{A}$. (I am sorry being a bit informal at this point, but I think it is clear what I mean). Also, $\nabla^k$ is supposed to denote any $k$-derivative of the metric. This is a sub-basis for the $C^k$-topology on $\mathrm{Lor}(M)$.

My first question is: is this topology somehow metrizable?

Further, if $M$ was compact (and $\chi(M) = 0)$, then $\mathrm{Lor}(M)$ is not empty, by imposing the $L^2$ metric (for some Riemannian metric) on $\mathrm{Lor}(M)$ induces the same topology as that I have defined? I mean, if we are interested on statements like:

(M,g) is locally causal if $g$ is close (on the sense of $C^r$-topology) to a causal metric $h$ on $M$,

Does this can be interchanged by

(M,g) is locally causal if $g$ is close to a causal metric $h$ on $M$ on the $L^2$-norm.

I am also sorry it these questions don't make sense at all, at the end, my question is: in general, introducing a Riemannian metric for comparing Lorentzian metric is somehow inappropriate, in the sense it leads to lost of some information?

Answer 1

First of all, there is a bunch of basic things that you need to write in a slightly clearer way. If you try to topologize the set of Lorentzian metrics as you did, you need first:

1. Restrict to the subset of Lorentz metrics of a given (say, $C^k$, $0\leq k\leq\infty$) regularity, otherwise your definition for the basic neighborhoods $\mathcal{V}(g,r)$ makes no sense.
2. Once you did the above (denote the resulting set by, say, $\mathrm{Lor}_k(M)$), replace $|\nabla^k g_{ij}(p)-\nabla^k h_{ij}(p)|$ by the sum $\sum_{0\leq j\leq k}|\nabla^j g_{ij}(p)-\nabla^j h_{ij}(p)|$ in the definition of $\mathcal{V}(g,r)$ for $k$ finite. If $k=\infty$, you must include such $\mathcal{V}(g,r)\doteq\mathcal{V}_k(g,r)$ for all $k\geq 0$ (or at least for all $k$ in an infinite subset of $\mathbb{N}\cup\{0\}$).

This is the bare minimum. Ideally, it would be better to do things in a coordinate-free way: denote by $\nabla^k g$ the iterated covariant derivative of order $k$ of $g$ w.r.t. some (say, torsion-free) covariant derivative operator $\nabla$ on $M$ and define the pointwise Euclidean norm $|T|$ of a tensor field $T$ by lifting some reference Riemannian metric $e$ on $M$ to the corresponding tensor bundle. This provides a way to write a fiberwise scalar product on the jet bundle of order $k$ of the fiber bundle of Lorentzian metrics over $M$ (more generally, on the jet bundle of order $k$ of the vector bundle of covariant tensors of rank 2 over $M$). One can write these things in more detail, but this is more or less standard. Anyway, I think you get the gist.

With the trivialities out of the way (at this point, if something about them is not yet clear, please do let me know), we can begin to address your question proper. What you wrote above (with the tacit understanding that the amendments 1. and 2. above are included) are the basic neighborhoods for the $C^k$ Whitney topology of $\mathrm{Lor}_k(M)$. In fact, this topology does not depend on the choice of a reference Riemannian metric $e$ as above (the reason will be explained below).

If $M$ is compact, this topology is even normable if $k$ is finite (as a subset of the normable vector space of $C^k$ covariant tensor fields of rank two) and still metrizable if $k=\infty$ for then it coincides with the compact-open $C^k$ topology. However if $M$ is non-compact (as you assumed, since you seem to be ultimately interested in causality theory for Lorentzian metrics and this theory is nontrivial only for non-compact $M$) this topology is non-metrizable for all $k$ (even $k=0$). In fact, this topology is not even first-countable in this case.

This is easier to visualize in the case of $C^k$ scalar fields (i.e. $C^k$ real-valued functions) on $M$ instead of Lorentzian metrics: the connected component of $f\equiv 0$ in the $C^k$ Whitney topology of $C^k(M)$ is the space $C^k_c(M)$ of $C^k$ functions with compact support on $M$ with the usual inductive limit (locally convex vector space) topology. This topology is not first-countable, hence non-metrizable. More generally, the connected component of any $C^k$ function $f$ in the $C^k$ Whitney topology of $C^k(M)$ is precisely $f+C^k_c(M)$. One sees from this remark that the Whitney topologies get so fine when $M$ is non-compact, they become extremely disconnected. A similar fact holds for the $C^k$ Whitney topology in $\mathrm{Lor}_k(M)$ - the metrics $h$ in the connected component of $g\in\mathrm{Lor}_k(M)$ in this topology differ from $g$ only inside some compact subset of $M$ (depending on $h$). This remark also makes it clear why the choice of the Riemannian metric $e$ is not relevant to the definition of the $C^k$ Whitney topology, despite the fact that $M$ is not compact.

For (many!) details on the Whitney topologies, you may want to check The Convenient Setting of Global Analysis by Andreas Kriegl and Peter W. Michor (AMS, 1997), specially Chapter IX (Manifolds of Mappings). The paper by Lerner quoted by Igor in his comment relates the Whitney topologies to structures which are natural to Lorentz metrics - e.g. conformal classes of Lorentzian metrics with representatives being $C^0$ Whitney-near to each other amounts to their light cones being close to each other, metrics which are $C^1$ Whitney-near to each other have their geodesics near to each other in some sense, and so on.