I figured if I was going to say anything more, I shouldn't keep spamming HT chat.

:)

One thing about this is the idea that any path can be approximated by timelike ones is ringing a vague bell.

like i might have heard it somewhere before

I have such a strong memory of hearing it. But I'm very confident your argument is correct...

A "family of light cones" will be a closed subset $L \subset X \times X$, considered up to germ equivalence along the diagonal, satisfying some axioms I haven't tried to work out. A curve $\gamma: I \to X$ is causal if for all $t \in I$ and all $s \in I$ which is $\epsilon(t)$-close to $t$, we have $(\gamma(t), \gamma(s)) \in L$.

For $X = \Bbb R^{n,1}$, if $L_0$ is the light-cone through the origin, then we have $L = \{(x, x+l) \mid l \in L_0\}$.

The fact that $L$ is closed means that being causal is a $C^0$ closed condition. This is precisely your argument.

seems about right

(I would like to point out that while it's clear what a good guess of morphisms are in your Lorentzian exit path category, I have no idea what to guess the n-cells should be. Surely not Delta^n-parameterized families of timelike curves.)

I think $\Delta^n$-parameterized families was basically what I had in mind.

That leaves open questions of regularity

But maybe there's something a bit more point-set to be recorded.

I think you shouldn't believe the regularity in the Delta^n direction matters.

I'm thinking -- in Minkowski space, the space of weakly timelike curves from x to y (where weakly timelike means it can have timelike or lightlike tangent vectors at any point) seems like it might be compact -- in the $C^0$ topology, maybe?

If it's already fiberwise C^1 you should be fine to do a smoothing argument which leaves the fibers C^1 close to where they started.

So this suggests there's a topologically interesting space of weakly timelike curves and the timelike curves are the interior -- and if this is the case, it seems a bit crude to treat either space as simply homotopy types.

@TimCampion I think not. Given a timelike curve, you can move it in the direction of any compactly supported vector field and obtain a new curve; we agree now that causal curves are a C^0 closed set.

I say causal to mean |c'(t)| <=0 for all t.

Or maybe the other way, whatever the convention is.

I'm happy with this convention!

Rephrased, timelike curves are an open set in the space of curves, which is already infinite dimensional.

Yeah, I guess I don't have very good intuition for infinite-dimensional things. This shows conclusively that the space is not locally compact at point corresponding to a timelike curve. Is there any hope that if you include all causal curves you get a compact space, though?

I guess my intuition is that to go from $x$ to $y$, you have to stay within the intersection of the forward lightcone of $x$ and the backward lightcone of $y$, which is a compact set... but I suppose that doesn't make the function space compact.

Yeah, I think you're out of luck. As far as I can tell, a compact Hausdorff space is locally compact, so you'd have to do some nasty identifications.

ah, right!

And if you want to think of these as infinite dimensional manifolds, then actually if I recall, inf dim Frechet manifolds are homeomorphic if and only if they're homotopy equivalent.

I want to say even diffeomorphic so long as the tangent spaces you're modeled on are isomorphic TVS's.

So my gut feel would be to just consider the homotopy type of these path-spaces. These homotopy types carry some interesting information.

But of the timelike path space, or the causal path space?

For instance, people care if there are closed causal loops; I think these can't be causal null-homotope.

@TimCampion Oh, I see, the causal path space isn't a manifold, because these maps are valued in a non-manifold.

I don't have a good guess how different the results are.

Actually, maybe the inclusion is a homotopy equivalence,

I believe that if we allow our timelike curves to have zero derivative, not if we don't

That's probably the simplest thing then -- consider the homotopy type of causal paths, which is hopefully homotopy equivalent to an appropriate space of timelike paths...

So then it's no harm to think simplicially and just take the $n$-simplices to be $\Delta^n$-families of such paths.

I'm trying to convince myself quickly that a strictly timelike closed curve is not null-homotopic, or at least not causal null-homotopic. Just to convince myself that this is an interesting homotopy type at the level of pi_0.

If you put a Riemannian metric on the disk paramterizing the nullhomotopy, the Lorentzian metric picks out a vector field flowing in the "time" direction. I think there are no nonvanishing vector fields on $D^2$ which are in the tangent direction at the boundary.

That's correct. Thanks, I was trying a totally different argument.

(Take the double. This works on any even-dimensional manifold with nonzero Euler characteristic, where taking the double indeed doubles the Euler characteristic. I haven't thought about the odd case.)

So then one thing I was thinking of doing with this $\infty$-category $Exit(M)$ would be to think about the topos-theoretic properties of the presheaf category $Fun(Exit(M)^{op}, Top)$ -- just like if $X$ is an ordinary space, you can think about the topos of functors $\Pi_\infty(X) \to Top$.

I am convinced now that this is an interesting invariant of Lorentzian manifolds. It's hard to say what to do with it, other than to start computing, I guess.

Oh, nice timing.

@TimCampion Well, now you're a bit out of my pay grade, but I believe you it's interesting.

:)

But I'm sure there are other interesting things to do. Weren't exit-paths invented for some kind of intersection theory thing?

The other thing is that I expect your sheafy definition should be equivalent to my light cone definition, which sounds a little more homotopical to me, maybe. Or at least a little easier to see which paths are timelike. But this isn't very important, just a curiosity to me.

Probably. I don't know anything about them, just the name and sort of the definition.

Yeah, initially I sat down and was like "'topological Lorentz manifold' -- that makes sense, right?" and then I realized that neither atlases nor sheaves of functions seem to work immediately

It would definitely be nice to be able to pass back and forth between different definitions.

Well, thanks for chatting -- let's keep in touch!

Sure, suddenly got busy - sorry for dropping out.

@TimCampion Since they are totally orthogonal to the question, I suggest deleting our comments. Let me know when you see this and I'll go ahead and get rid of mine.

I wouldn't mind leaving them -- I still feel a bit unsure about what exactly Gromov's theorem says about the situation.

well

maybe your

rigt

@MikeMiller I'll delete mine

I don't think it says much, but I still plan to work out what it does say when I get a chance, and I'll let you know here when I do

@TimCampion Pretty silly error in the end. When approximating in a neighborhood of a compact set $K$, you want the compact set to be a subpolyhedron of positive codimension. Then for the general form of the h-principle on noncompact manifolds, you approximate near a codimension 1 subset whose homotopy type is correct, and use a diffeomorphism to "expand outwards". In this expansion step you lose the C^0 denseness.

In other words, the only thing the h-principle argument can give here is that near any point you can approximate a curve by timelike curves. Pretty obvious!

I seem to have pinned down what I was remembering, too.

It's the exercise at the start of Chapter 17 in Eiliashberg-Mishachev, which shows that any timelike path can be approximated by a null curve.

This should be visually surpising yet also much more plausible than the previous claim, after struggling to come up with an obstruction for a few minutes.

Thanks to @BalarkaSen for pointers.

@MikeMiller Awesome, that's great to have pinned down!

I'm currently thinking that the Lorentzian exit-path $\infty$-category will require some further modification. For instance, in Minkowski space, one ends up with an $\infty$-category equivalent to a poset. Its points are the points of Minkowski space, and the poset relation $p \leq q$ just says whether there is a timelike path from $p$ to $q$. I was kind of hoping to get a category which doesn't have uncountably many isomorphism classes of objects, though...

Well, if you use causal paths $|c'(t)| \leq 0$, you can reverse of a causal path and then (I think) contract the loop to a point through a causal homotopy, so that any two points which are connected by a causal path are isomorphic.

My temptation is to think that it's not bad to have so many isomorphism classes anyway, though. Riemannian manifolds have infinitesimal invariants (most importantly, curvature), which give different information at every point; the set of closed geodesics emanating from any point depends interestingly on the point and the global geometry of the manifold. This feels sort of like that.

Eh, no, it seems very strange to allow both forward-time and backward-time paths. I'm a little uncomfortable with that.

I was thinking at some point of allowing both strictly-forward and strictly-backward timelike paths, but then composition gets messed up.

Igor Khavkine also commented on my question that in Lorentzian geometry, the conformal structure is determined by the lightcone structure, which messes up the definition I was using for a $C^0$ Lorentzian manifold. I think it also messes up your definition, but I'm not sure. The point is that if everything is locally isomorphic to Minkowski space, then it's actually conformally flat -- a big restriction.