# On the causal structure of spacetimes: piecewise $C^1$, $C^k$ or $C^\infty$?

This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the infinitesimal causal relation is the choice of a forward light cone at every point, i.e. a time orientation. Now the (global) causal relation would take two points $p$ and $q$ and call $q$ in the future of $p$ if there is a causal curve from $p$ to $q$. Of course you have different futures depending on whether you take timelike or just causal curves, their interplay is of crucial importance in understanding the global geometry of Lorentz manifolds.

Now my question is what types of curves one actually uses:

One option is to stay with $C^\infty$-curves, simple but sometimes difficult in technical aspects as one would like to have arguments with broken geodesics etc.

The other option is to use piecewise $C^\infty$ curves, where the jumps of the velocities at break points stay in the same (forward) light cone. This should make the above disadvantage mostly disappear.

The option I have also seen in the literature (O'Neill...) is to take piecewise $C^1$-curves, again with the same condition at the break points. Now this is probably the most general version where one has still a meaning of what is "future-directed". However, many important(!) arguments in exploring the causal structure use variational formulas for arclength etc which can handle finitely many break points but require higher differentiability, say $C^2$ or even $C^3$.

So does the above choice matter? If so, what is the good convention?

EDIT: For convenience, here the more precise definitions: a tangent vector is called timelike if it is in the open interior of the lightcone and causal if it is inside the closed lightcone. It is called lightlike if it is in the boundary. Correspondingly, one has timelike, lightlike and causal curves, by using this for the tangent vector to the curve.

The set $J^+(p)$ is the causal future (we assume to have a time orientation) of the point $p$ which can be reached by causal curves, while $I^+(p)$ is the timelike future of the point $p$, i.e. those points which can be reached by timelike curves. While the condition "timelike" seems to be rather robust (tangent vectors inside the open cone) for causal this is more touchy. Nevertheless, for normally hyperbolic pdes on the Lorentz manifold, i.e. those with principal symbol given by the metric, it is the causal future $J^+$ which controls the propagation of singularities etc.

To give a flavour of known results (see e.g. O'Neill): for a causal piecewise $C^3$-curve $\gamma$ which can not be reparametrized into a lightlike geodesic one finds a variation $\gamma_s$ such that for all $s > 0$ the curve $\gamma_s$ is timelike and has the same starting and end point. This result is important for the transitivity of the causal relations. But the proof uses very much $C^3$.

• Have you thought about approximating $C^k$ curves by $C^\ell$ curves for $\ell>k$? It seems to me that being causal is an open condition in $C^1$ topology and the forward timelike cone is convex at each point, so all this should be possible. In other words, the causal structure should be independent of the choice. Or am I missing some important point here? Apr 8, 2017 at 14:15
• @Sebastian: Hmmm. I'm not so sure about the open condition. For timelike curves, i.e. with tangents in the open forward light cone, this should be OK, but for causal ones? If they have tangents on the (boundary of the) lightcone, the usual smearing could tilt the tangent outside the lightcone into spacial directions. At least, I don't see that directly. It just wonders me that in the literature this is not discussed too well. The timelike curves are certainly robust enough. Apr 8, 2017 at 16:35
• I see. Maybe you should add this in your question (for people like me who don't know the difference between causal and forward timelike). Anyway, the cone still has strictly convex cross-section, and I have the impression that a causal curve that is not a lightlike geodesic should be approximable by timelike $C^\infty$ curves. But I don't have a proof or reference for that unfortunately. Apr 9, 2017 at 3:37
• @StefanWaldmann Who is O'Neill? Can you please provide a link to the reference? Thanks. Jul 18, 2017 at 12:28

The answer is that it doesn't matter, as long as the metric itself is sufficiently regular. In the following notes by Chrusciel, the basic assumption is that the metric is $C^2$:

[C] Chruściel, Piotr T., Elements of causality theory, arXiv:1110.6706 (2011).

I'm not up on the state of the art of how low you can push the regularity of the metric and keep all the standard results of causality theory, but in the following paper it is shown that it all works even for $C^0$ metrics (those that avoid what the authors call causal bubbles):

[CG] Chruściel, Piotr T.; Grant, James D.E., On Lorentzian causality with continuous metrics, arXiv:1111.0400 Classical Quantum Gravity 29, No. 14, Article ID 145001, 32 p. (2012). ZBL1246.83025.

The paths actually considered in these references are locally Lipschitz, which covers all the regularity classes that you've included in the question. Corollary 2.4.11 [C] shows that both the timelike and causal futures/pasts coincide, whether you use locally Lipschitz paths or piece-wise broken geodesics, respectively timelike and causal. If you deal with $C^\infty$ metrics, then your geodesics will also be $C^\infty$, so this result shows that it is indeed safe to stick with the piece-wise $C^\infty$ class. Then the result that you refer to, about deforming a non-null geodesic into a timelike curve with the same end points, is known as a push-up lemma in the literature. You can find this result for locally Lipschitz curves in Proposition 2.4.18 [C], which relies on the technical intermediate results Corollary 2.4.16 [C] and Lemma 2.4.14 [C].

• Thanks a lot. I will take a look at the references. In any case, I'm happy that this seems to be not completely innocent (the paper being of 2011...). Apr 9, 2017 at 10:59
• For the record, I think for regular metrics the actual results of this sort are rather old. The article [C] from 2011 is a review, which was conveniently time with respect to the actual novel results of [CG] about rough metrics. Apr 9, 2017 at 11:21
• @StefanWaldmann, the earliest relevant reference that I know from the classic literature on causality theory is Prop 2.19 of Techniques of differential topology in Relativity by R. Penrose (SIAM, 1972). Penrose proves that two points connected by a "broken" null geodesic are either connected by a single "unbroken" null geodesic or by a "broken" timelike geodesic. Since the individual pieces of these "broken" curves are all geodesics, their regularity is dictated by the regularity of the metric. May 30, 2017 at 10:12

Let me expand on Igor Khavkine's answer, especially:

"The answer is that it doesn't matter, as long as the metric itself is sufficiently regular."

In our recent paper: The future is not always open we clarified all these issues with the regularity of the curves vs. the regularity of the metric (and showed that there are some pathologies in low regularity). Maybe this is also of interest to you.