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Let $M$ be a sufficiently nice Lorentzian manifold of dimension $\geq 3$. It's known [1] (see also [2]) that the differential and even conformal structure of $M$ is completely encoded in the causal relation (i.e. the relation on points of $M$, where $p \leq q$ if there is a causal path from $p$ to $q$).

This suggests that one should be able to study smooth and conformal questions about $M$ synthetically. By this, I mean that one should be able to work with a sufficiently-nice poset $(M,\leq)$ (no longer assumed to be a Lorentzian manifold -- see [3] for more precise hypotheses on the poset), and develop notions of conformal geometry such as Weyl curvature, null geodesics... One should be able to describe what additional structure on $(M,\leq)$ is needed to encode a metric. One should be able to talk about differential concepts such as smoothness of functions, vector fields and differentiation of functions by vector fields. And so forth.

Question 1: Is the synthetic conformal geometry of (nice, at least time-oriented) Lorentzian manifolds developed in the literature anywhere?

I'd be happy to start by getting off the ground with something basic, for example

Question 2: In particular, can one directly define what it means for a function $f : M \to \mathbb R$ to be smooth, when $M$ is a sufficiently-nice poset?

[1] Peleska, https://doi.org/10.1007/BF02192656

[2] Hawking, King, and McCarthy https://doi.org/10.1063/1.522874

[3] Martin and Panangaden and https://arxiv.org/abs/gr-qc/0407094

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    $\begingroup$ The abstract notion of "causal spaces" as a special kind of preordered space goes back to work by Kronheimer and Penrose (1967). Look as well for the refined notion of "Lorentzian length spaces" as introduced by Kunzinger and Sämann (2017). $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented May 11, 2023 at 23:07
  • $\begingroup$ You deduce the differential and conformal structure from a fairly naive local argument, no? I doubt this would be a significant-enough departure from the traditional perspective to contribute anything new. $\endgroup$
    – Ryan Budney
    Commented May 11, 2023 at 23:28
  • $\begingroup$ @RyanBudney In the argument I've seen you need to construct local coordinates in a clever way. I'd like to see how to think about notions like differentiability without the awkward assumption that "the prescription for local coordinates given by Hawking-King-McCarthy in fact yields local coordinates". $\endgroup$
    – Tim Campion
    Commented May 12, 2023 at 0:16
  • $\begingroup$ @PedroLauridsenRibeiro That's great, thanks! Has anybody described what it means to have a smooth function, for instance, on a causal space? I don't see such a development in Kronheimer-Penrose. $\endgroup$
    – Tim Campion
    Commented May 12, 2023 at 0:18
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    $\begingroup$ I think the whole idea in this synthetic approach is to be able to work in the opposite direction, that is, to do "rough" Lorentzian geometry, just like in the Riemannian case (e.g. metric measure spaces). Particularly, how to define and estimate length and curvature in such scenarios, and so on. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented May 12, 2023 at 3:44

2 Answers 2

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I'm far from an expert on this area and hopefully someone more knowledgeable can answer, but it seems to me that it is impossible to give an answer unless you are more precise about what you would consider a "sufficiently nice poset".

Just to make this (somewhat) more concrete. If you take $(N,g)$ to be a four-dimensional Eguchi-Hanson metric and form a Lorentzian metric $(M,h)$ by setting $M=\mathbb{R}\times N$ and $h=-dt^2+g$ then you get a Lorentz manifold (that should satisfy the Einstein equations even).

Now consider the sequence $(M, \epsilon h)$ for $\epsilon\to 0$. The causal structures of all of these manifolds are the same, which suggests that in the limit the causal structure is passes to one on the manifold given by taking the cone over $\mathbb{R}\times C(\mathbb{R}P^3)$ where here $C(\mathbb{R}P^3)$ is the cone over the $\mathbb{R}P^3$ something that is manifestly not a manifold.

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  • $\begingroup$ I think what I mean by a "sufficiently nice poset" is as described in [3] above -- a bicontinuous poset (see Def 2.10 -- this is actually a notion imported from computer science!) which is globally hyperbolic (i.e. intervals are compact in the interval topology). $\endgroup$
    – Tim Campion
    Commented May 12, 2023 at 0:25
  • $\begingroup$ I guess your point is that from a Lorentzian manifold it's easy to construct spaces which still have nice causality relations without actually being smooth? So either the conditions on the poset will have to rule out deformations which look natural order-theoretically, or else they will have to give some "pseudo-smooth structure" to non-manifolds. This seems like an important point. I think I'm tending to think I want notions which will do the latter, but it does look a bit implausible when you put it this way. $\endgroup$
    – Tim Campion
    Commented May 12, 2023 at 0:26
  • $\begingroup$ Yes. A good example is convexity in one variable analysis. By this I mean, $f(tx+(1-t) y)\leq tf(x)+(1-t)f(y)$ for $t\in [0,1]$. When the function is $C^2$ this is equivalent to $f''\geq 0$, but the first notion makes sense even for less smooth things (and while it does impose some regularity properties on the function it includes a much broader class). I'm speculating here but related ideas in geometry are sometimes called synthetic notions of curvature and that is perhaps some source of confusion (as I understand this term is also used in categorical treatments for something else). $\endgroup$
    – RBega2
    Commented May 12, 2023 at 1:28
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I arrive a bit late to this question but here's my answer, I hope it is useful:

There is a very prolific synthetic approach to Lorentzian geometry (and GR) first introduced by Kunzinger and Sämann [1] where they define the notion of "Lorentzian (pre-)length space", which is a causal set with a chronological relation and a lower semicontinuous time separation function satisfying some extra properties.

There are a lot of results that can be translated from the smooth setting to this synthetic setting such as the causal ladder of spacetimes. This also has the advantage that is the "analogous" case to metric geometry of length spaces so people also study classic metric results (as in Alexandrov geometry) in the Lorentz case as a notion of timelike curvature bounds is available in Lorentzian length spaces.

There's also a notion of ricci curvature bounds to obtain synthetic energy conditions via optimal transport. A very interesting paper was published by Cavalletti and Mondino [2] where they prove Hawking type singularity theorems in this setting.

I understand that there is a program by a lot of people (Kunzinger, Sämann, Gigli, McCann, Beran, Rott, Ohanyan, Calisti, and maybe I'm forgetting someone) to develop non-smooth hyperbolic calculus in this Lorentzian length spaces in a similar manner to what optimal transport has allowed people to do in RCD spaces.

[1] Kunzinger and Sämann https://arxiv.org/abs/1711.08990

[2] Cavalletti and Mondino https://arxiv.org/abs/2004.08934

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