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David just pointed out an article that looks very interesting even after just reading the first few paragraphs. I thought I’d go ahead and start a discussion on this paper
Abstract: This article gives a conceptual introduction to the topos approach to the formulation of physical theories.
Here’s the first few paragraphs of the introductions:
Introduction
Over forty years have passed since I first became interested in the problem of quantum gravity. During that time there have been many diversions and, perhaps, some advances. Certainly, the naively-optimistic approaches have long been laid to rest, and the schemes that remain have achieved some degree of stability. The original ‘canonical’ programme evolved into loop quantum gravity, which has become one of the two major approaches. The other, of course, is string theory—a scheme whose roots lie in the old Veneziano model of hadronic interactions, but whose true value became apparent only after it had been re-conceived as a theory of quantum gravity.
However, notwithstanding these hard-won developments, there are certain issues in quantum gravity that transcend any of the current schemes. These involve deep problems of both a mathematical and a philosophical kind, and stem from a fundamental paradigm clash between general relativity—the apotheosis of classical physics—and quantum physics.
In general relativity, space-time ’itself’ is modelled by a differentiable manifold $\mathcal{M}$: a set whose elements are interpreted as ’space-time points’. The curvature tensor of the pseudo-Riemannian metric on $\mathcal{M}$ is then deemed to represent the gravitational field. As a classical theory, the underlying philosophical interpretation is realist: both the space-time and its points truly ‘exist’\footnote{At least, that would be the view of unreconstructed, space-time substantivalists. However, even purely within the realm of classical physics this position has often been challenged, particularly by those who place emphasis on the relational features that are inherent in general relativity.}, as does the gravitational field.
On the other hand, standard quantum theory employs a background space-time that is fixed ab initio in regard to both its differential structure and its metric/curvature. Furthermore, the conventional interpretation is thoroughly instrumentalist in nature, dealing as it does with counter-factual statements about what would happen (or, to be more precise, the probability of what would happen) if a measurement is made of some physical quantity.
It makes sense to distinguish what is purely topological from what is metrical. The metric is a tensor field sitting on the smooth manifold. Maxwell’s equations provide a good example.
The expression
$d F = 0$is purely topological. It is defined even if you don’t have a metric at all. This represents two of four Maxwell’s field equations ($\nabla\times E + \frac{\partial B}{\partial t} = 0$ and $\nabla . B = 0$).
The expression
$d G = j$is also purely topological. This represents the other two Maxwell field equations ($\nabla\times H - \frac{\partial D}{\partial t} = J$ and $\nabla . D = \varrho$).
The metric is needed to link $F$ and $G$ (via the constitutive relations)
$G = \star F$giving $D = \epsilon E$ and $B = \mu H$.
You probably don’t want to learn point-set topology - it’s full of gnarly things you wouldn’t want to meet in a dark alley (like the long circle - it’s sort of like a circle, but ’infinitely long’, so that no map from an ordinary circle can wrap around it. Or the Warsaw circle, which is connected, but has points that don’t have connected neighbourhoods, and again an ordinary circle can’t wrap around it, even though it cuts the plane into two regions). Hatcher’s book Algebraic Topology is free and quite good (you don’t need the stuff towards the end, I wager, but I used it to construct a toy spacetime in which to look at string theory’s T-duality in my honours thesis) Or Ronnie Brown’s book on topology is very cheap (£5!) for an electronic copy - you can buy a hardcopy as well - and this is very good for basic topology. I used it as a constant reminder of things for my PhD thesis that I was slow in absorbing.
I wouldn’t suggest it as a general reference for learning topology, but the first part of my dissertation was designed specifically to introduce (and motivate) topology to “scientists and engineers”.
PS: I received good feedback specifically on the motivating Introduction from Richard Bishop (yes, the Richard Bishop) and Stephanie Alexander (who was on my committee and still my all time favorite professor who taught me elementary differential geometry). I think you might like it too.
@Eric,
the link to your dissertation goes to the nLab at the moment, instead of your private web, where I imagine it should have gone.
Thanks David. Fixed.
@DavidRoberts:
You probably don’t want to learn point-set topology - it’s full of gnarly things you wouldn’t want to meet in a dark alley (like the long circle - it’s sort of like a circle, but ’infinitely long’, so that no map from an ordinary circle can wrap around it. Or the Warsaw circle, which is connected, but has points that don’t have connected neighbourhoods, and again an ordinary circle can’t wrap around it, even though it cuts the plane into two regions). Hatcher’s book Algebraic Topology is free and quite good (you don’t need the stuff towards the end, I wager, but I used it to construct a toy spacetime in which to look at string theory’s T-duality in my honours thesis) Or Ronnie Brown’s book on topology is very cheap (£5!) for an electronic copy - you can buy a hardcopy as well - and this is very good for basic topology. I used it as a constant reminder of things for my PhD thesis that I was slow in absorbing.
I disagree with you for the following reason: there are basic skills in point-set topology that one needs to understand before progressing. Things like the product topology, the quotient topology, at least the axioms T0, T1, and T2, the many equivalent formulations of continuity, closures, supports, compactness, convergence, metric spaces, topological equivalence of metrics, uniform equivalence of metrics, isometries, product and quotient metrics, retractions, injective homeomorphisms, and the list goes on. These are important technical aspects of point-set topology that are used all the time in the more “interesting” areas of topology.
@Ian: I suggested one of the following three books:
General Topology by John L. Kelley
Topology by Munkres
Topologie Generale by Nicolas Bourbaki.
I believe I ranked them as follows: Easiest read - Munkres, Most citations (over a thousand!) - Kelley, Sheer clarity of presentation - Bourbaki.
You’ve said in the past that you’re rather comfortable with elementary modern algebra, which leads me to believe that Bourbaki would actually be the best fit for you. My philosophy for books on modern algebra and general topology is that if you can read Bourbaki (without struggling too much), then you should, simply because the books are a work of art (the proofs are always clear and concise, but never slick).
Please e-mail me at harry.gindi ([{AT}]) gmail.com, and I can provide you with electronic copies of all three books.
Ian said:
I really do need to learn some topology.
and David answered:
You probably don’t want to learn point-set topology - it’s full of gnarly things you wouldn’t want to meet in a dark alley…
I think what David wanted to warn you about is that the notion of “topological space” is very general, and general point-set topology is about all the “pathological” situations that tend to fascinate mathematicians and are deemed irrelevant by physicists, much like the difference of smooth and analytical, or that of the Riemann and the Lesbegue integral.
Is there a specific topic that you would like to learn something about?
Like, for example: “I would like to understand Connes’ work on the standard model, would like to understand his book ’Noncommutative Geometry’ for that reason, keep running into ’fribrations’, what is that?”. It would be easier to point out some literature if I knew what you are up to.
@Tim: I’m not saying that Ian should read these books front to back. That would be crazy. However, many arguments in functional analysis and differential geometry (I assume that this is what Ian is really interested in from a physics perspective) rely on a fairly solid understanding of most of the things that I noted above.
Point-set topology is a fine thing to learn. Munkres’s book is a nice gentle introduction at an undergraduate level, but is solidly written and rather free from errors. I wish it were written in a more categorical spirit (which really explains why, e.g., products and quotients have the topologies they do), and I personally believe there is undue emphasis on things like necessary and sufficient conditions for metrizability, but that’s as may be. In fact, all the books Harry recommended are good.
It’s possible and maybe even tempting to teach point-set topology with rather a heavy emphasis on pathology, but that would be a mistake, and I don’t see that this is what the subject is actually about. On the other hand, core notions like connectedness, compactness, and separation axioms are indispensable, as are basic constructions – every mathematician should know them.
It’s possible and maybe even tempting to teach point-set topology with rather a heavy emphasis on pathology, but that would be a mistake, and I don’t see that this is what the subject is actually about. On the other hand, core notions like connectedness, compactness, and separation axioms are indispensable, as are basic constructions – every mathematician should know them.
Absolutely! Pathological examples should only be studied as far as they explain the necessity of hypotheses. It seems like point-set topology has gained a bad reputation for having so many such examples, but this really only serves to demonstrate how the early development of topology proceeded. The definition of a topological space is an abstraction of the definition of a metric space. Once they discovered this definition, the early topologists set to work finding sufficient conditions over and above the definition of a topological space from which we could derive the properties of euclidean space.
Contrast this with algebra. Most theorems in algebra are not proven first from minimal conditions over the core axioms. Rather, theorems are often proven in the simplest case (for example, over algebraically closed fields) then extended to more general cases by reducing the problem to the case that is already known.
Point-set topology doesn’t lend itself nearly as well to such reductions. Since we are trying to generalize the important properties of euclidean space, which is a very very special space, we can’t often prove things by reduction to the case of euclidean space, so we’re forced to move ahead and develop things from the bottom up. The pathological examples in topology are essentially justifications of the separation axioms. They were much more important to the development of point-set topology than to the study of the subject in the modern day.
I’ll be honest and say I’d never heard the word “margaritiferous” before.
Darn! I had hoped someone could explain that to me.
He says quantum gravity is lacking the experimental data portion of the “unholy trinity” as he calls it.
Sometimes you first need a theory to be able to formulate new questions, with answers that can support or falsify your theory. Take QM itself: If you are restricted to non-relativistic systems you will never find any facts that contradict it: you would need to find a physical system and prove that there cannot be any Hamiltonian at all such that the time evolution is described by the Schrödinger equation. I don’t think that this is possible. If you take special relativity into the account, you see that 1.QM violates causality and 2. cannot describe particle creation and annihilation. To see that QM cannot be the end of the story you don’t need any experiments that show 1., the theoretical fact of a contradiction should be enough to motivate the development of a new theory…
While he ultimately may be correct in saying that we need to dispense with real-numbered assumptions, he seems to gloss over the fact…that we have good reason to rely on them since that is how we experience the world…
But we can measure rational numbers only, not real ones.
From Chris Isham’s paper:
The Kochen-Specker theorem is equivalent to the statement that the spectral presheaf has no global elements.
Now that statement should go to a nLab page, shouldn’t it? (Yeah I know, but I don’t think that I understand both the classical theorem and its Isham-reformulation well enough yet).
We had had some discussion of the sheaf-theoretic interpretation of Kochen Specker in old blog discussion, such as here. I agree that it would be nice to turn this stuff into nLab entries. I don’t have time for it right now, though…
the theoretical fact of a contradiction should be enough to motivate the development of a new theory...
But we can measure rational numbers only, not real ones.
measure irrational numbers as well via ratios
???? ratios of what? All we measure are finitely long decimal expansions, and ratios of these are also finitely long. This does depend on one’s choice of units, of course. There nothing to say one can’t define the length 1 Michael* = $\pi$ cm, and by measuring 1.254336 cm, then this is an irrational number of Michaels. But this is silly (*Michael is my middle name, ’Davids’ or ’Robertses’ sounds silly)
I’m not following this discussion closely; just writing to mention there was Café discussion on this topic here, including some discussion of Kochen-Specker. I don’t think this blog post was mentioned yet.
???? ratios of what? All we measure are finitely long decimal expansions, and ratios of these are also finitely long.
Todd said:
I’m not following this discussion closely; just writing to mention there was Café discussion on this topic here, including some discussion of Kochen-Specker. I don’t think this blog post was mentioned yet.
Urs did, although I have to admit that I did not have a look at Isham’s papers mentioned there - so the Kochen-Specker theorem fails in two dimensions just like Gleason’s theorem? I’m sure that is connected in some way or another…
http://arxiv.org/abs/1005.4172 - the article aims at recovering special relativity from the purely causal structure of spacetime i.e. as a poset (perhaps with some structure). This certainly seems up Eric’s alley, and possibly Ian’s
@Ian: I just want to note another very cool approach taken by Bourbaki:
Chapters 1 and 2 cover topological and uniform structures respectively and give axiomatic descriptions of such spaces.
Chapter 3 covers the theory of topological groups.
Chapter 4 applies the results of chapters 1 and 2 to construct from scratch the real line, which is not used explicitly or implicitly in any of the preceding chapters or even the preceding books. I find this extremely appealing.
@David: Yeah, I haven’t looked at the paper yet, but the idea is not new. It goes back at least 20 years. I’ll try to find some references.
The ingredients you need to construct Minkowski space are (roughly)
Thanks to the Leinster measure, I think you really only need a poset now. This is so cool, I wish it was taught to undergrads.
@Eric: That’s not true unless you’re using a nonstandard definition of a Minkowski space.
Ingredients you need to construct a Minkowski space are: An R-vector space of dimension n and a symmetric nondegenerate bilinear form of signature (n-1,1,0). Note that Sylvester’s classification law does not hold over general fields (for instance, it is patently false for the rational numbers, and over the complex numbers, the signature is even simpler).
Here are some references
One of the earliest (if not the earliest) paper to discuss this is
Eric said:
Thanks to the Leinster measure, I think you really only need a poset now.
Unfortunatly I did not follow that discussion, I suppose that the nCafé thread that Leinster measure redirects to is still the best place to start?
Harry said:
Ingredients you need to construct a Minkowski space are:…
I think Eric alluded to the game “taking the classical limit” of some “quantum system”, which has floating rules, usually you are not restricted to use the ingredients of your quantum system only, you may take all sorts of approximations :-)
I’m pretty sure you can construct Minkowski space from the causal structure (poset) and a measure. If it were trivially obvious, it wouldn’t be “so cool” :)
The poset itself gives you Minkowski space up to a conformal transformation. The measure ties down the conformal transformation. I don’t know if anyone has gone ahead and proved it with Leinster measure, but I have very little doubt the math will go through as promised.
Roughly, you start with $\mathbb{R}^4$ and define a (causal) relation between points. This determines the Minkowski metric up to a conformal transformation. If you specify a measure, you get the Minkowski metric. It would be interesting to determine the Leinster measure of this poset and see if it gives the correct Minkowski metric.
I would lose faith in the universe if the Leinster measure does not give the correct metric :)
Here is a blurb Urs wrote on smooth Lorentzian space:
Being causal means being a poset
Precisely if the Lorentzian space is causal in that there are no closed future-directed curves is the relation
- $(x \leq y) \Leftrightarrow$ “$y$ is in the future of $x$”
a poset, hence a category with at most a single morphism between any two objects:
The objects of this category are the points of $X$. A morphism $x \to y$ is a pair of points $x \leq y$ with $y$ in the future of $x$. Composition of morphisms is transitivity of the relation. The identity morphism on $x$ is the reflexivity $x \leq x$.
The anti-symmetry $(x \leq y \leq x) \Rightarrow (x = y)$ is precisely the absence of closed future-directed curves in $X$.
Conversely, from just knowing $X$ as a smooth manifold and knowing this poset structure on $X$, one can reeconstruct the light cone structure of $(X,\mu)$, i.e. the information about which tangent vectors are timelike, lightlike, etc.
One can see
(…reference…)
that the pseudo-Riemannian metric $\mu$ may be reconstructed from the lightcone structure and the volume density that it induces. In this sense a Lorentzian manifold without future-directed curves is equivalently a smooth poset equipped with a smooth measure on its space of objects.
It looks like I might be confusing “conformal structure” with “light cone structure”.
This is a very pretty factoid and I’d highly recommend tracking some references to get the story straight. I’m going from memory and I didn’t completely understand it the first time around :)
Oh right. Here is the Discussion from smooth Lorentzian space
Discussion
A previous version of this entry started the following discussion.
Toby asked: How does this relate to a (smooth) Lorentzian manifold? if at all.
Eric says: Good question. I took the statement from a comment Urs made here. I chose to use the word “space” instead of “manifold” simply because it seemed to fit into a theme here about generalized smooth “spaces”. The definition definitely needs fleshing out, but its a start.
Urs:
The point is: there is a theorem
that says that a map between two Lorentzian manifolds which preserves the causal structure, i.e. which is a functor of the underlying posets, is automatically a conformal isometry. There is, I think, another related theorem which says that from just the lightcone structure of a Lorentzian manifold, one can reconstruct its Lorentzian metric up to a conformal rescaling.
Both theorems suggest that a Lorentzian metric on a manifold is in a way equivalent to a pair consisting of a measure on the manifold and lightcone structure. The latter in turn can be encoded in a poset structure on the manifold. If true, it would seem to suggest that a good foundational model for relativistic physics might be posets internal to Meas.
Somebody should sort this out.
Eric says: I like this idea. The measure could be the Leinster measure, which would be neat. We discussed this before at the nCafe I think.
Urs: Yes, exactly. There was the idea that, since many finite categories come with a canonical measure on their space (set) of objects, maybe we somehow need to merge this idea of Leinster measure with the idea of modelling a Lorentzain spacetime by something like a poset. Playing around with this observation was the content of this blog entry. But I am not sure if it works out…
You know, I have no problem with the other direction: All that is objective in Minkowski space are the causal relationsships of events (poset) and - after agreeing on a clock device - the proper time of observers (measure?). Intuitively it seems perfectly obvious. But on the other hand I never got my intuition to tell me something as simple as what the commutator of two observables in QM should be, so I don’t trust it.
But on the other hand I never got my intuition to tell me something as simple as what the commutator of two observables in QM should be, so I don't trust it.
It would be interesting to determine the Leinster measure of this poset and see if it gives the correct Minkowski metric.
The poset takes care of the signature :)
No time, but a small comment:
to construct a Minkowski spacetime you need no information at all. Because there is a unique Minkowski spacetime for each dimension.
The statement is that given just the poset structure and the volume density you can reconstruct a general spacetime (subject to some conditions). Such that the poset is the relation “is in the future of” with respect to its metric, and the volume density is the volume density of its metric.
The poset takes care of the signature :)
Found an online version of Bombelli’s thesis from the Wikipedia page, which has a lot of other interesting looking references.
PS: Check out the section on “‘t Hooft’s Proposal” in Bombelli’s thesis on page 25
The signature of the pseudo-Riemannian space corresponding to some poset is always (-1,1,1,1,…). The morphisms in the poset connect points where one is in the future of the other, with respect to the given metric. The very notion “being in the future of” makes good sense and is defined only for this signature.
@Ian: Bombelli outlines how this works on pdf page 51 in “Flat Space-Times”.
This thesis is absolutely beautiful. I can’t believe I never saw this. I’ve always used Bombelli’s published paper for a reference. This is one of those dissertations that gives me hope for humanity :)
Because it is insufficient to merely say that just because one element lies in the future of another they are causally linked (i.e. you can have a space-like separation in which one time coordinate is greater than another). Or maybe he’s talking about proper time or the spacetime interval here.
All answers are at Lorentzian manifold in the section “Causal structure”.
@Ian: Before we can communicate we have to agree on the semantics or we will just talk past each other. Let’s try to avoid it here and set some terminology upfront.
First, the words “in the future” are synonymous with “in the future lightcone”. If an element/event is not in the future lightcone of another element/event, it is not in the future of that event. This is what it means to be causally connected. If two events are not causally connected this way, you cannot tell which one came first. The order would depend on the reference frame. In some reference frame they can even appear to be simultaneous. This is explained pretty nicely on Wikipedia: Relativity of simultaneity. I know you know this, so my purpose for saying it is just to fix the terminology. “In the future” means “in the future lightcone”. I could be wrong, but I get a sense you might be thinking in terms of time coordinates, which is a reasonable way to express a sense of future, i.e. “$x$ is in the future of $y$ if $t_x \gt t_y$”, but this would be reference frame dependent. Proper time is only defined for events that are causally connected, i.e. where one event is in the future lightcone of another, so I suppose you could alternatively define “future” in terms of proper time but it would mean the same thing, i.e. “in the future lightcone”.
This also has ramifications for the maths terminology. We do not say “Spacetime is a totally ordered set”. Only a “partially ordered set” :) For some elements, you cannot determine whether $x\lt y$ or $y\lt x$. This inability to decide happens precisely when the two events are not causally related, i.e. one is not in the future of the other, i.e. one is not in the future lightcone of the other. Only when events share a lightcone, can you give them an order.
This is why giving points of a manifold a partial order is equivalent to defining the lightcone. Once you’ve defined the lightcone, you’ve defined the Lorentzian metric up to a “scaling”, i.e. conformal factor. The volume element sets the scale. It is pretty cool. I wish I could check myself, but it’d be neat to check whether the Leinster measure on the poset of spacetime sets the scale so the only input needed would be the poset.
By the way, Tom and Simon (and I and anyone else who finds their work on “cardinality of a category” fascinating) often wonder aloud about the meaning of “weights”. Skimming Bombelli’s thesis, I think I know what the weight is now. It is (related to) the “probability of having a link” and its integral is “expected number of links”. I’ll need to look up references to match the terms correctly, but the meaning is clear now. See Equation 2.5.1. It is also related to Alexandrov neighborhood.
Conversely, from just knowing X as a smooth manifold and knowing this poset structure on X, one can reeconstruct the light cone structure of (X,μ), i.e. the information about which tangent vectors are timelike, lightlike, etc.
I apologize if this sounds obvious or trivial, but my brain works in funny ways. So, having read this section, I’m still a little unclear on how one starts with X and the poset structure on X and, from only that, obtains the light cone structure. Are the tangent vectors defined as part of X or part of the poset structure?
We do not say “Spacetime is a totally ordered set”. Only a “partially ordered set” :)
Ah, OK, that explains a lot.
In terms of terminology, I’m used to three notions of “time:” coordinate time, proper time, and spacetime intervals. I will admit that I fall back on the colloquial notion of “future” as specifically referring to coordinate time. But as long as I know you take it to imply a reference to the lightcones, I’m OK with that.
I’m still a little uncertain about constructing the lightcone from the poset (see comment above).
@Ian: Since we posted within 4 minutes of each other, I hope my 4 minute-earlier-comment helps answer your question. We must have been outside each others lightcones :)
$FutureLightcone(x) = \{y\in X | x \lt y\}$ $PastLightcone(x) = \{y\in X | y \lt x\}$I’m still a little uncertain about constructing the lightcone from the poset (see comment above).
Edit: Once you see it, you will feel cheated because it is so obvious :) But the obviousness is good news because we can now think of “lightcone structure” and “poset” as (almost) synonymous, i.e. given one you can determine the other.
Edit^2: And posets (or preorders really) are nice because they form a particularly nice kind of category. The symbol $x\lt y$ means there is a morphism $x\to y$ and for any two $x,y$, there is at most one such morphism :)
Right, but how would you be able to distinguish a poset that describes a Newtonian version of time from one that describes an Einsteinian one? Or do we just assume it is Einsteinian, i.e. we’re assuming everything is either proper time or spacetime intervals?
First, the words “in the future” are synonymous with “in the future lightcone”.
Only on Minkowski space. On a general spacetime it means: connected by a future-directed path.
Let me clarify my previous comment since it might seem a bit muddled.
So, let’s take the simple Minkowski metric in 4-D spacetime,
$\Delta s^{2} = \Delta t^{2} - \Delta d^{2}$ (where I’ve chosen the +,-,-,- signature over the -,+,+,+ one)
where $d$ represents the spatial separation of two events. Call $t$ the coordinate time and $s$ the spacetime interval. We can also think of $t$ as corresponding to ‘Newtonian’ time. $s$ is invariant under Lorentz transformations while $t$ is not. When $\Delta s^{2} \ge 0$, the events separated by $\Delta s$ are causally connected, i.e. they are either timelike or lightlike separated.
How does a simple poset capture the detail in that structure? And couldn’t you develop a poset that only described $t$? Then how could you distinguish between the poset describing $s$ and the poset describing $t$? And how could you recover the notion that two events might not be causally connected to one another, but both could be causally connected to some third event?
Does anyone see what I mean? I’m trying really hard! :/
Right, but how would you be able to distinguish a poset that describes a Newtonian version of time from one that describes an Einsteinian one? Or do we just assume it is Einsteinian, i.e. we’re assuming everything is either proper time or spacetime intervals?
Good question :) I don’t know :)
The difference between Newtonian version and Lorentzian (don’t give Einstein credit for everything :)) version has to do with the speed of light. Newtonian is the limit of Lorentzian as $c\to\infty$, so your question could be rephrased as “What determines the speed of light?”
I’m sure the answer lays in some of the references Bombelli points to, but I’m sure it has something to do with the follow…
Given a set $P$ and a partial order $\lt$ on $P$, then any element $p\in P$ partions $P$ into three pieces
The speed of light must be related to the relative size of $Unrelated(p)$. In the Newtonian case, $Unrelated(p)$ is empty since the cone angle is 90 degrees (speed of light is infinite) and all points are ordered. In other words, a Newtonian spacetime is a totally ordered set. Given any two points in a Newtonian spacetime, you can always tell which is in the future of the other.
For a finite speed of light, $Unrelated(p)$ is not empty and we have a partially ordered set. As the speed of light decreases, the relative size of $Unrelated(p)$ grows.
First, the words “in the future” are synonymous with “in the future lightcone”.
Only on Minkowski space. On a general spacetime it means: connected by a future-directed path.
I think this is another semantic issue and I could be the culprit :)
The collection of all future-directed paths, is what I meant by future lightcone. It is not a Minkowski cone, but should still be cone-shaped. If not, I have no problem with sticking to “connected by a future-directed path”, but I thought this was precisely how you’d define a lightcone in a general Lorentzian manifold, but I admit my use of terminology may not be standard either.
How does a simple poset capture the detail in that structure?
Do you see how knowledge of the light-like vectors together with the volume element at any point determines a Lorentzian metric at that point?
When you see that, the remaining step is to extract from the causal structure alone the light-like vectors. But the collection of all points in the future of a given point is the light-cone. So all you need to do is find the tangents to it, at your given point.
Ah, ok, it is much clearer now. Eric’s description makes complete sense. To be honest (just being honest here!), if I had only Urs’ most recent comment to work from, I would still have wanted to better understand the statement “[b]ut the collection of all points in the future of a given point is the light-cone,” but I think Eric’s description clears it up.
So, in short, I think I get it now. Many thanks to everyone for your patience.
Cool :)
Now ponder the far reaching implications… :)
@Eric: By the way, you said you didn’t know the answer to my question about Newtonian versus Lorentzian time, but in the exact same post I think you explained it quite well. So you really did know. :-)
Eric said:
The collection of all future-directed paths, is what I meant by future lightcone.
You may use chronological or causal future as defined now on smooth Lorentzian space.
It is not a Minkowski cone, but should still be cone-shaped.
I don’t know what “cone-shaped” means for a general manifold, but there is a “niceness” theorem about the boundary of the chronological future of a point that I also added, below the definition of chronological future.
The speed of light must be related to the relative size of Unrelated(p).
Just an addendum: For spacetime to be locally Lorentzian the important fact is that there is an upper bound on the velocity of signal propagation at all. How “fast” that is or if it is a maximum or only a supremum is secondary (i.e. there need not be a physical process that actually propagates with that speed, like light in the vacuum does in general relativity). But if you change the poset structure you will probably get different Lorentzian manifolds associated to it, I would guess.
but there is a “niceness” theorem about the boundary of the chronological future of a point that I also added, below the definition of chronological future.
Thanks, Tim! Very useful.
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