4 ms·

An example of a partial order that blew my mind recently are the ordering of "events" in special relativity. Two events can be ordered in time (i.e A happened b

An example of a partial order that blew my mind recently are the ordering of "events" in special relativity. Two events can be ordered in time (i.e A happened before B) if their spacetime interval is positive. But for two events that are separated by a distance than light can't travel in the time by which they are separated, then nobody can give a before/after order to these events. This fits exactly into the partial ordering system.

gugagore 6y ago
> But for two events that are separated by a distance than light can't travel in the time by which they are separated As written, this sounds circular to me. How do the know how far apart in time the two events are?
- d_tr 6y ago
  The separate space and time interval measurements between two events can vary among observers, but they will all measure the same spacetime interval, which in turn determines whether light is "fast enough".
  - caminocorner 6y ago
    Is there a somewhat longer (maybe illustrated) example of this principle somewhere? You've captured my interest and I want to read more
    - jfoutz 6y ago
      maybe this one? https://en.wikipedia.org/wiki/Ladder_paradox https://en.wikipedia.org/wiki/Ladder_paradox depending on where you stand, the ladder can fit in the garage, or it can't. If I remember right, it's from the original special relativity paper, which is super accessible. General was always beyond me, but special is just high school math.
    - alephu5 6y ago
      https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity
    - d_tr 6y ago
      The links provided are nice but I will leave a comment here in case it is useful. If I measure the temporal (dt) and spatial (dx) distance between two events A and B, then I can calculate what another observer would measure (dt' and dx') using the so-called Lorentz transformation, provided that I know his velocity relatively to me (v). The Lorentz transformation is a linear operator, written down as a matrix. Now, the spacetime interval (ds) between A and B, is computed with the formula ds^2 = dx^2 - c^2 dt^2. The interesting property here is that the Lorentz transformation leaves ds^2 unchanged, i.e. dx^2 - c^2 dt^2 = dx'^2 - c^2 dt'^2. So, it also does not change the sign of ds^2, which determines whether light is fast enough to travel a distance dx within time dt. The animation in the Wikipedia link by alephu5 shows the Lorentz transformation in action for a smoothly varying value of relative velocity. The events A B C are all separated by positive ('spacelike', light not fast enough) intervals, which graphically means that the line connecting them has a slope of less than 45 degrees in that graph, and the Lorentz transformation can tilt that line both ways and change the ordering of the events in the t axis. If two of these events on that graph were separated by a negative ('timelike') interval, the line connecting them would have a slope larger than 45 degrees and the Lorentz transformation could not alter their relative ordering in the t axis, meaning that all observers would agree on the ordering.
    - shaunpersad 6y ago
      Light cones helped me understand causality from a slightly different perspective: https://en.wikipedia.org/wiki/Light_cone https://en.wikipedia.org/wiki/Light_cone https://www.youtube.com/watch?v=OZv3ycr6Jxg https://www.youtube.com/watch?v=OZv3ycr6Jxg
  - 6gvONxR4sf7o 6y ago
    How's this work with multiple paths between things? Like in special relativity, it's all uniquely defined, but how's it work in GR? Shortest spacetime path?
    - deleted 6y ago
      [deleted]
    - contravariant 6y ago
      In short, yes. To expand on that a bit, this shortest path is known as a geodesic and one of the more important axioms of general relativity is that all laws of physics are preserved locally when traveling along a geodesic. In particular all such observers should measure the same speed of light, since it's a simple property of electrodynamics. Interestingly they won't measure the same CMB, showing that the local part is important.
      - 6gvONxR4sf7o 6y ago
        That's neat, thanks. When you say it's an axiom that it's all preserved locally, does that mean that all the laws only state local properties (i.e. it's all just differential equations over spacetime)? If so, is that why stuff like QM has trouble, where something like entangled things at a distance might be hard to express locally, or is it actually easy to transform spooky stuff into a local statement and QM issues are something else entirely?
        contravariant 6y ago
        It's a bit too simple to say that that's the problem, it's not that general relativity can't deal with global features of reality, those will simply not look the same to all observers. The reason I mentioned CMB is that space necessarily seems to have a preferred speed (even with general relativity you can't have a distribution of speeds that looks the same from all frames of references), and measuring the blueshift in CMB allows you to measure an absolute speed in some sense (it might still be local to our part of the universe, but that includes pretty much everything we can interact with). The obstacle to combining general relativity and quantum mechanics is, in short, that general relativity is a classical theory of physics (e.g. exact positions, energy, momentum and all that), whereas quantum mechanics expands classical mechanics to get quantum mechanical laws of physics. Now for whatever reason the techniques we used to turn electromagnetism etc. quantum mechanical fail to work on the (classical) theory of general relativity. And IMHO that's about as far as we've gotten, a lot of work's gone into it but it's honestly hard to tell if we've gotten a better grasp on why general relativity refuses to 'quantize'. Personally I blame the fact that the mathematical foundations of quantum mechanics aren't strong enough to support a quantum mechanical description of geometry itself, but I may not be the most qualified person to judge this.
- contravariant 6y ago
  That's a good question. The answer is that the time between the two events will vary for different observers, however the property that light can't travel fast enough between the events is invariant and can be agreed upon by all observers. It's somewhat important that this property remains invariant, simply because all observers should be able to agree whether a flash produced at one point in spacetime is visible at another point in spacetime.
  - walleeee 6y ago
    How does this work, if you can spare a hand-wavy explanation? If the perceived time between events varies by observer, and if for every observer there is a precise threshold at which events become casually disjoint, naively I would imagine there to always be disagreement.
    - contravariant 6y ago
      Just as there is a difference in perceived time there's also a difference in perceived distance. As it turns out these cancel out in a way that ensures the speed of light is the same for all observers. Together with the axiom that everyone agrees whether something travels in a straight line you can figure out most of special relativity. It's somewhat hard to explain why this works without handwaving or just pointing to the maths, but the youtube channel minute physics at least has some good visualizations: https://www.youtube.com/watch?v=Rh0pYtQG5wI https://www.youtube.com/watch?v=Rh0pYtQG5wI
      - walleeee 6y ago
        Thanks!
captaindiego 6y ago
If anyone is interested in digging into this more, there's an absolutely fantastic Leslie Lamport paper related to clocks and ordering of events: https://lamport.azurewebsites.net/pubs/time-clocks.pdf https://lamport.azurewebsites.net/pubs/time-clocks.pdf
libraryofbabel 6y ago
And if you’re an engineer this idea from relativity can be applied in distributed systems theory as well! See Leslie Lamport’s famous paper Time, Clocks, and the Ordering of Events in a Distributed System (1978)[0]. I believe Lamport wrote a book on General Relativity before getting into computer science, so the connection runs deep. [0] https://lamport.azurewebsites.net/pubs/time-clocks.pdf https://lamport.azurewebsites.net/pubs/time-clocks.pdf
- proc0 6y ago
  Cool. Distributed and asynchronous programming made me think of non-locality a lot, just looking at the clock isn't good enough to determine a sequence of events.

gugagore 6y ago

> But for two events that are separated by a distance than light can't travel in the time by which they are separated As written, this sounds circular to me. How do the know how far apart in time the two events are?

d_tr 6y ago
The separate space and time interval measurements between two events can vary among observers, but they will all measure the same spacetime interval, which in turn determines whether light is "fast enough".
- caminocorner 6y ago
  Is there a somewhat longer (maybe illustrated) example of this principle somewhere? You've captured my interest and I want to read more
  - jfoutz 6y ago
    maybe this one? https://en.wikipedia.org/wiki/Ladder_paradox https://en.wikipedia.org/wiki/Ladder_paradox depending on where you stand, the ladder can fit in the garage, or it can't. If I remember right, it's from the original special relativity paper, which is super accessible. General was always beyond me, but special is just high school math.
  - alephu5 6y ago
    https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity
  - d_tr 6y ago
    The links provided are nice but I will leave a comment here in case it is useful. If I measure the temporal (dt) and spatial (dx) distance between two events A and B, then I can calculate what another observer would measure (dt' and dx') using the so-called Lorentz transformation, provided that I know his velocity relatively to me (v). The Lorentz transformation is a linear operator, written down as a matrix. Now, the spacetime interval (ds) between A and B, is computed with the formula ds^2 = dx^2 - c^2 dt^2. The interesting property here is that the Lorentz transformation leaves ds^2 unchanged, i.e. dx^2 - c^2 dt^2 = dx'^2 - c^2 dt'^2. So, it also does not change the sign of ds^2, which determines whether light is fast enough to travel a distance dx within time dt. The animation in the Wikipedia link by alephu5 shows the Lorentz transformation in action for a smoothly varying value of relative velocity. The events A B C are all separated by positive ('spacelike', light not fast enough) intervals, which graphically means that the line connecting them has a slope of less than 45 degrees in that graph, and the Lorentz transformation can tilt that line both ways and change the ordering of the events in the t axis. If two of these events on that graph were separated by a negative ('timelike') interval, the line connecting them would have a slope larger than 45 degrees and the Lorentz transformation could not alter their relative ordering in the t axis, meaning that all observers would agree on the ordering.
  - shaunpersad 6y ago
    Light cones helped me understand causality from a slightly different perspective: https://en.wikipedia.org/wiki/Light_cone https://en.wikipedia.org/wiki/Light_cone https://www.youtube.com/watch?v=OZv3ycr6Jxg https://www.youtube.com/watch?v=OZv3ycr6Jxg
- 6gvONxR4sf7o 6y ago
  How's this work with multiple paths between things? Like in special relativity, it's all uniquely defined, but how's it work in GR? Shortest spacetime path?
  - deleted 6y ago
    [deleted]
  - contravariant 6y ago
    In short, yes. To expand on that a bit, this shortest path is known as a geodesic and one of the more important axioms of general relativity is that all laws of physics are preserved locally when traveling along a geodesic. In particular all such observers should measure the same speed of light, since it's a simple property of electrodynamics. Interestingly they won't measure the same CMB, showing that the local part is important.
    - 6gvONxR4sf7o 6y ago
      That's neat, thanks. When you say it's an axiom that it's all preserved locally, does that mean that all the laws only state local properties (i.e. it's all just differential equations over spacetime)? If so, is that why stuff like QM has trouble, where something like entangled things at a distance might be hard to express locally, or is it actually easy to transform spooky stuff into a local statement and QM issues are something else entirely?
      - contravariant 6y ago
        It's a bit too simple to say that that's the problem, it's not that general relativity can't deal with global features of reality, those will simply not look the same to all observers. The reason I mentioned CMB is that space necessarily seems to have a preferred speed (even with general relativity you can't have a distribution of speeds that looks the same from all frames of references), and measuring the blueshift in CMB allows you to measure an absolute speed in some sense (it might still be local to our part of the universe, but that includes pretty much everything we can interact with). The obstacle to combining general relativity and quantum mechanics is, in short, that general relativity is a classical theory of physics (e.g. exact positions, energy, momentum and all that), whereas quantum mechanics expands classical mechanics to get quantum mechanical laws of physics. Now for whatever reason the techniques we used to turn electromagnetism etc. quantum mechanical fail to work on the (classical) theory of general relativity. And IMHO that's about as far as we've gotten, a lot of work's gone into it but it's honestly hard to tell if we've gotten a better grasp on why general relativity refuses to 'quantize'. Personally I blame the fact that the mathematical foundations of quantum mechanics aren't strong enough to support a quantum mechanical description of geometry itself, but I may not be the most qualified person to judge this.
contravariant 6y ago
That's a good question. The answer is that the time between the two events will vary for different observers, however the property that light can't travel fast enough between the events is invariant and can be agreed upon by all observers. It's somewhat important that this property remains invariant, simply because all observers should be able to agree whether a flash produced at one point in spacetime is visible at another point in spacetime.
- walleeee 6y ago
  How does this work, if you can spare a hand-wavy explanation? If the perceived time between events varies by observer, and if for every observer there is a precise threshold at which events become casually disjoint, naively I would imagine there to always be disagreement.
  - contravariant 6y ago
    Just as there is a difference in perceived time there's also a difference in perceived distance. As it turns out these cancel out in a way that ensures the speed of light is the same for all observers. Together with the axiom that everyone agrees whether something travels in a straight line you can figure out most of special relativity. It's somewhat hard to explain why this works without handwaving or just pointing to the maths, but the youtube channel minute physics at least has some good visualizations: https://www.youtube.com/watch?v=Rh0pYtQG5wI https://www.youtube.com/watch?v=Rh0pYtQG5wI
    - walleeee 6y ago
      Thanks!

captaindiego 6y ago

If anyone is interested in digging into this more, there's an absolutely fantastic Leslie Lamport paper related to clocks and ordering of events: https://lamport.azurewebsites.net/pubs/time-clocks.pdf https://lamport.azurewebsites.net/pubs/time-clocks.pdf

libraryofbabel 6y ago

And if you’re an engineer this idea from relativity can be applied in distributed systems theory as well! See Leslie Lamport’s famous paper Time, Clocks, and the Ordering of Events in a Distributed System (1978)[0]. I believe Lamport wrote a book on General Relativity before getting into computer science, so the connection runs deep. [0] https://lamport.azurewebsites.net/pubs/time-clocks.pdf https://lamport.azurewebsites.net/pubs/time-clocks.pdf

proc0 6y ago
Cool. Distributed and asynchronous programming made me think of non-locality a lot, just looking at the clock isn't good enough to determine a sequence of events.