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Frames vs Lines of Simultaneity

  1. Jul 12, 2010 #1
    ------------Frames and Lines of Simultaneity----------------

    Is there any difference between the two???

    If there is what is it??

    I may be missing something obvious but as far as I can see they are just two ways of graphing and conceptualizing a singular entity.

    Thanks
     
  2. jcsd
  3. Jul 12, 2010 #2
    A line of simultaneity just links more than one event that are considered simultaneous according to the observer. So on a spacetime diagram they'll be horizontal lines all along the time axis.

    Usually they are used to show differences in simultaneity between 2 different observers moving relatively to each other. So in this case the observer who is being considered stationary might plot a line of simultaneity which maps a second, moving, observers notion of simultaneous events. In this case the line will be sloped.

    In this case the initial observer can see that what the other observer considers simultaneous events, the first considers to happen one after the other.
     
  4. Jul 12, 2010 #3
    Hi Aaron_Shaw
    I was not asking about the meaning and application of L's of S.

    Regarding the horizontal line of the rest frame. Those lines are purely and simply the rest frame itself. The point used to locate the timeline in the graph is is simply a single point on the extended coordinate frame.

    Similarly the sloped timelines of the "moving" frame are simply that frame as an extended coordinate system at different points in time. The point in that frame which is used to locate the worldline is also just a point on the total frame.

    The slope of the line is simply a graphic convention. In the real world they are congruent with the path of motion.

    The information they contain [relative time relationship and spatial locations] where they intersect another worldline is the same as what would be found if you did a parallel analysis.

    For example the train and tracks. The relative simultaneity [relative clock desynchronization] can be directly calculated from the frames coordinates with the Lorentz transform. This will tell the relative time at the points of the lightning flash. The relative simultaneity of clocks at those points and will be exactly consistent with the diagrams for the same point in time.

    This will hold true no matter how long the train and tracks are. SO the lines of simultaneity that are sloped in the diagram are just long trains indefintely extended in space.
    ANd any points of intersection represent observers and clocks on the train and tracks at that point.

    L's of S in the diagrams are simply a wonderful convention telling you the relative clock readings at distant locations without having to do the specific calculations for the point.
    Of course to get the location of this colocation in the moving frame you still need to transform with gamma.

    So does any of this make sense???

    Thanks for your responce
     
  5. Jul 12, 2010 #4
    Lines of Simultaneity only have meaning relative to a particular frame. It's hard to call them the same thing though, when spacelike separated frames can agree on certain lines of simultaneity in some cases.
     
  6. Jul 12, 2010 #5
    Each frame has a plane of simultanety (in 2D space + 1D time) associated with it. The PoS is just a property of the frame, and is only non-trivial when compared to another frame.

    Frames which travel in the same direction with the same velocity have parallel PoS's.

    Since the PoS is parallel to the spatial axes, their use in spacetime diagrams are to, as you said, find out when an event occurs in a paticular frame.
     
  7. Jul 12, 2010 #6
    What do you have in mind by a frame? What do you have in mind by a line of simultaneity?

    This is complicated to say, but the pictorial and intuitive idea isn't really too hard:

    If you have a 4-d Minkowski manifold, 3 of space and 1 of time, then you can naturally partition the points of this manifold into a sequence of 3-d `simultaneity' spaces. Think of these as a kind of slicing up of space-time, into a series of `moments'. Each element of the sequence corresponds to space at a moment in some inertial frame. Lines that are orthogonal to the surfaces are timelike, and represent the inertial paths of things that are stationary with respect to that frame.

    Unlike the Newtonian case, where there is an absolute notion of simultaneity, there is no one special way of slicing up Minkowski space-time - different partitions, or slicings, correspond to different ways in which various objects would slice up space time.

    In a 2-d Minkowski manifold - which corresponds very closely to a MInkowski diagram which I know you've been focussing on - a frame's `moment' would correpond to a simultaneity line, and the frame could be sliced up (partitioned) into a sequence of such lines. I think this is what you had in mind? For the realistic 3+1 case, we have hyperplanes of simultaneity.

    What's a frame? If a frame is an inertial rectangular coordinate system, then it's not quite true that frames and simultaneity partitions correspond. In a coordinate system, we have to choose an origin, set t = 0, focus on a particular inertial line, whereas a slicing gives us a whole set of parallel inertial lines. However, we of course feel that choice of origin is a purely arbitrary choice, that this kind of difference is of only mathematical rather than physical importance. If we get rid of such arbitrary differences between coordinate systems and consider instead equivalence classes of coordinate systems, coordinate systems that differed only by choice of origin, then, I think, there is a 1-1 correspondence between such sets of frames and a partitioning of space-time into a sequence of simultaneity hyperplanes.

    So - the intuition that, in Minkowski space time, there's not much real difference between an inertial frame and a way of partitioning space-time into simultaneity equivalence classes, you'd more or less be right. There's a very natural correspondence between them.

    That took way longer to explain than it should have. I blame myself.
     
  8. Jul 12, 2010 #7
    Wouldn't a frame have a sequence of planes of simultaneity associated with it? Not just one?

    Wouldn't they have the same planes of simultaneity?
     
  9. Jul 12, 2010 #8
    Yes, but what I had in mind was a plane moving with time along the time axis.

    Yes, I would think so.
     
  10. Jul 13, 2010 #9
    Exactly! But doesn't the PoS have an exact one to one mapping to the actual coordinate plane it is a property of???

    Isn't the term PoS simply a useful semantic distinction with no real difference or meaning???

    Don't misunderstand me ; I understand the usefullness and am not suggesting that they should be called by the same name.

    I am simply interested in finding out if there is a real difference that I have missed.

    Thanks
     
  11. Jul 13, 2010 #10

    Exactly my understanding and conceptualization


    .
    Me too

    Wouldn't this only apply to the rest frame. The lines of S of the moving system in the diagram are sloped relative to the spatial axis??

    Isn't the PoS actually congruent to the spatial axis at any specific point on the time line?
     
  12. Jul 13, 2010 #11
     
    Last edited: Jul 13, 2010
  13. Jul 13, 2010 #12
    600px-Minkowski_diagram_-_simultaneity.png

    In this Minowski diagram, the thin blue lines show the blue S' frame's simultaneity planes. The thin black lines show the black S frame's simultaneity planes.

    In a Minowski diagram, the usual rules of coordinates apply. If you have x'=3 at t=t'=0 and want to find the location of x'=3 in S at time t', you simply translate the point (3,0)' along the t' axis. In this sense, the simultaneity planes are copies of the spatial coordinate plane, or at least a temporal extension, however you want to define it.
     
  14. Jul 13, 2010 #13
    We could describe lines, planes and hypersurfaces of simultaneity as (loosely speaking) subspaces of frames of reference.

    A frame of reference is an imagined system of rigid rods forming a grid throughout all space and ideal synchronized clocks attached to points of the grid. We can attach to this reference frame a system of coordinates of our choice. For an observer at rest in any particular frame of reference his line/plane/hypersurface of simultaneity is just the set of points, selected by some convention, which he considers to be his now. For a line of simultaneity on a two dimensional spacetime diagram the line of simultaneity corresponds to his spatial axis. The same applies, mutatis mutandis, to planes and hypersurfaces of simultaneity.

    Matheinste.
     
  15. Jul 13, 2010 #14
    From this can I infer that there is no disagreement on our definitons and conceptions of the coordinate systems in question?

    But why a subspace? Why would the PoS be less indefinitely extended than the frame?

    ((1)) Is n't this exactly equivalent as applied to the frame itself? The spatial coordinates and clocks as selected by convention which is then considered "now"?

    Thanks
     
  16. Jul 13, 2010 #15
    Because a simultaneity space doesn't have a temporal extension, it is a subspace of the 4D frame which has.
     
  17. Jul 13, 2010 #16
    I am not sure I understand you. At an instant of time the 4-D frame is indefinitely extended in space but would seem to be unextended in time , yes?
    This would be exactly the same for the 4-DofS wouldnt it??
    They are both only extended temporally as they move along the worldline, no?
     
  18. Jul 13, 2010 #17
    Thanks for the diagram
     
  19. Jul 13, 2010 #18
    No, moving along the temporal axis is not equivalent to being extended along that axis. The simultaneity space doesn't have temporal extension. Why would it? The simultaneity space has the same ammount of dimensions as the number of spatial axes. That's why we have simultaneity lines and not simultaneity surfaces in 2D minowski diagrams.
     
  20. Jul 13, 2010 #19
    You're right here - I shouldn't have used the word frame that second time - I meant to say it's the space-time that's being partitioned. But I wasn't talking about the extended worldline.

    Unfortunately, because I'm still not sure how you're using the concepts, I don't know how to answer this without repeating my last answer.

    I think espen's diagram says it all - but notice that on the lines and planes he's drawn no *numbers* are attached. Coordinate systems typically involve actually assigning numbers to events, but since this obviously involves nothing more than pure conventional choice of unit, we tend to forget about this.

    I prefer to think of splitting up spacetime into a sequence of simultaneity (hyper)planes, rather than thinking of this as a single plane which `moves' through an observer's time - we can treat Newtonian space time as a sequence of copies of the very same plane at different times, because there's an absolute notion of sameness at place at different times, i.e absolute rest, in his theory. But provided we don't push the heuristic too far, and we understand there's no true identity, we can talk of various planes at different times as another way of thinking about the sequence of simultaneity planes.

    I thought I'd explained this - but maybe I don't get your 'outside of difference in name'. A plane of simultaneity is just a plane of simultaneity. Think of it purely geometrically. There's no number associated with it. That requires a choice - setting the clocks at zero rather than 3. It has no origin either. To get a coordinate system going, numbers associated with points, we have to choose spatial axes too. But I doubt that this kind of thing really matters for any conceptual/physical purpose and can be counted as a mathematical nicety (though, for my own part, I am partial to a few mathematical niceties!)

    As I've been using it, a plane of spacetime has no origin, just as the lines on espen180's diagram have no numbers attached to them.

    I think most of us agree that any two observers travelling at the same velocity will partition the planes of simultaneity in the same way; that any partition of the planes of simultaneity will have associated with it a set of time like lines that correspond to inertial observers with the same velocity; etc etc, and so tend to think of these as different ways of describing the same situation.

    The only thing that you've said that concerned me was this:
    I don't know what you're contrasting this to, but I would urge that a sequence of simultaneity classes corresponds to genuine geometric structure on the Minkowski space-time: they're the surfaces orthogonal to inertial lines. Minkowski space-time privileges no unique sequence of simultaneity classes - in this respect, it is very different from Newtonian space-time, but their existence is not a purely semantic issue.
     
  21. Jul 13, 2010 #20
    Isn't the reason we have lines instead of surfaces is simply the limitations of graphing a 4-D reality onto 2D?? The frames also appear as lines.

    Maybe if you explained how you see a frame as having temporal extension I could get it?
     
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