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Visualisation of Brian Greene's concept.

  1. Jun 15, 2012 #1
    I thought it would be interesting to do a visualisation of Brian Greene's concept that "everything moves at the speed of light". For a preamble see this old thread https://www.physicsforums.com/showthread.php?t=398590

    First I should warn that speed or velocity is not defined in the conventional way here. It uses a concept of 4 space dimensions comprising 3 of the normal spatial dimensions and a fourth dimension with units of cTau where Tau is the proper time of a particle. Speed is here defined as dU/dt where U = √(c^2Tau^2+x^2+y^2+z^2) and t is the conventional coordinate time. It is important to notice that this is not the conventional four velocity. (Note that all the signs are the same). For lack of a better term I will call dU/dt the G-velocity and all the dimensions other than the coordinate time collectively as G-space after Brian Greene.

    For simplicity I will only consider the x spatial dimension.

    For a particle moving relative to the observer, the G-velocity through through G-space is:

    [tex] \frac{dU}{dt} = \frac{\sqrt{ (c d\tau)^2+d x^2}}{dt} = c [/tex]

    (See the green path in the attached 3D graph for a particle with a 3 velocity of 0.7071c.)

    For a particle that is at rest with respect to the observer this reduces to:

    [tex] \frac{dU}{dt} = \frac{c d\tau}{dt} = c [/tex]

    This is the idea that a stationary particle moves purely through the (proper) time dimension at the speed of light.

    (See the blue path in the attached 3D graph)

    For a photon the equation reduces to:

    [tex] \frac{dU}{dt} = \frac{d x}{dt} = c [/tex]

    This is the idea that a light particle moves purely through the spatial dimensions (of 3 space) at the speed of light. Note that there is no component of the photon's path in the proper time dimension.

    (See the red path in the attached 3D graph)

    The full graph would require plotting in 5 dimensions, (3 spatial + coordinate time + proper time) so only the x spatial dimension is shown along with the two time dimensions in the attached chart. Although it is not very clear from the image, all the paths lie on 45 degree cone, of which a quarter is shown in the image, with the apex at the origin.

    In summary, nothing can remain stationary in G space. You are either moving through 3 space or moving through the proper time dimension or a combination of both.
     

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    Last edited: Jun 16, 2012
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  3. Jun 15, 2012 #2

    PeterDonis

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    Two questions:

    (1) Shouldn't there be square root signs in the numerators of your formulas for dU/dt? There is one in the formula for U.

    (2) What happens when you change frames?
     
  4. Jun 15, 2012 #3
    Good observation. :approve: Thanks for spotting that while I was still able to edit the post. Fixed now!
    Good question. Not worked that out yet, but the maths should not be too difficult. Visually, if we switched to the rest frame of the green particle, it would look like it has the position of the green particle (i.e. lie on the new cTua, t plane), the green particle path will be rotated about 45 degrees anticlockwise looking from the top and the red photon path will of course remain in the x,t plane. All the paths will still lie on an a 45 degree cone.
     
    Last edited: Jun 15, 2012
  5. Jun 16, 2012 #4

    A.T.

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    See this interactive diagram for a visualization of this idea:
    http://www.adamtoons.de/physics/relativity.swf

    It doesn't have the the t-axis because it is somehow redundant: As your formula shows the coordinate time is the Euclidean path integral in "space-propertime" (in natural units). I think the idea is older than Greene's though. I know it from Epstein's "Relativity Visualized"

    This one shows how transforms look like in "space-propertime" compared to "space-coordinate time". Just move the slider called "Observes velocity in A' frame" back and forth.
    http://www.adamtoons.de/physics/twins.swf
     
    Last edited: Jun 16, 2012
  6. Jun 17, 2012 #5
    Thanks AT for your interactive diagrams. I have seen them previously a long time ago, but I think I understand them a bit better the second time around, especially in the context of this thread. I like that you have factored gravity in and would like to discuss that more. Brief question. Is the curvature of spacetime due to gravity in your animation accurate or just a first order approximation? Would you also agree that all objects move at c through space(proper)time (as per Brian Greene) is no longer true when gravity is involved?
     
  7. Jun 17, 2012 #6

    A.T.

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    The gravity part in this is just to give a qualitative idea. It lacks the distortion of the spatial dimension. This one is about gravity.
    I prefer the interpretation that the advance rate through space(proper)time is still the same, but the distances between coordinates are increased.
     
  8. Jun 23, 2012 #7

    Saw

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    Thanks to both for explaining how this idea works (i.e. that the speed of all objects through space-propertime is c). I had thought so far that it stemmed from the usual four-velocity, I now realize it is a different thing.

    I was wondering now about its utility.

    In principle, it looks to me as if in the definition of the so called Greene or rather Epstein space you combined (vectorially) X + (T – X), which gives T. If you then derive over T (divide by T), you get logically 1… So what…?
     
  9. Jun 24, 2012 #8

    A.T.

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    I think the main reason why Greene and Epstein use the space-propertime concept in their popular-scientific books, is that it offers a much more intuitive visual model, than the pseudo Euclidean Minkowski space-time.

    - In the Epstein-diagrams you see both: propertime and coordinate time directly as distances. You can visually see how speed in space, clock rates, length contraction are related. In fact you can go directly from the moving light clock to a space-propertime diagram, when you identify the vertical light movement-component with proper-time.

    - The speed limit in space of c seems easier to grok, when you see how it follows from an universal constant advance rate in space-propertime.

    - The relation between rest-mass, momentum and total-energy is the same as between delta proper-time, delta space and delta coordiante-time (in natural units). So once you have drawn the space-propertime diagram for an object, you can relabel the axes and visualize those relations in a geometrical way as well.

    - In GR you can visualize very directly the relation between gravitational time-dialtion and the geodesic-paths of free-fallers. And why it takes infinite coordinate-time (distant observer), but finite proper-time to fall into a black hole.
     
  10. Jun 24, 2012 #9

    Saw

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    Well, I was wondering rather only about the utility of the idea that everything travels at c in this space.

    As to the space-proper time diagrams, I fully agree that they can be very revealing and didactic.

    In fact you can see in this recent thread a discussion that was based on and made possible by this sort of diagrams.

    You may find it curious that I started inspired by Epstein diagrams, continued with yours, to which I expressly linked, and ended up, guided by bobc2, discovering that I was aiming at a Loedel diagram.

    A Loedel diagram is one where the T axis in one frame is perpendicular of the X’ axis of the other frame (just as X is at right angle to T’).

    Look at this picture. Just eliminate the blue X axis and you have the Epstein-like diagram of your site (where such blue axis is represented, I think, by the yellow stick).

    HeightIntervals.jpg
     
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