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Flowing time velocity

  1. May 16, 2006 #1
    In Brian Greene's last work (The fabric of cosmos - chapter 3) I've found an interesting approach to special relativity. It's very simple and straightforward, and I'm very surprised I've never heard of it before. But maybe it's just my fault and my ignorance. Since I'm a teacher it seems very useful to broach the themes of special relativity to young students.
    It goes (more or less) like this:
    All bodies have a fixed unique dynamic reserve, a velocity. Let's call it c. Every body lives in spacetime. Time always flows. So this velocity reserve can be spent totally in the time dimension. In that case the time flows with velocity c and the body is at rest in space.
    But if the body moves (with respect to some frame of reference) then it spends some of the stock-velocity in the spatial dimension. The flowing of time velocity must slow down.
    If the spatial velocity reach c the flowing of time velocity will come to a rest.
    The relevant equation follows the simple pythagorean theorem (no need to introduce hyperbolic norm):
    It can be derived from the invariance of 4-velocity vector norm, just interpreting c^2 as the square of the flowing of time velocity for a body at rest and the velocity of proper time for a moving body as dtau/dt = 1/gamma.

    I'm just wondering if it's possible to follow this approach to derive, starting from this simple axiom, the full frame of Lorentz transformations, and if there is some existing didactic material that goes through the same route.
    What do you think?

  2. jcsd
  3. May 17, 2006 #2

    Andrew Mason

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    It seems to add more complexity to a rather simple concept and obscures the underlying physical basis for relativity: the equivalency of all inertial reference frames and the speed of light being c for all inertial observers. What is the physical reason for this dynamic reserve?

    One can create many conceptual models that fit a mathematical relationship. One can think of light as little billiard balls of light stuff to explain the photo electric effect. But that does not mean that light bears any physical resemblance to a billiard ball. This 'dynamic reserve' might provide a workable conceptual model. But it doesn't appear, at least from what you have written, to help the student to understand the physics of relativity.

  4. May 17, 2006 #3
    This is not a new result; the physical reason for this dynamic reserve is that the four-velocity,

    [tex]U^\mu = \frac{dx^\mu}{d\tau},[/tex]

    is always of length 1 (in geometric units) due to the way [itex]d\tau[/itex] is defined (simple exercise: check for yourself). All this means is that the sum of the components of velocity squared must equal the same: i.e. they must equal 1.

    I do not think that you could derive all of the results of special relativity from this one fact. The essence of special relativity is that all quantities are naturally tensors of some rank where the metric is the Minkowski metric and the co-ordinate transformations are given by the Lorentz transformation; this cannot be derived from the simple fact that

    [tex]U^\mu U_\mu = 1.[/tex]

    Obviously this does not work at age 16-18 level for the machinery required is too extensive (although it depends on little more than calculus and some knowledge of vectors), but I think this approach should be absorbed into all/most undergraduate courses.
    Last edited: May 17, 2006
  5. May 17, 2006 #4

    I don't believe it would be possible, as you can't get the minkowski metric from that fact. That the four velocity is always equal to one is a consequence of the Lorentz transforms, not the other way around. Personally, I think this description is a very poor one. It works to explain the nature of time dilation to people who lack the background to really understand it, but as Andrew pointed out it obscures far more than it reveals. You start with two principles, the laws of physics are the same for all observers (the extension of which, all observers agree on events that occur at the same place and time), and the speed of light is the same for all observers regardless of their motion. If these two points are accepted to be true, then Special Relativity has to be right. There is no way around it.
  6. May 17, 2006 #5
    This approach forms the basis for "Euclidean special relativity", a not-so-often used mathematical framework as alternative to the Minkowski approach. Characteristics are the use of proper time [itex]\tau[/itex] as fourth Euclidean dimension and the universal velocity [itex]c[/itex] for all object in space-time. There are some inconsistencies with classical special relativity though. This thread in the Independent Research forums gives more background info.
  7. May 18, 2006 #6
    Thanks to everyone for your useful comments.
    Thanks Mortimer for your link. That's what I was looking for.

    I'll have a look at "Euclidean Special Relativity" and at the material picked up following your thread.

    As I see it, the physical reason for this dynamic reserve is the simple fact that time flows. It's something we may not deeply understand but just fully experience in our own consciousness. So we can assign time a "flowing speed". It could be a reasonable, intuitive and "self-evident" axiom.

    From a didactic perspective I'd like to build an approach that start from the very essence of spacetime. Just like in geometry. I mean, light speed is something that must be "hardwired" in spacetime.

    Spacetime properties don't come in the wake of Maxwell equations, even if historical developement followed this thread. Logically it's just the other way around, and Maxwell equations reflect spacetime properties.

    So it seems natural to me that light speed should be introduced after some more basic principle, ingrained in the set of properties of spacetime. Light should be introduced after that. It is energy with no mass and its only admitted speed is the speed of time.

    I'm an italian teacher (rather unexperienced yet). Here, the current practice is that special relativity can be (sometime) introduced to young students (15-18 yo) after electromagnetism. Here high school physics courses follow more or less the sequence of physics' historical evolution, from Galileo to Einstein.
    But I'm trying to find out if its possible to (correctly) explain the main features of special relativity to students who haven't heard yet about electromagnetic waves and Maxwell theory, and that cannot fully grasp its mathematical subtleties. Just from a geometric perspective.
    In this regard, Euclidean SR seems to have a promising starting approach, but it also seems to recover complexity in the following steps.

    So I must consider the possibility to draw some didactic material from it with very "special" care.
  8. May 18, 2006 #7


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    From the following quote, note that you won't get Special Relativity from "Euclidean special relativity":

    In that case, definitely don't do a disservice to your students by teaching them "Euclidean special relativity". (If you wish, you can study it as an alternative theory later... after thoroughly teaching "Special Relativity" and consulting the vast amount of experimental support for it.)

    Light speed is "hardwired" in spacetime through the Light Cone that sits at each event in spacetime.
  9. May 25, 2006 #8

    Andrew Mason

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    You may wish to have a look at this paper published by an Indian physicist trying to develop a way of teaching relativity to school children. It impressed me as being a very easy way to teach the basic concepts of relativity to young children.

    I have to thank ZapperZ for locating it a few years back.

  10. May 26, 2006 #9

    Euclidean Special Relativity is not an "alternative" theory. It is strictly based on the Lorentz Transformation which is strictly based on the equivalence of inertial frames.
  11. May 26, 2006 #10


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    Although it may be "based on the Lorentz Transformation which is strictly based on the equivalence of inertial frames", the last sentence of the abstract in the page Mortimer posted: https://www.physicsforums.com/showthread.php?t=103977
    says "The velocity addition formula shows a deviation from the standard one; an analysis and justification is given for that." In particular, look at the figures 6 and 7 in "dimensions6-10.pdf" (from the above page)

    So, to me, if a theory disagrees with SR (in its realm of application) then it is an alternative theory.

    Said another way, two theories may be based on similar things... but if they yield inequivalent results, then they must have diverged somewhere... that is to say, one has taken an alternate route from the other.

    If you want to study "Euclidean Special Relativity", go for it. It's sometimes a worthy cause to carry out to completion a line of thinking.

    But in any case, it is not equivalent to SR. (I'd be curious to see how experiment compares with the different [i.e. alternative] velocity addition formula.)
  12. May 26, 2006 #11
    I looked at that paper. It's definitely not based on Special Relativity! I retract my statement.
  13. May 26, 2006 #12
    I thought 'Change' was the only constant that was absolute..

    If U burn a piece of paper the fundamental stuff is still there..

    Nothing has changed other than the way it was assembled...or collpased
  14. May 26, 2006 #13
    'Time' is an expression of 'Change'....

    its not a literal thing or entity...

    Seems almost a metaphor
  15. May 26, 2006 #14
    A lot of things are very slippery, eggman. Like mass, which is an expression of "resistance to change", or energy, which is an expression of "ability to change". And if you chuck out time nothing can move, so you have no distance and you've chucked out space too. Tricky, tricky.
  16. May 26, 2006 #15

    If u take a pile of paper and set it on fire...u will see it

    burn..smoke and dissappear...

    But in the basic universal sense...

    'Nothing' has changed...

    ..thats what I was meaning
  17. May 26, 2006 #16
    the same 'stuff' remains
  18. May 26, 2006 #17
    And tricky "stuff" it is, which is why we're all here...

    Q: What is stuff made out of?
    A: Other stuff.
  19. May 26, 2006 #18
    It is the apparent movement that captures our attention...
  20. May 27, 2006 #19
    I've read some of the papers in van Linden site and I think that Euclidean Special Relativity can be made consistent with SR, as showed in this article by Alexander Gersten.

    "In this method the Lorentz transformation is described
    by 4-rotations in a four dimensional Euclidean space which we will call the mixed space

    Above article shows that the new proposed frame of reference lead to transformation that are perfectly equivalent to Lorentz's, including velocity addition.

    So the good thing is that in this approach you can recover orthogonal transformation for different inertial frames.
    The bad one is that you have to fumble and jumble with two possible time dimensions: the usual one and that of proper time.
    The consequence is that some paper places ESR in a 5 dimensional space, with a 4 Euclidean space and a jolly extra-coordinate (or parameter) to put things together.
    I like the very first approach of ESR, interpreting the length of the four-velocity vector as an invariant speed that can be "orthogonally projected" in the time or in the space dimensions.
    But apart from that, it's not easy to re-build all SR features with more clearness than the traditional approach.

    Actually time doesn't flow. In the physical sense. It's just our perception of change (or the change in our brain's state, in our sensations) that suggest us this metaphor.
    Time is.
    Like a spatial dimension.
    We must just accept the fact that every material object, like a fidgety quantum particle, can never be at rest in spacetime.
  21. May 27, 2006 #20
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