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Featured Insights Relativity using the Bondi k-Calculus - Comments

  1. Apr 10, 2017 #1

    robphy

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  2. jcsd
  3. Apr 10, 2017 #2
    Great insight.
     
  4. Apr 11, 2017 #3
    To the author, you have exposed different approaches to this body of knowledge. What approach or ways would you use and sequence say for undergraduate instruction.
     
  5. Apr 11, 2017 #4

    robphy

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    Any approach I use must use the spacetime diagram
    because I think it is difficult to represent the relativity-of-simultaneity using boxcars as "moving frames of reference".

    Any approach I use must use radar methods
    to motivate measurements and the assignment of coordinates.
    I think radar methods are more straightforward than lattices of "clocks" and "rods".
    (For inertial motions in special relativity, they are equivalent.
    However, for more general motions in special and general relativity, they may differ....
    and would require more advanced discussion to address.)

    In my opinion, the Bondi k-calculus method (with its emphasis on radar measurements) is the best starting point, especially for algebra-based physics. With the k-calculus methods, the standard textbook formulas are straightforward to derive and fall out naturally.

    A related but even less well known approach by Geroch (in his General Relativity from A to B) is also a good starting point. Geroch uses radar methods to emphasize the square-interval and give operational interpretations of the geometry of spacetime (e.g., what simultaneity means to an observer) in both Special Relativity and Galilean Relativity. My AJP article (which inspired the Insight https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ ) was my attempt to combine Bondi's and Geroch's approaches.

    From here, I would go on to develop the geometry of Minkowski spacetime, while comparing and contrasting with Euclidean geometry, using the [unappreciated] geometry of Galilean spacetime (e.g., https://www.desmos.com/calculator/ti58l2sair ... play with the E-slider) ...something I call "Spacetime Trigonometry", a large ongoing project with many aspects which generates lots of posters for me at AAPT meetings. (I should really write this up soon... but it would have to be broken into a series of AJP articles.) These are examples of Cayley-Klein geometries, which includes the deSitter spacetimes.. This "unification" can help formalize the numerous analogies mentioned in the literature. In addition, I can develop vector and tensorial methods (algebraically, graphically, and geometrically) in order to make contact with traditional intermediate and advanced presentations of relativity.
     
  6. Apr 11, 2017 #5
    Please keep us posted, I will enjoy following your work.
     
  7. Apr 12, 2017 #6

    pervect

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    Thanks for posting this. A lot of times I want to refer people to Bondi's approach, as I also feel it's one of the best elementary treatments for the person new to relativity. I can and do refer interested people to his book, but it's nice to have a more accessible source.
     
  8. Apr 12, 2017 #7

    robphy

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    Thanks. I was torn between making it as elementary as possible for a beginner (which would only be a tweak on Bondi or just the equations already provided by Wikipedia) or making clarifications and connections to geometry (the logical next step).
     
  9. May 3, 2017 #8
    Why does he assume the velocity of light is the same for all inertial observers?
     
  10. May 3, 2017 #9

    Ibix

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    About forty years before Einstein, Maxwell published equations describing electromagnetism. One solution to the equations was a wave, which turned out to have the properties of light. One weird thing was that the speed of the wave came out the same always. Naturally everyone assumed that the equations weren't quite right and the hunt was on to find the problem.

    The next forty years were a bit confusing as no one could find anthing wrong. Experiments that were expected to help (e.g. Michelson and Morley) didn't work as predicted, but did provide some ad hoc patches. Einstein had the insight that if the (apparently daft) prediction that light always travels at the same speed for all inertial observers was correct then he could explain all of the confusion. So he made the assumption.
     
  11. May 3, 2017 #10
    Alice's movie is seen by Bob to be in slow motion, and Bob's movie is seen by Alice to be in slow motion. That is similar to SR in which Alice's and Bob's clocks would measure to each other to run slow ... but all your diagrams are presenting the case of increasing separation of the inertial travelers.

    To the degree that the diagram tends to suggest that movie duration is a proxy for time dilation... it looks like it only works with cases of increasing separation, not cases of approach. Students would notice this...

    If these movies were youtube videos, there would be a time indicator rolling at the bottom of the screen, so for example, both Alice and Bob could see that Alice's movie indicates that it starts at 00:00:00 and increments to 00:60:00 at the end. Although Bob can't necessarily "view Alice's clock" , he can see by the video time index that in comparison to his own clock her video is running slow... suggesting that her time is slower relative to his (and likewise his to hers when he sends video to her).

    When Alice and Bob approach each other, it looks like Bob is going to see Alice's movie running faster (shorter time), and Bob's movie will be seen by Alice to be running faster... so this is not similar to SR which would maintain that each measure each others clocks running slow.
     
  12. May 4, 2017 #11
    That's difference between "see" and "observe". We see Doppler shifted light as it enters our eyes in the same way as we see the movie running slow as its images enter our eyes.

    But if you want to observe what is really happening you have to allow for the light travel time. That will lead you to time dilation.

    Note that even for the case of increasing separation the time dilation factor is not the same as the Doppler factor.

    If the relative speed is ##\beta## then the time dilation factor is ##(1-\beta^2)^{\frac{1}{2}}## whereas the Doppler factor is ##\big(\frac{1+\beta}{1-\beta}\big)^{\pm\frac{1}{2}}##.
     
  13. May 4, 2017 #12

    robphy

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    These viewings of movies are not proxies for time-dilation... they are descriptions of the Doppler effect for light.
    For observers receding from each other, each observes a "redshift" (or, in the case for sound, a lowering of frequency).
    For observers approaching each other, each observes a "blueshift" (or, in the case for sound, a raising of frequency).
    In some sense, the Doppler Effect needs the time-dilation factor in order to satisfy the principles of relativity.
    Indeed, in the derivation of receding sources and receding receivers,
    one gets expressions involving the Galilean-Doppler factor and the time-dilation factor:
    ##\gamma(1+\beta)=\left(\frac{1}{\sqrt{(1-\beta)(1+\beta)}}\right)(1+\beta)=\sqrt{\frac{1+\beta}{1-\beta}}=k##
    and ##\frac{1}{\gamma}\left(\frac{1}{1-\beta}\right)=\left(\sqrt{(1-\beta)(1+\beta)}\right)\left(\frac{1}{1-\beta}\right)=\sqrt{\frac{1+\beta}{1-\beta}}=k##.

    It might be useful to point out a distinction between time-dilation and the Doppler effect for light.
    For two inertial observers Alice and Bob that met at event O,
    • time-dilation involves two spacelike-related events,
      say "event P on Alice's worldline" and "event Q on Bob's worldline that Alice says is simultaneous with P"
      (so, ##\vec{PQ}## is a purely-spatial displacement vector according to Alice... it is Minkowski-perpendicular to ##\vec{OP}##).
      The time-dilation factor measured by Alice is ##\gamma=\frac{OP}{OQ}##.
    • Doppler-effect involves two lightlike-related events,
      say "event P on Alice's worldline" and "event S on Bob's worldline which is in the lightlike-future of P"
      (so, ##\vec{PS}## is a future-lightlike displacement vector).
      The Doppler factor measured by Alice is ##k=\frac{OS}{OP}##.


    In the case of approaching, one has a diagram like this [based on reflecting the original diagram from the Insight]:
    Bondi-approach.png
    where I have used a "factor" ##\kappa## (kappa).
    So, as you said, Bob would view Alice's T-hour broadcast "sped up", in only ##\kappa T## hours (where ##\kappa<1##).
    By similar triangles, ##\displaystyle\frac{\kappa T}{T}=\frac{kT}{k^2T}##, which implies that ##\kappa=\frac{1}{k}##.
    Note that since ##k=\sqrt{\frac{1+\beta}{1-\beta}}##,
    we have ##\kappa=\frac{1}{k}=\sqrt{\frac{1-\beta}{1+\beta}}##, which is the original expression for "##k##" with "velocity ## -\beta##".
    Thus, there's no need to use ##\kappa##... "receding and approaching" are handled by ##k##.
     
    Last edited: May 4, 2017
  14. May 19, 2017 #13
    The theory of special relativity was derived from a simple fact based on the right triangle as follows :
    Imagine a light signal is sent from a point to an observer moving with a velocity "v". This signal will be received by this observer moving with velocity "v" after a time delay with respect to the initial position of this observer that forms the hypotenus of the right triangle on which the velocity of light "c" is the same as the one of the right sides of the right triangle while for the other right side the velocity of the observer is "v". If you multiply these velocities with the same time difference "dt", addition of squares of the two right sides would be greater than the square of the hypotenus that would violate the pytogoran theorem for which it becomes necessary to denominate the time intervals with different indices as : :
    (cdt*)^2 = (vdt*)^2 + (cdt)^2 which after a simple algebra becomes dt* = dt / [ 1 - (v/c)^2 ]^1/2
     
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