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I was trying to explain special relativity to a few friends in a simple way and wound up with an analogy using a c unit circle.

I was using y as travelling in time, x moving in space; move in space and you are borrowing from your clock-speed. E.g. the twin paradox: Your travelling twin has borrowed clock-speed. Twins clock was slower -> twin is younger.

What is the problem with representing relativity using a (c) unit circle from an observers point of view? Anything Lorentz or Minkowski would be out of the question of course, I'm looking for a layman friendly description that is close to true.

The circle discussed, was in my mind a half circle $y>=0$. Where $y=0$ would be a photon.

John Rennie
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Captain Giraffe
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  • What were you trying to explain using this 'image'? – Danu Aug 07 '14 at 19:53
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    The relativistic "circle" (locus of points a set interval from a given point) is actually a hyperbola, or hyperboloid in $3+1$ dimensions, because the Lorentzian analogue of the Pythagorean theorem has a minus sign in it. – Stan Liou Aug 07 '14 at 19:53
  • @Danu See edit about twin paradox. – Captain Giraffe Aug 07 '14 at 19:54
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    @StanLiou I think OP was trying to work in $1+1$ dimensions for the sake of his argument. – Danu Aug 07 '14 at 19:54
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    @Danu: I know; that's why I said it was a hyperbola first. – Stan Liou Aug 07 '14 at 19:58
  • @Stan. So how does the circle break things? In (1 + 1) dimensions. – Captain Giraffe Aug 07 '14 at 19:59
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    @CaptainGiraffe: Lorentz transformations rotate along a hyperbola in spacetime in the exact same way that Euclidean rotations rotate along a circle in space. The Lorentz boost is just an addition of hyperbolic angles $\alpha = \tanh^{-1}(v/c)$. Beyond that, I'm not clear on what your question is. – Stan Liou Aug 07 '14 at 20:02
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    @CaptainGiraffe: circles come back around. If you could represent transforms using circles, then it would be possible to exchange all of your time for space, or vice versa. In a lorentz transformation, the best you can do, at infinite energy, is to get ${\dot t} = {\dot x}$, which is why a hyperbola works, because it has the asymptote, rather than closing back in on itself. – Zo the Relativist Aug 07 '14 at 20:04
  • @JerrySchirmer I'm looking for a layman friendly discussion. I can't discuss anything nonlinear with these friends besides a circle. Edited the question. – Captain Giraffe Aug 07 '14 at 20:08
  • @CaptainGiraffe: then you're better off describing the quarter circle going from -45${}^{\circ}$ to +45${}^{\circ}$ and not including the endpoints. But it's still definitely a hyperbola. – Zo the Relativist Aug 07 '14 at 20:22
  • @JerrySchirmer Now you are just going at a space/ time diagram. light cones and all. I wanted to avoid that. So no time-like events. – Captain Giraffe Aug 07 '14 at 20:24
  • If analogies with circles worked, we wouldn't teach SR with hyperbolas. Things would be much better for the layman if hyperbolic trig got half the attention of "regular" trig in high school. –  Aug 07 '14 at 21:07
  • You may find this video interesting. Unfortunately, the author doesn't touch on how this representation relates to the more common hyperbolic representation, though he certainly is aware of that too. – Tamlyn Oct 30 '23 at 21:53

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Your model is not only useful for layman, but it does also have physical importance.

One thing needs to be clarified: Your diagram is not a Minkowski diagram (permitting Lorentz transforms). In particular, your y-axis is not coordinate time (as in the Minkowski diagram) but proper time. I proposed a similar scheme in Minkowski spacetime: Is there a signature (+,+,+,+)?

By the way, the advantage of such a diagram is that it permits an improved description of time (because any time derives from proper time). Currently we describe time only by the means of Minkowski diagrams – however, Minkowski diagrams were made for Lorentz transformation and not for a description of what time is.

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Moonraker
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Einstein (i think) used a simple example. Travelling upon a photon and looking himself in a mirror. Would he be able to look at his mirror-image or not? Einstein (and Lorentz et al) said yes, as such light (should) travels with same constant speed $c$ in all (inertial) frames of reference.

Then read Einstein's original 1905 paper (On the electrodynamics of moving bodies) which derives all Relativistic/Lorentz transformations based on this principle (of constancy of speed of light) only.

i think this is good intuitive explanation and analogy.

Nikos M.
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