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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.

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    $\begingroup$ 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. $\endgroup$ – Stan Liou Aug 7 '14 at 19:53
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    $\begingroup$ @StanLiou I think OP was trying to work in $1+1$ dimensions for the sake of his argument. $\endgroup$ – Danu Aug 7 '14 at 19:54
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    $\begingroup$ @Danu: I know; that's why I said it was a hyperbola first. $\endgroup$ – Stan Liou Aug 7 '14 at 19:58
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    $\begingroup$ @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. $\endgroup$ – Stan Liou Aug 7 '14 at 20:02
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    $\begingroup$ @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. $\endgroup$ – Jerry Schirmer Aug 7 '14 at 20:04
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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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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.

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