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- Thread starter StationZero
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http://mathworld.wolfram.com/HyperbolicGeometry.html

or just the fact that γ

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What common(?) circular geometrical framework are you specifically referring to?common circular geometrical framework

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And here's a supplementary reference, just scroll down about halfway tot he hyperbolic geometry section. Thanks.

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you are familiar with standard presentations in terms of relative-[dimensionless] velocities ([itex]\beta[/itex]) and time-dilation factors ([itex]\gamma[/itex])

and are wondering about

presentations involving "rapidity" ([itex]\theta[/itex]), where [itex]\beta=\tanh\theta[/itex] and [itex]\gamma=\cosh\theta[/itex]. (Note that, in his preface, Tevian Dray is describes this as

In my opinion, the rapidity ("Minkowskian angle") approach allows one to more clearly draw analogies to your Euclidean intuition... and possibly make special relativity less mysterious. It will reveal the geometrical origin of the numerous "formulas" of special relativity.

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These are just two different but equivalent ways of describing SR.

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You'll see that what are usually called "Lorentz boosts" are really just the analogue for rotations compared to the usual Euclidean framework.

In general, Euclidean rotations are generated by some object [itex]i[/itex] that squares to [itex]-1[/itex]. Taking [itex]e^{i \theta}[/itex] generates (circular) sine and cosine through power series. [itex]e^{i \theta} = \cos \theta + i \sin \theta[/itex].

"Rotations" involving a timelike and spacelike direction in Minkowski space are generated by some object [itex]\epsilon[/itex] such that [itex]\epsilon^2 = +1[/itex] yet [itex]\epsilon \neq 1[/itex]. A similar power series argument sowing that [itex]e^{\epsilon \phi} = \cosh \phi + \epsilon \sinh \phi[/itex] generates the hyperbolic trig functions. Just as [itex]e^{i \theta}[/itex] is at the heart of rotations in the 2d Euclidean plane, so is [itex]e^{\epsilon \phi}[/itex] at the heart of "rotations" in the 1+1d plane.

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