In the Introductory Special Relativity book, by W G V Rosser, page 19, the author is formulating the relativistic kinetic energy equation. In his derivation, he writes the following equation: $$u=c\sin\theta$$ I have two questions regarding Equation (2.7), the above equation, in the book:

  1. How did the author prove this equation?

  2. Could anyone please explain the physical interpretation of this equation?

Any help is much appreciated. Thank you so much.

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closed as unclear what you're asking by Kyle Kanos, Jon Custer, GiorgioP, HDE 226868, ZeroTheHero Jun 17 at 17:02

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  • $\begingroup$ You might have to provide more context (e.g., quoting the passage) as this really only will make sense to those who have the book in front of them. $\endgroup$ – Kyle Kanos Jun 12 at 0:18
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    $\begingroup$ The equation is simply the definition of the angle $\theta$. It's a definition so it does not need a derivation or proof. He did it so as to simplify the equation just as you make substitutions when doing integrals. $\endgroup$ – mike stone Jun 12 at 0:56
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    $\begingroup$ Note that I said quoting the passage, not taking picture with your cell phone and uploading it. $\endgroup$ – Kyle Kanos Jun 12 at 2:08

In this passage by Rosser, $u=c\sin\theta$ is the spatial velocity, where $\theta$ is a parameter used in a "Loedel diagram" (see Wikipedia entry and Paul Gruner's Elementary geometric representation of the formulas of the special theory of relativity), which is a limited variant of the Minkowski spacetime diagram that tries to use Euclidean trigonometry instead of the more appropriate hyperbolic trigonometry (which uses the rapidity parameter $w$, geometrically interpreted as the Minkowski-arc-length of the unit hyperbola).

So, $$\text{spatial velocity}=u=c\sin\theta=c\tanh w.$$ [Incidentally, $\theta=\text{gd}(w)$ where $\text{gd()}$ is the Gudermannian function which could be defined as $\text{gd}(w)=\arcsin(\tanh w)$.]

With the above definitions, the time-dilation function is $$\gamma=(\text{Rosser's}\ \alpha)=\frac{1}{\sqrt{1-u^2/c^2}}=\sec\theta=\cosh w,$$ and the spatial-component of the 4-momentum [the relativistic momentum] is $$\begin{align*}p &=\alpha mu =\sec\theta\ m (c \sin\theta)=mc\tan\theta\\ &=\gamma mu =\cosh w(mc\tanh w)=mc\sinh w\end{align*}.$$

As noted by one of the commenters, the $\theta$-parameter is a convenient choice of variables that make the equations of special relativity easier to analyze with Euclidean trigonometry.

There is a limited geometric interpretation of $\theta$ using the Loedel diagram.

A fuller explanation of this aspect would require a detailed treatment of the Loedel diagram.

In my opinion, the Minkowski diagram and its hyperbolic trigonometry using rapidity $w$ is superior (and does not suffer from the limitations of the Loedel diagram). But I can't elaborate on this point here.

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    $\begingroup$ +1 for the preference for rapidity. $\endgroup$ – garyp Jun 12 at 2:32

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