# The Meaning of $u=c\sin\theta$ Equation in Special Relativity [closed]

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.

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 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. – Kyle Kanos Jun 12 at 0:18
• 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. – mike stone Jun 12 at 0:56
• Note that I said quoting the passage, not taking picture with your cell phone and uploading it. – 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.

• +1 for the preference for rapidity. – garyp Jun 12 at 2:32