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After studying the concept of rapidity and the associated formulation of special relativity in terms of hyperbolic trigonometric functions of rapidity, I've come to understand the elegance of this formulation and its applications. However, I fail to see how it truly simplifies things from a practical point of view. For an example, calculating the addition of velocities using the standard algebraic formula can be done quickly with a simple calculator, whereas doing the same using hyperbolic functions requires you to go translating velocities to rapidities by looking for inverse hyperbolic tangents, adding them up and then converting back to velocities with hyperbolic tangents.

I get that many things can be expressed in a more elegant manner using this formulation: Lorentz transformations look more symmetrical and more closely resemble their Euclidian rotation counterparts and the formula for the Doppler shift becomes a simple $e^w$. However, when actually solving problems, I find the extra steps described above to get in the way instead of helping.

The first edition of the classic Spacetime Physics by Taylor and Wheeler introduces rapidity and the hyperbolic trigonometric functions, but they dropped them entirely in the second edition. According to a Physics Forums comment, Taylor said they dropped them because teachers rarely used the concept. I have several textbooks on relativity, and it seems that whenever I look for the word "rapidity" I get at most one or two results, where the author usually introduces the idea briefly and then never mentions it again. In Ta-Pei Cheng's Relativity, Gravitation and Cosmology, it is literally a footnote. Even MTW's Gravitation has no mention of the word.

With all that being said, as someone in the midst of a journey trying to learn all this, I have the following question:

Is it necessary to understand special relativity in terms of hyperbolic trigonometric functions of rapidity in order to move on to general relativity?

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  • $\begingroup$ There’s really not much more to rapidity than what you’ve already said, so you’re not missing anything. Indeed, like a lot of little things in physics, it’s nice but you certainly won’t need it day to day. $\endgroup$
    – knzhou
    Aug 1, 2022 at 7:15
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    $\begingroup$ This question (v4) seems opinion-based. $\endgroup$
    – Qmechanic
    Aug 1, 2022 at 10:06
  • $\begingroup$ I find the Bondi k factor more convenient. $k=e^\varphi$ $\endgroup$
    – PM 2Ring
    Aug 1, 2022 at 10:40

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It might help to consider the Euclidean analogues of your questions:

  • How important is the concept of angle in Euclidean geometry?
  • Is it necessary to understand Euclidean geometry in terms of circular trigonometric functions of angle in order to move on to Riemannian geometry?

When first learning Euclidean geometry, did you reason with angles and lengths?
Or did you reason with vectors and dot-products? Or vectors and tensors?

  • Since you probably learned with angles and lengths, you probably can refine your intuition by adjusting your circular-trigonometry-reasoning to hyperbolic-trigonometry-reasoning. So, you can import analogues of what you learned in Euclidean geometry and apply them to the hyperbolic-trigonometry of special relativity.

    Certainly, it would be good to develop reasoning and computation skills with vectors and dot-products and tensors... but (in my opinion) it's good to be able to use aspects of Euclidean geometry [suitably generalized] to understand special relativity.

Realize that many textbook problems in special relativity are essentially problems in hyperbolic trigonometry, often solved with techniques analogous to problems in Euclidean geometry. Collision problems can solved like free-body diagrams. Although invariant techniques with dot-products are preferred in relativity, such techniques are not developed or emphasized for free-body diagrams.

Here are some non-trivial examples of using rapidity to reveal and interpret the meaning of an equation:

Are angles necessary for Riemannian geometry?
No [as others have said], but (in my opinion) it couldn't hurt that much to learn it?


By the way, I was the one on physicsforums.com who asked Edwin Taylor about rapidity in the first and second editions during the AAPT topical workshop Teaching General Relativity to Undergraduates (2006) https://www.aapt.org/doorway/TGRU/
See:

As I reported, he told me that he dropped it from the second edition because "physics teachers" reported that they didn't use it.
However, (as I also reported) there were several of us "relativists interested in pedagogy" who encouraged him to put it back in the next edition.


Note that MTW (Gravitation by Misner, Thorne, Wheeler) does use the concept of "rapidity" (named by A.A. Robb), e.g. on p. 67-68 [and p. 109, 122, 481... in topics related to special relativity] , but it uses the term "velocity parameter" $\alpha$ instead of "rapidity".

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I did work in particle physics years ago, so special relativity was one of our everyday's tools, and frankly I never saw velocity either mentioned or used.

The way I see it, you have two ways to approach special relativity:

  • the "old" approach, that didn't rely (much or at all) on tensor algebra and tried to have formulas that look like classical mechanics as much as possible (with things like velocity and relativistic mass),
  • the "modern" approach that uses tensors everywhere, where those notions aren't useful.

The old approach was obviously more natural at first, because people like what they're already familiar with, but that doesn't mean it's fruitful. The modern approach is prefered because it's more powerful, in the sense that it handles invariants more smoothly and it could be generalized to larger theories later on.

So no: while you can say interesting things about velocity, it's nothing more than a curiosity to help students find their bearing because it looks a bit like something they know.

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  • $\begingroup$ do you have an intro reference that uses the "modern" approach? $\endgroup$
    – Casey Chu
    Nov 23, 2023 at 4:59
  • $\begingroup$ @CaseyChu It's been many years since I had one of those books in hand, so I'd take time to track them. Let's say that one way to make reasonably sure a book will use tensors is to see if it covers electromagnetism with special relativity (and, more generally, covariant objets in Minkowsky space). $\endgroup$
    – Miyase
    Nov 23, 2023 at 7:57

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