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In school we learn that we can add velocities together, and then later on we learn that it's not correct and that there is a speed limit. Why create all this confusion when we could just use rapidity to begin with?

Rapidity is defined as $w = \mathrm{arctanh}(v / c)$, where $v$ is velocity and $c$ is the speed of light in a vacuum. Rapidities can be summed and have no upper bound. At non-relativistic speeds it acts proportional to velocity.

In fact, at non-relativistic speeds, we could substitute $v$ for $wc$ (rapidity times speed of light), and one could hardly tell the difference. The ISS moves rather fast at a velocity of 7660 m/s (27,576 km/h), and has a $wc$ of about 7660.0000016667 m/s. Why can't we just substitute velocity for rapidity in real-world and classroom use, and end the confusion about why there is a speed limit once and for all?

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    $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Commenters, please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$ – David Z Apr 28 at 9:10
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    $\begingroup$ Please explain why you think there is any confusion about there being a relativistic speed limit. Seems as though anyone who knows anything about physics understands this, while it doesn't affect the daily lives of people (remote tribes in the Amazon, perhaps?) who don't. $\endgroup$ – jamesqf Apr 29 at 16:39
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    $\begingroup$ We could just skip it all and teach Minowski space time in kinder garten? There is a hierarchy of models being taught, each one being more complex than the previous. Would not this create unecessary confusion describing an everyday phenomenon? $\endgroup$ – Stian Yttervik Apr 30 at 8:30

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In everyday life, we experience the universe in a non-relativistic classical way. We are familiar with the concept of time and space.

Defining the velocity as the ratio between a distance traveled in a given time interval is a much more natural choice instead of defining the rapidity.

If we all lived at relativistic speeds, or close to the event horizon of a black hole, or if we were small as an atom, we would use other tools to describe the universe around us.

However, in our case, the quantities of classical mechanics work quite well and we can have a direct intuitive grasp of their meaning.

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    $\begingroup$ This is true, but for non-relativistic speeds we can cheat and define rapidity to be the same as velocity. It's no worse than the cheating we're doing by adding velocities together, but the difference is that we can add an infinite number of positive rapidities together and get an infinite result, rather than $c$. In my opinion that's more intuitive. $\endgroup$ – Fax Apr 28 at 10:11
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    $\begingroup$ @Fax Objectively I think you're right, but as a layman who enjoys reading this forum, I assume it's intuitive to you because of your more in depth understanding of physics, where velocity is more intuitive for us casual observers. Either way, thanks for the post, TIL rapidity is a thing ;) $\endgroup$ – TCooper Apr 28 at 18:15
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    $\begingroup$ @Fax: Because then everyone would grow up used to the fact that you can simply add one rapidiy to another and then when physics students reached college, they'd learn this isn't true, and that they'd have to use arctanh. To avoid mixing concepts, they'd probably give it some other name, like "fastness", and then propose on stackoverflow that everyone should just always use "fastness". $\endgroup$ – Mooing Duck Apr 28 at 18:33
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    $\begingroup$ @Fax if we can cheat and define it to be the same, why does it even matter? At non-relativistic speeds, velocity is accurate enough. It is unnecessary to complicate the math for cases which don't typically apply. $\endgroup$ – eques Apr 28 at 20:31
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    $\begingroup$ My point was that if you always use rapidity, people will stop using arctanh, and will start to just always add them. People will grow up doing that, and being taught to do it in schools, because it's correct for all non-relativistic situations. At which point we're back where we started. $\endgroup$ – Mooing Duck Apr 29 at 15:54
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It is not particularly easy to add rapidities pointing in different directions.

e.g. Suppose B moves with rapidity $\rho_{1}$ with respect to A in the common $x$-direction of both of them. And suppose that C moves relative to B with rapidity $\rho_{2}$ in the common $y$-direction of B and C (we are taking it that their axes are aligned in this way). Now what is the rapidity of C relative to A?

You will not find it easy to answer.

The fact that this sort of thing is non-trivial is another reason (in addition to some good ones given in other answers) why it does not help to replace velocity with rapidity.

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  • $\begingroup$ Related: physics.stackexchange.com/questions/178551 Doesn't seem to be harder to remember than the velocity addition formula. $\endgroup$ – WorldSEnder Apr 29 at 0:25
  • $\begingroup$ @WorldSEnder It is difficult because a rotation of axes also comes in, and as a result the spatial part of the direction of the 4-vector that one obtains is not easy to interpret. You have to think carefully about the directions of the various systems of coordinate axes. $\endgroup$ – Andrew Steane Apr 29 at 8:14
  • $\begingroup$ Good answer. While rapidity makes 1-dimensional relativistic problems easier (e.g. how long does it take an accelerating probe to reach 0.9c), it makes higher-dimensional problems harder. $\endgroup$ – Fax Apr 29 at 8:44
  • $\begingroup$ @Fax What do you think about velocity composition in (1+1)-spacetime? How about (2+1)- or (3+1)- spacetime? $\endgroup$ – robphy Apr 29 at 14:08
  • $\begingroup$ @robphy Not familiar with the notation, but I assume (1+1)-spacetime is one spatial dimension and one time dimension. From what I understand the velocity-addition formula still applies to some reference frame in (1+1)-spacetime, though I'm not sure about the (2+1) and (3+1) cases. $\endgroup$ – Fax Apr 29 at 14:48
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enter image description here

People understand what velocity means, and how it is used. Anyone who needs to use relativistic models knows enough to distinguish between which model they are using. No additional words were needed when it was created, and trying to invent new words now would just cause more confusion, not less.

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    $\begingroup$ What words are propsed to be invented? Rapidity is already an established term used in physics. $\endgroup$ – Chechy Levas Apr 29 at 7:36
  • $\begingroup$ @ChechyLevas If we supose rapidity appeared with Special Relativity, the word is roughly 100 years old and used in very specific context. Velocity is thousands of years old... In this context rapidity is still quite new. $\endgroup$ – Crowley Apr 29 at 19:51
  • $\begingroup$ Your image is not formatted properly. $\endgroup$ – Wai Ha Lee May 1 at 8:29
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While this varies by country, it's fair to say that kids have a qualitative understanding of speed before kindergarten.

By grade 8, they're talking about speed, acceleration, and related issues of energy quantitatively.

Vectors are introduced during high school.

Most students won't get a quantitative discussion of Special Relativity until freshman physics.

Hyperbolic trig functions won't appear until AP Calculus or freshman year as well.

Therefore, introducing rapidity in grade 8 is only really going to be feasible in a non-relativistic way. As you stated in your question, one could present rapidity in its product form with c. However, all you've achieved here is a change of terminology; students are still using velocity in its non-relativistic form.

We're already teaching them about velocity in its non-relativistic form. We call it velocity.

What seems intuitive, in retrospect, is a lot less intuitive as you're learning it. Most physics students will never be exposed to SR. Those that are should be advanced enough to understand that SR corresponds to classical mechanics at low speeds, in the same way that they will understand that quantum mechanics corresponds to classical mechanics at high energies. This understanding is an essential step in their growth as physicists.

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    $\begingroup$ This is the best and most straight forward answer imo. In basically every subject we first teach kids concepts that are overly simplified and not technically correct. This is a good thing because teaching the "correct" thing far before they are ready to understand it doesn't help with understanding and clarity and in fact just the opposite: it confuses and frustrates. I don't remember exactly when I first thought about how speeds add but u wouldn't be surprised if it was as early as 4th grade. I didn't become truly comfortable with trig functions until late high School in at the earliest $\endgroup$ – eps Apr 28 at 20:37
  • $\begingroup$ True, but an 8th-grader is still capable of asking "why can't I go faster than light?". You could tell them "light goes infinitely fast" (rapidity) or "there is a universal speed limit, and light is already going at that speed" (velocity). I think it's easier for an 8th grader to understand that you can't go faster than infinitely fast. $\endgroup$ – Fax Apr 29 at 9:17
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    $\begingroup$ @Fax And what would you answer to that 8th grader when they ask why there's a 20-minute delay when communicating with our colony in Mars? $\endgroup$ – JiK Apr 29 at 9:26
  • $\begingroup$ @JiK Indeed, that is a good question, and perhaps the best counterargument to the intuitiveness of rapidity. If the choice is between the unintuitiveness of a speed limit and the unintuitiveness of simultaneity, I concede that speed limit wins. $\endgroup$ – Fax Apr 29 at 9:31
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    $\begingroup$ @Fax conversely, most people have trouble with grasping the concept of infinity, and get doubly upset if you try to introduce the countable/uncountable distinction. As for the speed limit, that is no weirder than all sorts of other physical limits? $\endgroup$ – pjc50 Apr 29 at 12:48
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As well as issues of practicality, it doesn't answer the question of why there's a speed limit. (It can't as it's just a mathematical transformation.). The question becomes 'in the formula for $w$, why do you take $c=3 \times 10^8$ m/s'?

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  • $\begingroup$ Indeed, although would not the only seemingly arbitrary mathematical constant. It does however allow you to say "there is no speed limit, only weird simultaneity". But is it true? Or is there no practical way to explain relativistic effects without a speed limit? It seems the answer to that would also answer my original question. $\endgroup$ – Fax Apr 28 at 11:09
  • $\begingroup$ @Fax I don't know why you want to do away with speed limits. It's a physical fact, in science education, you teach students to deal with them, not hide them. $\endgroup$ – Dvij D.C. Apr 29 at 10:46
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    $\begingroup$ @Fax You can do infinitely many such tricks to hide the physical content of theories. For example, you can define a "modified commutator" of $x$ and $p$ as $[x,p]-i\hbar$ and then claim that there is no uncertainty principle. Do you see how it is not helping? $\endgroup$ – Dvij D.C. Apr 29 at 10:52
  • $\begingroup$ @DvijD.C. That's probably where I am mistaken. I see two models - velocity and rapidity, and one has a seemingly arbitrary speed limit. It's not clear to me how the speed limit is a physical fact rather than an artifact of the model. It seems I must do some reading to understand it better. $\endgroup$ – Fax Apr 29 at 12:48
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    $\begingroup$ @Fax I'll try to draft an answer explaining how speed limit is not an artifact of using a misguided variable to describe nature. $\endgroup$ – Dvij D.C. Apr 29 at 12:57
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There are really 2 questions here, so let's answer them separately.

Why don't we use rapidity in daily life?

Simple, most people don't even know what relativity is don't understand its implications or how to reason about them. They are not capable of using relativistic quantities.

If you are suggesting they simply don't worry about relativity and just be mindful of the $arctan$, the answer doesn't change. Most people do not understand trigonometry and cannot use it; including many people who have a need to understand and manipulate velocities. For example, how many people would still have a driver's license if this was required to pass driving exams?

If you are saying they should just call it "rapidity" and then act exactly as if they were speaking about non-relativistic velocities, then sure that could work. But it would get in their way if they do start learning advanced physics and must differentiate between the "old rapidity" and the new one. Also, "rapidity" sounds kind of funny.

Why don't we teach rapidity in school?

Most students who learn about velocities do not go on to learn relativity, so they would never get the pay off. On the other hand, those that do learn relativity, probably do not stop at scratching the surface of its definition. They would go on to learn more advanced topics. If we are asking these students to comprehend those advanced topics, I think the simple matter of velocity in a relativistic context is not that confusing and arguably not worth the extra terminology.

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  • $\begingroup$ I agree in principle, but some common and seemingly simple questions have complicated answers because they touch on relativity. Questions such as "why can't I go faster than the speed of light?" and "if I go at 90% speed of light and then double my speed, why aren't I going at 180% speed of light?". Those are questions that I think should be possible to answer by scratching the surface. $\endgroup$ – Fax Apr 29 at 9:50
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I think the main reason, as already captured in part by other answers (e.g. Davide Dal Bosco's), is the following: velocity is a physical quantity, it tells us how far something goes in a given time.

Rapidity may be mathematically convenient due to its relativistic addition properties, but what does it tell us?

As an example, the rapidity of light is $w = \textrm{arctanh}(1) = \infty$. Isn't it much more useful to know that light moves at $c=299 792 458 \frac{m}{s}$ through space?

Mathematically, we can transform everything back and forth as we wish to simplify our calculations. But in the end, we will want to know something physical: the velocity.

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    $\begingroup$ Why the approximation sign? The speed of light is an exact value. $\endgroup$ – Sandejo Apr 29 at 2:03
  • $\begingroup$ @Sandejo of course you are right, the speed of light is nowadays exact since the meter is defined in terms of it. Edited. $\endgroup$ – Wolpertinger Apr 29 at 10:37
  • $\begingroup$ It's the fuel consumption of a rocket that accelerates up to that velocity! $\endgroup$ – MaudPieTheRocktorate May 10 at 9:36
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In retrospect, having seen special relativity, what you suggest is reasonable. In fact, one can unify the geometries of Euclid, Minkowski, and Galilean-relativity after one distinguishes that "Galilean rapidity" is different from "[Minkowskian] rapidity". So, I think we should use at least introduce it (or aspects of it) if one is going to discuss special relativity... which is one way to reveal that we are reaching the limits of an approximation. (Why not use GM/R^2 instead of g?)

Here are some reasons why we don't use rapidity.

  • Historically, we didn't [and primarily still do not] view a position-vs-time graph as a "[Galilean] spacetime diagram" and thus make connections to Euclidean geometry.
  • Rapidity is an angle (a spacelike arc-"length" [with the appropriate metric] or a sector area in an appropriate "circle"), whereas velocity is a slope... and we tend to think of "rates of change".
  • One could argue that velocity is "more physical" (in our non-relativistic upbringing) than rapidity which is arguably too abstract. (Edwin Taylor told me that he dropped rapidity from the 2nd edition of Spacetime Physics because its users (teachers) reported to him that they didn't use it. A few of us politely protested and suggested that he put it back in a future edition.)
  • Some could argue that trigonometry is hard... especially hyperbolic trigonometry.
  • There are likely others... but I keep some of these in mind as I develop an approach to using a unified viewpoint suggested above to teaching physics, dropping such hints along the way... with the goal of making relativity more understandable, and less mysterious.

It's probably not too far off to suggest that the following is an analogue of your suggestion.

  • Why not use aspects of spherical trigonometry (e.g. angles on the sphere to measure distances) instead of Euclidean geometry of the plane?

UPDATE: In trying to motivate rapidity (in relativity) vs velocity (in relativity)

  • Rapidities add but velocities don't... just like saying that angles add, but slopes don't: the slope between two lines is not the difference of their slopes. (To rotate images, does Photoshop use slopes or angles?)
  • A lot of these cryptic calculations (e.g. Momentum in center of mass-frame out of knowledge kinetic energy in lab-frame ) are easier to interpret with rapidity... and appealing to one's intuition with ordinary trigonometry and Euclidean geometry.
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There are several answers to this question.

Rapidity is not taught from the outset in introductory physics classes in part because it would unnecessarily confuse students, and besides, you only need to worry about it when you are dealing with relativistic speeds (or precise-enough measurements that can detect relativistic effects). But there are other reasons, too.

Even when working in SR (special relativity), rapidity is not as useful or (in some sense) as fundamental as velocity. It's true that, in SR, for relative motion in 1D, one simply adds rapidities. But what if you want to know the position of an object after a certain amount of time has elapsed (given its initial position)? For that, you'll need a velocity.

Even then, in most cases in which rapidity would come in handy, it is better to deal with momentum (or 4-momentum), since this is what is actually conserved. This is connected with rapidity very simply: the 3-momentum is a vector pointing in the direction of motion, with a magnitude equal to the rest mass times the rapidity (and the time-component of the 4-momentum is the total relativistic energy).

Also, generally, practicing physics is the art of keeping things as simple as possible. Life is complicated enough. There is no reason to introduce relativity, quantum mechanics, warped spacetime, etc etc, if you don't really need to.

Finally, historically, Einstein's special relativity, Lorentz transforms, Minkowski spacetime, etc, were developed long after Galilean relativity and Newtonian mechanics. In most physics courses it makes sense to pursue a pedagogical track that more-or-less follows the historical development because then students can see how the accumulation of evidence and lines of reasoning led, historically, to improvements on what came before. Physics, like all sciences, does not arise from pure reasoning from a set of postulates handed down on stone tablets. It is important that students understand that the theory of SR arose out a failure of classical physics (Newton + Maxwell) that led to contradictions that needed to be resolved.

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My guess would be that people like to use the simplest tools for doing specific tasks, until it turns out that there is a special task for which they require a more sophisticated tool. Non-relativistic speeds do behave as if they were very simple vector quantities, so why go through the rapidity overhead just to get a negligible improvement of your results?

Pretty much everything we believe and calculate in the real world is based on approximations (e.g. small angle approximation, Taylor series, the derivative) and numerical solutions to differential equations (Fourier series, finite element simulations), so the takeaway would be to simplify where possible and to complicate only where it's absolutely necessary.

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  • $\begingroup$ Fair enough, but what creates an overhead here? Why can't we just substitute "its velocity is X km/h" with "its rapidity is X km/h" and be done? I'm not suggesting we should use rapidity to get more accurate results, but to get more intuitive results. $\endgroup$ – Fax Apr 28 at 10:06
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    $\begingroup$ I understand where you're coming from, it would make sense numerically that adding a bunch of positive numbers yields an arbitrarily large number. But since you cannot directly measure the rapidity (you need to know the total velocity or momentum vector), converting from velocities to rapidities will always feel like an artificial procedure to 'make the the maths work out' in one way or another. People are used to thinking and living in a Newtonian way, so everything outside of that scope, no matter how much more sense it seems to make numerically, will feel weird so to speak. $\endgroup$ – JansthcirlU Apr 28 at 10:45
  • $\begingroup$ @Fax So imagine asking a 6-year old: if a car moves with a rapidity of 60 km/h for 1 hour, how far has it moved? If this were a speed, they would easily be able to answer it. But since it's a rapidity, they would have to know about hyperbolic trigonometric functions. $\endgroup$ – knzhou Apr 29 at 3:17
  • $\begingroup$ @knzhou Imagine asking this to a 6-year old: If Charlie rides his motorcycle at 30 km/h, and Emma passes Charlie at 30 km/h, how fast is Emma going? Does your 6-year old need to know the velocity-addition formula? Of course not; Emma is going at 60 km/h, and the car moved 60 km. $\endgroup$ – Fax Apr 29 at 9:06
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    $\begingroup$ Which is kind of my point @Fax, simple scalar addition works for your example, so we don't need to complicate things using relativity. On the other hand, if you use rapidity by default, you are required to complicate things with inverse hyperbolic functions to get to the same result. $\endgroup$ – JansthcirlU Apr 29 at 10:48
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Whatever you choose as you everyday quantity, you'll eventually have to use velocity. And that's needed in quite a natural context: to predict how much time it would take for an object you observe to cover some distance.

An example of a practical question: What is the minimum time for a space probe to reach a planet in Alpha Centauri and return back with a sample of its surface? Rapidity is useless here, and starting from it would needlessly add complexity to the calculations, which with velocity amount to division of double distance by mean speed.

On the contrary, adding velocities of two objects that both move at relativistic speeds is far from being a common practical problem, so exchanging the ease of use of velocity in everyday problems for (seeming) intuitiveness of rapidity doesn't really look useful.

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  • $\begingroup$ Can you expand on why rapidity is useless? Why can't I use the double distance by mean speed formula with rapidity? $\endgroup$ – Fax Apr 28 at 10:46
  • $\begingroup$ @Fax because if we want to take the smallest time, the probe would go with speed close to the speed of light. But whether we take rapidity 5 or 50 doesn't matter much for travel time: although the latter rapidity is 10× larger, the relative speed difference is 0.01%, which is nothing, and directly translates to negligible difference in travel time. $\endgroup$ – Ruslan Apr 28 at 11:04
  • $\begingroup$ A practical followup question: How much time elapsed for the folks in the probe? $\endgroup$ – robphy Apr 28 at 18:57
  • $\begingroup$ @Ruslan In some cases it may be more useful to think of velocity rather than rapidity, but for a physical probe it does indeed take ten times as long to accelerate from rapidity 5 to rapidity 50. You can say that you wish to go there at 0.99c, but you might find that you've arrived before you get there. $\endgroup$ – Fax Apr 29 at 13:27
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Speed of light isn't a limit on abstract velocity, only on the velocity attainable by a body. $2c$ is a mathematically valid velocity. A spot of light created by a laser pointer shining upon a surface can move faster than the speed of light. Velocities are in fact perfectly summable.

If I observe some object moving $0.8c$, I can think about adding another $0.8c$ to make it go at $1.6c$. That will turn out to be energetically impossible, but the velocity itself is cheerfully ponderable and expressible.

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Why does the US uses imperial system and not metric system?It's just that the population understands what a mile is better than a kilometre because people have used it since birth. NASA and most scientific community still use metric system.If you know what metric equivalent is then surely you can convert it.Its better to use formulas to convert than to ask whole population to change to metric.

Similarly,most of people can understand what velocity is.Someone who can understand the difference between velocity and rapidity can learn to convert from one to other easily. Surely, it would be easier than asking a whole generation of people to understand rapidity, its difference from velocity and why does not have a upper limit?

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