Talk:Rapidity

Unnecessary minus signs
As presented today, the matrix has minus signs in front of two of its components. These minus signs disappear when the parameter &phi; is made negative because sinh is odd and cosh is even. I realize that elsewhere this minus sign may be used but that is an insufficient reason to include it here. The use of any unnecessary minus signs tends to lead to confusion when one compares the matrix operator with a comparable matrix that represents ordinary rotation in a plane by ordinary complex multiplication. In that case there is one minus sign on the counter-diagonal. Using two minus signs does not improve the analogy. There should be no minus signs when none are necessary.Rgdboer (talk) 22:44, 14 November 2008 (UTC)
 * The minus signs are there simply to follow the convention that the Lorentz transform expressed in terms of velocity usually has a minus sign: see for example Lorentz transform which uses
 * $$\begin{cases}

t' &= \gamma \left( t - v x/c^{2} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{cases}$$
 * Anyone trying to compare this article with that one will be able to make more sense if the v in that article is measured in the same direction as the &phi; in this one. --Dr Greg (talk) 13:32, 17 November 2008 (UTC)
 * Thank you for your reply and frankness Dr Greg. Does anyone else have any comments on this convention, or on its needless perpetuation ? Rgdboer (talk) 01:27, 24 November 2008 (UTC)
 * I would just add that although the minus sign is "needless" from the technical point of view, it is, in my view, "needed" from the expositional point of view to make this article as easy to understand as possible for the inexpert reader. It depends on whether it's the rapidity of A relative to B or of B relative to A (which isn't explicitly spelled out in this article, just assumed to be the same as the Lorentz transform article). --Dr Greg (talk) 12:52, 24 November 2008 (UTC)

The minus sign is not found in the use of the rapidity parameter in E.T. Whittaker's History of Electricity, neither in the 1910 edition nor in the 1953 edition. The superfluous symbols have then an evidence of needlessness.Rgdboer (talk) 23:20, 14 July 2009 (UTC)

The problem of sign can be looked at in this way: Einstein 1905 had two reference frames S and S' with S' moving away from S along the x axis with uniform velocity v. He called S the stationary frame. Looking at frame S from S' (as Einstein did) we get the minus sign and looking at frame S' from S we get a plus sign. The books tend to follow Einstein but is it not more natural to view from the stationary system S and get the plus sign? (PS Notation: Einstein had frames K and k for S and S')JFB80 (talk) 20:02, 11 October 2010 (UTC)

The issue here is that of active and passive transformations. As that article says, the terms "alias" and "alibi" have been adapted from detective fiction. The use of the minus sign means that "another place" is the frame of reference being brought to rest. Use of a plus sign refers to a new identity of a moving frame compared to the one at rest. As also mentioned in that article, the two points of view often arise from two different disciplines of study. Preference for one over the other then is a matter of taste. However, the issue has implications for the article on hyperbolic angle as the use of the minus sign here confuses the mathematical idea. See Talk:Hyperbolic angle. Comments to improve this article are sought.Rgdboer (talk) 20:55, 8 April 2014 (UTC)


 * The minus signs have been removed. See Ronald Shaw (1982) Linear Algebra and Group Representations, §6.7 Canonical forms IV: Minkowski space, usage of rapidity on page 231.Rgdboer (talk) 01:26, 11 March 2015 (UTC)

Pre-multiply or concatenate
As the article stands today, events are represented by column vectors and the transformation matrix operates on the left of this column vector. Successive transformations are then built up by placing transformation matrices to the left. On the other hand, as originally posted 11 November 2008, events were represented by row vectors so that transformation matrices may be written in the ordinary left-to-right direction. This stylistic convention of having further developments arise to the right in text is a strong tradition in European languages. Since the matrices for various &phi; form a multiplicative group, it seems more natural to apply them in an ordering like group multiplication. For these reasons, in this fundamental article in relativity theory, I recommend the left-to-right notation be adopted (restored). From the greater transparency here, editors may adapt these reasonable measures elsewhere.Rgdboer (talk) 21:28, 24 November 2008 (UTC)


 * Again, this is a matter of convention and what readers are likely to find easier to understand. (It was I who made the change.) Putting the operator on the right is something pure mathematicians, especially group theorists and other algebraists, like to do, but in my experience physicists and most non-mathematicians, as well as some applied mathematicians, put operators on the left. Technically, either way is correct; there are pros & cons for each approach. As this is an article about physics rather than pure maths, my opinion is we stick with a convention that most readers will be familiar with. --Dr Greg (talk) 18:07, 25 November 2008 (UTC)

Peculiar concluding sentence
I've removed the last line, as I can't see that "constrains the future to a quadrant of a spacetime plane" actually have any meaning in this context. Apologies if I'm missing the point. The future is constrained to a light-cone in a normal spacetime plane - an isosceles triangle for one spatial dimension. Are we now envisioning a new measure of spatial extent defined as rapidity × speed of light × time? If so, and if it's relevant, it probably warrants some explanation. As it stands, it seems a bizarre, glib, confusing line; whereas the previous line seems to be a fine conclusion. Thoughts welcome - Bobathon (talk) 20:51, 12 January 2009 (UTC)
 * In pre-relativity kinematics, sometimes called Galilean spacetime, velocity was unlimited. Any spatial separation could be crossed in a given time provided speed was grand enough. In that context the future included any event in the universe with a positive absolute time coordinate. The sentence you removed referred to the modern future, one cut off from events that are space-like with respect to the here and now origin. Thus the re-linearization associated with rapidity, which replaces the potential future t = 1 with a hyperbola constrained by its asymptotes, amounts to a re-identification of the future events.Please replace the line or help, perhaps with relativitistic wiki-links, to improve it.Rgdboer (talk) 23:46, 12 January 2009 (UTC)
 * What I'm saying is - as a target reader, ie. a reasonably intelligent person curious about rapidity - I don't really understand the point of that line. If I did, I'd have attempted a rewrite. If it is relevant at all, it's opening up another subtopic to do with the redefinition of the whole space-time plane based on rapidity, which isn't a concept introduced in the article. It was a conclusion to a subtopic that isn't there. The preceding line nicely sums up the point of rapidity. - Bobathon (talk) 08:57, 13 January 2009 (UTC)

Please Explain
Rgdboer, I am a little perplexed by your comment.

Ever since Einstein first published his special theory of relativity, people have been looking for appropriate examples, situations, and thought experiments to illustrate the unusual consequences of this theory. To my understand, the effort to come up with better examples of an existing theory is a pedagogical exercise and not original research.

Why do you consider this thought experiment to be original research? The consequences described in the story are precisely those predicted by special relativity, and special relativity is over 100 years old and is almost universally accepted. —Preceding unsigned comment added by Unitfreak (talk • contribs) 05:14, 12 July 2009 (UTC)


 * Unitfreak, in Wikipedia, the phrase "original research" has a rather broad definition that may confuse newcomers. Basically, it means presenting something, such as an argument or a piece of terminology, that does not appear the same way in published literature. See No original research for the details. I haven't seen the terms incremental velocity and relative velocity used the same way anywhere else, and if this usage is original to you, then that qualifies as "original research". If, on the other hand, you read it in a textbook or something, then it is not original research, but you should cite the source where you read it. —Keenan Pepper 21:04, 12 July 2009 (UTC)


 * Keenan, thanks for the explanation.


 * In retrospect, using the terms “incremental velocity” and “relative velocity” as tools to help the reader understand the difference between velocity and rapidity was poor judgment on my part. And as I have mentioned below, I agree that my explanation was far too long and too “textbookish” for an encyclopedia.


 * I also can see the need for this “rather broad definition” of original research in many fields of investigation. I’m sure Wikipedia’s policy developers wouldn’t want to see this project degenerate into a forum for outlandish our unsubstantiated theories and opinions.


 * However, this particular policy seems to be unnecessarily restrictive and counter productive for articles in the hard sciences. I believe the universally accepted rule for literature in the hard sciences, as stated above, is that “the effort to come up with better examples of an existing theory is a pedagogical exercise and not original research”, provided that all conclusions are shown to be immediate consequences of the existing theory.


 * Who is responsible for developing wikipedia’s policies? Writing without creativity is like eating without chewing!  —Preceding unsigned comment added by Unitfreak (talk • contribs) 05:48, 13 July 2009 (UTC)


 * Unitfreak, you may have misinterpreted both the comments about your contribution and the policy. I personally don’t believe that the policy was written to forbid “pedagogical” improvements.  The policy, as quoted below, forbids using unpublished analysis or synthesis to advance a position.  There is a difference between a “pedagogical” improvement, and a “position.”


 * “any unpublished analysis or synthesis of published material that serves to advance a position.”
 * “Do not combine material from multiple sources to reach a conclusion not explicitly stated by any of the sources.”

Definition
In particle physics rapidity is usually defined as

$$y=\frac{1}{2}\ln\frac{E+p_z}{E-p_z}$$,

where z is the beam axis. See for example eq 38.39 in

http://pdg.lbl.gov/2009/reviews/rpp2009-rev-kinematics.pdf

Maybe this definition should be presented as an alternative, especially since it is never called "longitudinal rapidity" or something else that it would make it distinguishable from the rapidity defined in the current article version. —Preceding unsigned comment added by 128.141.103.146 (talk) 14:50, 5 November 2009 (UTC)
 * For particle physics, please see pseudorapidity. Thank you for noticing the two similar terms. Using the search button often helps one find the meaning you have in mind.Rgdboer (talk) 01:34, 7 November 2009 (UTC)
 * There are the three different quantities of rapidity, "longitudinal rapidity" and pseudorapidity. -- KlausFoehl (talk) 15:44, 4 November 2010 (UTC)

Metric Tensor corresponds to acceleration?
The article says "possible values of relativistic velocity form a manifold, where the metric tensor corresponds to the proper acceleration (see above)." I don't find anything "above" in the article to explain this correspondence. What does it mean for the metric tensor to "correspond to" the proper acceleration?Flau98bert (talk) 14:33, 17 September 2012 (UTC)
 * This is a bad wording, of course ☺
 * It is not the acceleration vector which corresponds to the metric tensor. But the metric tensor on the hyperbolic "velocity space" defines the magnitude of acceleration. I mean, $$a^2 = \dot v^k \dot v^l g_{kl}$$, where $$\dot v$$ is the derivative of velocity as a function of the proper time. Incnis Mrsi (talk) 15:46, 17 September 2012 (UTC)
 * Since we agree that the article wording is not correct, I suggest we fix the wording to make it correct. It seems to me the article ought to say "The possible values of relativistic velocity form a manifold, with a metric that defines the (relativistic) relative speeds between the points of the manifold."  If someone thinks it's notable and/or important, we could add that, dividing through by dtau^2, gives the squared magnitude of the acceleration - although I personally think that is rather obvious and not particularly notable.Flau98bert (talk) 16:57, 1 October 2012 (UTC)
 * If metric tensor, then relative velocities must be infinitesimal. Otherwise, metric (mathematics) between arbitrary points is the rapidity. Incnis Mrsi (talk) 17:49, 1 October 2012 (UTC)
 * Hmm. This whole section ("In more than one spatial dimension") seems to be an attempt to summarise the Rhodes and Semon reference, but the summary doesn't seem to make much sense as currently worded. The reference makes no mention of proper acceleration at all. The "rapidity space" defined in the reference is a new invention of that article paper. Is it a concept that has been taken up by other sources since its publication? Maybe the best course of action would be to delete the whole section. --  Dr Greg   talk  18:59, 1 October 2012 (UTC)
 * A "rapidity space"? Do you really see this in the article? Or, maybe, on the talk page (except your own posting, of course ☺)? Incnis Mrsi (talk) 19:06, 1 October 2012 (UTC)
 * Not in the article. In the Rhodes and Semon reference (which seems to be what that section of this article is attempting to describe). --  Dr Greg   talk  20:17, 1 October 2012 (UTC)
 * I see "relativistic velocity space" there. Do you perceive something wrong with this, or ? Maybe, very classical books of fathers-founders lack "velocity spaces", but it is perfectly comprehensive. I knew such things many years ago, it's trivial. Incnis Mrsi (talk) 10:11, 2 October 2012 (UTC)

Difference between rapidity w and Minkowski angle φ (=iw)
At present the article is quite confused over this difference. It needs correction.JFB80 (talk) 19:19, 18 January 2013 (UTC)
 * Perhaps you will find clarification in the section hyperbolic angle where the use of imaginary numbers in the context of hyperbolic angles is explained.Rgdboer (talk) 21:20, 18 January 2013 (UTC)
 * Yes I am aware of that - it is just school maths. What I am saying is that the article is unclear.JFB80 (talk) 21:51, 18 January 2013 (UTC)JFB80 (talk) 22:13, 18 January 2013 (UTC)
 * Later: I misunderstood you since you referred to hyperbolic angle which is on the use of imaginaries with hyperbolic functions. Instead you were probably wanting to draw attention to the definition of hyperbolic angle. This term though was not used by Minkowski, Robb etc and its use here has given rise to the inappropriate notation of denoting rapidity by the angle variable φ which Klein, Sommerfeld and others used for imaginary Minkowski angle. Robb, Varicak and others used w for rapidity which is appropriate as rapidity is a velocity so I think it should be used in this article.JFB80 (talk) 17:04, 19 January 2013 (UTC)
 * The units of rapidity are not meters per second like velocity. Rapidity is a real number which is interpreted as a hyperbolic angle in contexts using Lorentz transformations. It forms a one-parameter group, but it is not a velocity. For a deeper history of the rapidity concept see versor and its links. The interpretation of a mathematical structure as a mathematical model is an evolutionary happening. The terminology of Sophus Lie and continuous groups became standard.Rgdboer (talk) 01:37, 20 January 2013 (UTC)
 * The viewpoint of this article is interesting but it is a purely mathematical one and overlooks physical interpretations. It is astonishing that there occurs the statement to the effect that although in English rapidity has the meaning of speed, in physics it means something quite different. To the contrary rapidity is of fundamental importance as a measure of relativistic velocity
 * Of course it is not a normal velocity but neither is hyperbolic angle a normal angle. Both are measures of the respective quantities. Further rapidity is only dimensionless because it depends on v as a fraction of c. It can easily be considered in an equivalent form which is a velocity. All you have to do is multiply by c and get it as c artanh(v/c) which approximates v when v<<c.
 * I added early references to Varicak and Borel which take the point of view of physics.JFB80 (talk) 18:41, 27 January 2013 (UTC)

Hyperbolic velocity
The following text was removed:
 * A more direct relation with ordinary velocity comes from rescaling rapidity so as to define hyperbolic (or relativistic) velocity $V$ as $cw$. Then hyperbolic velocity reduces to ordinary velocity at speeds small compared with light.

The introduction of new terms hyperbolic velocity and relativistic velocity in this article without reference and without contribution to understanding of the topic calls for discussion here. — Rgdboer (talk) 02:47, 6 March 2016 (UTC)


 * This quantity was used by Varićak 1910 although he gave it no name. It is natural and almost essential to use it in physics because rapidity is exceedingly small at ordinary speeds. Scaling it up like this makes it coincide with velocity when this is small compared with that of light. It could be called just 'scaled rapidity' but, because it has a clear physical meaning, it is more descriptive to use one of the terms I have used.JFB80 (talk) 22:01, 6 March 2016 (UTC)


 * So which book or paper did you get the names from? --  Dr Greg   talk  23:42, 6 March 2016 (UTC)


 * I didn't get the names from any book or paper. I first used the name 'hyperbolic velocity' in a 1992 paper and I have since then used it in other conference papers. As I said above, Varićak used it but didn't give it a name. Also Borel 1913 used it implicitly when he said that in Special Relativity the kinematic space is hyperbolic with negative radius of curvature equal to the velocity of light. Again he did not give a name for relativistic velocity. But it needs a name being different to ordinary velocity. So I just used an obvious name. Now for over 20 years no-one to my knowledge has talked about it being content to use rapidity which makes the theory less close to the classical form and less physical.  JFB80 (talk) 22:25, 7 March 2016 (UTC)


 * How about mentioning explicitly in the article that the quantity has no name in the literature and that it in this article is called such and such. This could be put in a footnote (distinguished from ordinary footnotes by a label such as "nb"). YohanN7 (talk) 15:31, 9 March 2016 (UTC)


 * Of course I can't change past writings but I could do something like that in future. Incidentally checking up I find that on my first use in 1992 I called it scaled rapidity (unquestionably correct) changing to hyperbolic velocity after 1999. May I ask who has the right to give names in your opinion? JFB80 (talk) 20:24, 9 March 2016 (UTC)


 * In Wikipedia articles no-one. It is very different to define terminology locally in order to be able to talk about things in a well-defined manner. In Minkowski space there is an example of this, the "Minkowski inner product". What is intended is the content of footnote #3 there (nb 3). YohanN7 (talk) 08:54, 10 March 2016 (UTC)


 * That is not what I asked. It was said that the quantity I talked about was not given a name in the papers referred to. Who then has the right to give it a name (not only in Wikipedia)? I gave it a name (and not in Wikipedia). I am sure I am not the only one to do this. It is being done all the time. — Preceding unsigned comment added by JFB80 (talk • contribs) 17:43, 10 March 2016 (UTC)

You asked who gets to name things. The subject title is nomenclature, in this case scientific terminology as exemplified by the international scientific vocabulary. New terms are subject to naming conventions as in a systematic name. The encyclopedia only reports on usage as found in primary and secondary reliable sources. Notability of terminology is determined by standard metrics on educational and scientific texts. — Rgdboer (talk) 03:28, 11 March 2016 (UTC)


 * Thank you for this information. I am uncertain where to go from here. The sources you quote are mainly referring to the naming of established ideas of wide circulation. What happens to new ideas? In this connection I think there is a problem with Wikipedia in that an encyclopedia traditionally makes available established knowledge and this is the official policy of Wikipedia, But being online and easy to edit and change, it is ideally suited for keeping up to date with new ideas. And this does happen. For example gyrovector theory is very recent and almost entirely the work of one person. It cannot be called established but it treated as such in Wikipedia: it has its own article and is used in other articles, You yourself have introduced interesting ideas related to hyperbolic angle which I never saw in textbooks and could not be called standard. On the other hand the very classical ict relativity interpretation of Minkowski does not appear at all being considered old-fashioned and out of date. JFB80 (talk) 15:56, 12 March 2016 (UTC)

Thank you for your thoughts. On new terms see neologism which is the target of the redirect at coining (linguistics). As for hyperbolic angle and its applications, references have been given and the material is classical though neglected in some study sequences. The application in this article provides a motive to investigate the concept more fully, such as in the area of a hyperbolic sector. But the fact that your textbooks missed some applications does not mean they are not standard. Your idea to accelerate the use of neologisms in this encyclopedia is contrary to policy and would tend to discredit the project as going off in its own direction rather than reporting on usage as is the policy now. — Rgdboer (talk) 23:26, 12 March 2016 (UTC)

Non-standard trig function
$${\rm artanh}$$ is uncommon. Usually the hyperbolic arc tangent is either $${\rm arctanh}$$ or $${\rm atanh}$$. Please provide a reference where $${\rm artanh}$$ is commonly used. iou (talk) 18:59, 16 January 2017 (UTC)
 * Perhaps slightly less common, but certainly not uncommon. See our article Inverse hyperbolic functions and Our-Friend-Google:
 * {| class="wikitable" style="text-align: center"

! Google !! Scholar !! Books
 * arctanh
 * 7540
 * 5030
 * artanh
 * 1700
 * 3460
 * atanh
 * 3980
 * 5960
 * }
 * - DVdm (talk) 19:47, 16 January 2017 (UTC)
 * 5960
 * }
 * - DVdm (talk) 19:47, 16 January 2017 (UTC)
 * - DVdm (talk) 19:47, 16 January 2017 (UTC)

Recent reverts
I deleted a section beginning with
 * In more than one spatial dimension rapidities lie in a hyperbolic space having unit radius of negative curvature...

Rapidities aren't confined to a space of unit radius. Velocities are (with $$c = 1$$). One has $$-\infty < w < \infty$$. The rest, copyedited for errors, might belong in Velocity-addition formula, but the relevant material is there already.

It was all promptly reverted with motivation that I should provide a reference claiming the incorrectness of the content. To be honest, finding reliable sources confirming that $$2+2=99$$ is wrong are hard to come by. You might find one saying $$2+2=5$$ is wrong with a bit of luck. No, I rely mostly on references giving what is right and take anything else as false. This should suffice.

As for the sentence in the lead deleted by me, a thread just above highlights what is going on. Even if someone 107 years ago wrote $$V = cw$$, and even if the contributor wrote a conference paper 1992, it is not notable and to my knowledge not used to the extent that it warrants mention in the lead of this article.YohanN7 (talk) 10:07, 18 January 2017 (UTC)


 * It seems to me that you are systematically deleting more or less everything that you havn't written yourself. For example you deleted the law of addition of rapidities under 'rapidity' (deleting several different statements at first without source and then with a vague mention of Landau & Lifschitz) and reinstating the very same law of addition under 'velocity-addition' which is an article you write yourself. The law you deleted is quoted by many sources (some old but does that matter? Einstein is also old now) and I feel pretty certain that Landau and Lifschitz would not say differently because they do not write nonsense. JFB80 (talk) 20:09, 18 January 2017 (UTC)


 * I deleted it because you say
 * In more than one spatial dimension rapidities lie in a hyperbolic space having unit radius of negative curvature...,
 * which is wrong. This very same article (not a line written by me) says $$-\infty < w < \infty$$. Then you write
 * ''... two rapidities must be added sequentially "head to tail".
 * No, that wont work either. If you want to write something good about this, you have to make clear that the space of rapidities has two addition operations, one inherited from the Lie algebra (ordinary addition) and one induced by the map sending rapidities to velocities and pulling back the velocity addition formula. Until something at least not even wrong is in place, the section and your OR in the lead will have to go. YohanN7 (talk) 08:00, 19 January 2017 (UTC)


 * It is you who is not understanding something which is quite simple and it is not necessary to mystify the subject by bringing in Lie algebras and pullbacks. What I say is perfectly correct since it is well known to geometers (but apparently not by you) that an infinite hyperbolic space can be represented in the interior of a finite sphere. I will have to expand this entry to explain. May I suggest that you are not quite so fast in deleting other people' contributions. Wikirules say that when there is disagreement there should be discussion to reach a consensus. You do not keep to that but delete, delete, delete even when you have little justification. JFB80 (talk) 19:18, 19 January 2017 (UTC)


 * It is good that you finally get the message that you'll have to expand, explain, and correct at least the worst errors (though you'd never recognize errors for errors). But I object to that reference to Lie algebras is mystifying. After all, this article identifies rapidities with the coordinates on the Lie algebra. So for the internal consistency, and for easiest access for the reader, keep that notion. Then "pulling back the metric" is exactly what geometers do when they "represent" a space. See the notion of isometry (Riemannian geometry). This is important since Riemannian geometry is precisely the study of properties that are invariant under isometries. The representation space of unit radius (with the appropriate metric) is precisely the space of velocities. Then, again, the appropriate place to do this is really velocity addition formula where the velocity space arises. YohanN7 (talk) 09:46, 20 January 2017 (UTC)


 * You accuse me of not understanding "something which is quite simple". Well, I don't think Riemannian geometry is simple. It takes the equivalent of a couple of graduate courses to get there. You obviously think it is simple. But you object violently when I suggest to you to use standard terminology because it is "not necessary to mystify the subject". This is mystifying indeed.


 * Then don't give the impression that I am mass-deleting your contributions. It is, as far as I can remember, this one and a particularly goofy one at velocity addition formula (one over which you are still to this day furiously arguing over with editor User:DVdm on the talk page). You might have noticed that other editors as well have disagreed with your edits, e.g. at Minkowski space where you made outrageous claims with reference to something being obvious. The reaction on your part has been the same each time. You promptly go to edit-warring, furiously (like here, "What I say is perfectly correct since...") defending, even refusing to acknowledge grammatical errors.


 * You also misunderstand pretty badly who has the burden of proof (read verification). You have the burden of proof. You should supply a reference saying rapidity space has radius one. You should supply reference for "adding head to tail" being correct (and also explain what it means because the ordinary head-to-tail operation is precisely ordinary vector addition). You should explain what your prehistoric concept of equipollence has to do with this. There is no way that you can insert your interpretations in the articles and lay the burden on others to disprove them.


 * You know, when your edits are questioned, recognize that there is at least a slim chance that they are suboptimal. YohanN7 (talk) 10:18, 20 January 2017 (UTC)

Composition of rapidities
I have not examined the dispute in detail, but in all fairness it would be interesting to discuss rapidity for boosts in different directions somewhere on WP, provided it is clear and correct. Stated as a problem: if you have
 * $$\boldsymbol{\zeta}_1 = \mathbf{n}_1\tanh^{-1}\beta_1\,, \quad \boldsymbol{\zeta}_2 = \mathbf{n}_2\tanh^{-1}\beta_2\, , \quad \boldsymbol{\zeta} = \mathbf{n}\tanh^{-1}\beta$$

then how do ζ1 and ζ2 combine into ζ? Alternatively, is there a function z such that
 * $$z(\boldsymbol{\zeta},\boldsymbol{\zeta}_1,\boldsymbol{\zeta}_2) = 0 \quad ? $$

I am guessing the dispute is related to the content of this:
 * Translation:On the Non-Euclidean Interpretation of the Theory of Relativity by Vladimir Varićak

but we need modern books with references to the original sources. That would be stronger for WP. 'M'&and;Ŝc2ħεИτlk 11:34, 20 January 2017 (UTC)


 * Since you know $$\boldsymbol \beta$$ in terms of $$\boldsymbol \beta_1, \boldsymbol \beta_2$$ (which are known in terms of $$\boldsymbol \zeta_1$$ and $$\boldsymbol \zeta_2$$ respectively) from Velocity-addition formula, use $$\boldsymbol{\zeta} = \tanh^{-1}\boldsymbol \beta \quad( \equiv \mathbf{\hat n}(\boldsymbol \beta_1(\boldsymbol \zeta_1), \boldsymbol \beta_2(\boldsymbol \zeta_2))\tanh^{-1}\beta(\boldsymbol \beta_1(\boldsymbol \zeta_1), \boldsymbol \beta_2(\boldsymbol \zeta_2)))$$. YohanN7 (talk) 11:58, 20 January 2017 (UTC)


 * As you know, there is a rotation involved here as well. YohanN7 (talk) 12:25, 20 January 2017 (UTC)


 * I know of the velocity addition formula and rotation in terms of relative velocities, what I mean are closed form expressions (or at least a relation between them), which are cited in the literature.
 * As an extra aside, it may be interesting to connect angles inside a hyperbolic triangle traced out by the rapidities to the rapidity composition itself. (I do not mean to add the "rapidity vectors" above according to vector addition).
 * I am not obligating anyone to do anything, this is purely out of interest. I'll try to find it in the literature. 'M'&and;Ŝc2ħεИτlk 17:49, 20 January 2017 (UTC)
 * I would like to reply to the second writer Maschen first. The law of combination you look for is precisely the equation which was recently deleted by YohanN7. The equation was stated by Varicak (1910) and Robb (1911) who are the original sources. Later it was discussed by Borel (1913), Silberstein (1914) and some others mentioned in Scott-Walter's (1999) article. It was also used in atomic physics - Smorodinski (1964). All except the last are referenced in the article and are available online from Wikipedia. I don't know of other modern published papers and books which talk about the equation. The accepted standard books make no mention of it. Ungar has numerous papers and books on the hyperbolic view but I am unsure if he anywhere talks about this equation as he has his own approach using 'gyrovectors'. You could look at the following paper which gets close but does not actually mention the equation: Am. J. Phys., Vol. 72, No. 7, July 2004 J A. Rhodes and M D. Semon 946. JFB80 (talk) 20:18, 20 January 2017 (UTC)


 * Original (= primary) sources are not suitable for references on wikipedia (they are as external links, this is not the same thing). They are for historical reasons, but see below. I am very aware of Ungar also. WP needs secondary sources. At the top of my head, Sard may be the closest modern (and secondary) reference relevant to the dispute. What you added is not what I am looking for. I mean relating rapidities and directions of boosts (given by unit vectors) as shown above.
 * There were a few errors (or at least things which make no sense). E.g.
 * "Note that the two rapidities must be added sequentially "head to tail""
 * which makes it look like the rapidities are vectors (they are not), you really mean hyperbolic rotations through hyperbolic angles, and for two angles subsequent hyperbolic rotations do not commute because boosts in different directions do not commute.
 * "The Thomas precession is equal to minus the angular deficit of a triangle, or to minus the area of the triangle."
 * which seems to be wrong, when you switch the order of the boosts from ζ1 followed by ζ2 to ζ2 followed by ζ1, the rapidities of each composite boost are each in a different direction. This angular deficit is the Wigner or Thomas rotation between the frames moving with the composite velocities. For accelerated motion, the continuously changing relative velocity leads to rotational motion, which is the Thomas precession. M'&and;Ŝc2ħεИτlk 08:56, 21 January 2017 (UTC) Amended M'&and;Ŝc2ħεИτlk 16:14, 22 January 2017 (UTC)


 * This formula alone (which you did add)
 * $$\cosh w=\cosh w_1\cosh w_2 +\sinh w_1\sinh w_2 \cos \theta $$
 * may be part of what I am looking for in terms of relating magnitudes and the angle between the rapidities. This should be in Sard, will check later. 'M'&and;Ŝc2ħεИτlk 09:49, 21 January 2017 (UTC)


 * (1) I think it is far better to quote the original sources, even if old, than to give your 'own-research' version of it as you started doing. I cannot believe that original source references are not usable (please supply exact Wikilaw) Einstein (1905) is of the same period. Mathematical proofs and equations loose nothing by being old. I think the point here is that original sources are not always easy to read so a current explanation is sometimes desirable. This can easily be arranged by someone understanding the original and re-presenting the proof in a pop-up as YohanN did in velocity-addition. The proof is not difficult. (2) Just like ordinary magnitudes, rapidities can indeed be interpreted as vectors (in hyperbolic space) having a certain direction. Unlike in ordinary Euclidean space where vectors can be added either "head to tail" or by the parallelogram rule, in hyperbolic space only the first method is permitted. Silberstein 1914 explains this very well. It is due to the failure of 'equipollence' which property YohanN7 took such trouble to delete when I mentioned it in the velocity-addition article. The addition formula gives the magnitude of the resultant. If you add the vectors by parallel transport you will get the Thomas rotation which can be related to the area of the triangle as I say. (3) No I am not talking about hyperbolic rotations. It is possible to think in this way but if you are not careful you get a result which is origin-dependent. You also have to bear in mind that boost and rotation are ambiguous terms (boost+rotation or rotation+boost). This is not so with vectors.JFB80 (talk) 16:16, 21 January 2017 (UTC)
 * You say 'Original (= primary) sources are not suitable for references in Wikipedia' I don't see anything about this in Wikipedia:Citing Sources. Is it just your own opinion? JFB80 (talk) 04:58, 22 January 2017 (UTC)


 * Above I should have said primary sources can be referenced for historical reasons.
 * Otherwise, the link says at (time of posting):
 * "Policy: Unless restricted by another policy, primary sources that have been reputably published may be used in Wikipedia, but only with care, because it is easy to misuse them.[4] Any interpretation of primary source material requires a reliable secondary source for that interpretation. A primary source may only be used on Wikipedia to make straightforward, descriptive statements of facts that can be verified by any educated person with access to the primary source but without further, specialized knowledge. For example, an article about a novel may cite passages to describe the plot, but any interpretation needs a secondary source. Do not analyze, evaluate, interpret, or synthesize material found in a primary source yourself; instead, refer to reliable secondary sources that do so. Do not base an entire article on primary sources, and be cautious about basing large passages on them. Do not add unsourced material from your personal experience, because that would make Wikipedia a primary source of that material. Use extra caution when handling primary sources about living people; see WP:Biographies of living persons § Avoid misuse of primary sources, which is policy."
 * So there you go. The point of adding secondary sources is because they offer independent interpretations and formulations of the subject. The people who first formulate a theory (or make early contributions) do not know everything, nor are automatically correct, and there are gaps to fill in by other scientists. They are not always the best places to refer to about the subject, because interpretations, formulations, terminology, etc. changes, usually for the better (as more scientists in the area study the subject, misunderstandings which were not known before can be corrected). 'M'&and;Ŝc2ħεИτlk 16:13, 22 January 2017 (UTC)
 * That may be appropriate for descriptive subjects like history and medicine but I don't quite see how it necessarily applies to mathematics. If you get the idea of a proof from the original source what is to prevent you from reproducing it (quoting the original source of course)? I have usually found that the original proof is more clearly explained than later versions because it is the insight that matters. (An extreme example would be Fermat's Last Theorem where the proof Fermat must have had would have been incredibly simple compared with Wiles' more recent formidable proof using higher transcendental functions.) JFB80 (talk) 20:03, 22 January 2017 (UTC)


 * You reason precisely like editor Chjoayagame. He relied on primary sources and the oldest possible textbooks, and for the exact same reasons you state. He also managed to get it all wrong, down to every sentence and even every single word. Fortunately, he is now topic-banned. As for sources, see Primary Secondary and Tertiary Sources. This discussion is closed on my part. YohanN7 (talk) 07:35, 23 January 2017 (UTC)


 * By your logic JFB80, all scientists and engineers should use Newton's principia for mechanical calculations, as well as treatises/papers/books written by Euler, Lagrange, Laplace, Hamilton, Cauchy, Fourier, Navier, Stokes, Zhukovsky, Prandtl, Reynolds, etc. for particle and/or continuum mechanics. It doesn't matter to you that their work is 1-3 centuries old.
 * I'll try and find the composition of rapidities (in any direction) myself in secondary sources, regardless of what you say. 'M'&and;Ŝc2ħεИτlk 10:44, 23 January 2017 (UTC)
 * No I am not saying that. By all means use the current formulation. What I say is, go to the original source if you want to get the thinking clear. I would be interested to know if you do find a recent source for composition of rapidities. I tried but did not succeed. It seems to be something which was forgotten about after the early papers quoted. Silberstein (1914) is a secondary source and I think quotable. JFB80 (talk) 16:21, 27 January 2017 (UTC)

Comments
The section In more than one space dimension has versions done by two editors. The one by JFB80 includes the link hyperboloid model which is pertinent. The version by YohanN7 introduces the Lie algebra of the Lorentz group with no mention of bivector (complex). Though the Lorentz group is a Lie group, presuming the reader can follow Lie theory is a stretch. Therefore I prefer the JFB80 version. Furthermore, YohanN7 asserted on January 25 that I removed something which I did not touch. Derogatory comments in an edit description are offside, and a falsehood is easily confirmed by History comparisons. We work here with just ourselves and software, with hope for uplifting but sometimes subject to degradation. Such is the state of man on earth.Rgdboer (talk) 01:05, 27 January 2017 (UTC)


 * Derogatory? You did here remove the origin (reason) that rapidity space has the geometry it has. In your edit you simply state out of the blue that the geometry is hyperbolic. I don't think giving that as a reason to revert is derogatory.


 * If you assert problems with mentioning the Lie algebra, then you have to rewrite the whole article: From Rapidity;
 * ..., Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra.


 * Thus my version is in line with the rest of the article. If you think the incorrect version by JFB80 is better, then gather support for it. The first blunder (first sentence) in it is confusing raipdity space with velocity space. It flat out contradicts
 * ...and so the interval $− c &lt; v &lt; c$ maps onto $−∞ &lt; w &lt; ∞$
 * from the lead. Some other blunders are stated above.
 * If did add mention of the hyperboloid model with a link to all gory details. YohanN7 (talk) 10:25, 27 January 2017 (UTC)


 * As an aside, I can inform you that while dealing with biquaternions (with regard to your request for bivectors) in the context of the Lorentz group, while probably perfectly consistent, is far from mainstream. Like it or not. Involving that would certainly spook off 99% of the readers, whereas at least every math and physics undergraduate past their sophomore year will be familiar with Lie algebra. Biquaternions is a dead end, since it doesn't generalize to other groups or other spacetime dimensions, and introduces an unnecessary structure, particularly for the purposes of this article. YohanN7 (talk) 10:44, 27 January 2017 (UTC)


 * Thank you Rgdboer for giving some support for my version. I more or less gave up because YohanN7 works and deletes so fast it is quite impossible to keep up, let alone have a proper discussion as should be the practice in Wikipedia. Now he has supplied a version with sophisticated mathematics quite unsuitable for physicists and apparently mostly own research not supported by source references. He is completely wrong in his criticism of my version which was his justification for deleting everything and replacing it with his own version.
 * For example he thinks that my first sentence: In more than one spatial dimension rapidities lie in a hyperbolic space having unit radius of negative curvature and they may be combined by the hyperbolic law of cosines. - is nonsense. But that must be because he doesn't understand the negative curvature of a hyperbolic space or the vectorial nature of rapidities.
 * Then he objects to: Note that the two rapidities must be added sequentially "head to tail"; it is not possible to combine rapidities starting from a common origin as can be done with velocity vectors in Euclidean space. This is the failure of equipollence in hyperbolic space. He didnt accept failure of equipollence and has deleted remarks on it elsewhere, etc etc. But what can be done? He is deleting and rewriting everything in his own style without consultation. JFB80 (talk) 21:00, 27 January 2017 (UTC)

Thank you both for responding. Look at that History comparison YohanN7 since it shows more than a reversion, there was an insertion you made, and I did not remove it. Please stop perpetuating the falsehood.

It should be recalled that hyperbolic geometry was developed in the nineteenth century and that it is a metric geometry not applicable to spacetime. But A Treatise on Electricity and Magnetism was using t,x,y,z variables so some manifold was needed to make sense of it. Also from the nineteenth century came quaternions and biquaternions of Hamilton. In the struggle to achieve mathematical physics another contribution was hyperbolic quaternions. The stampede in the twentieth century was led by Ludwik Silberstein using biquaternions. Admittedly, the contributions of Sophus Lie and Elie Cartan are huge, but this article should be pegged to the likely reader, and he or she needs to learn examples of continuous groups. Advanced terminology is not helpful to the novice reader. Furthermore, disdain for mathematics of the past is unjustified.Rgdboer (talk) 00:28, 28 January 2017 (UTC)
 * There is of course Klein's book from 1872 treating hyperbolic geometry projectively.JFB80 (talk) 05:17, 28 January 2017 (UTC)

Yes, my point is that hyperbolic geometry and quaternions preceded spacetime relativity and enabled the physical theory. They amount to "off-the-shelf" technology that the conceptual engineers apply to physics. While there is no doubt that rapidity is a Lie group parameter, that feature does not define it. As we discuss a particular section aiming to expand the concept, some historic aspects have been brought up to highlight several decades of development that were necessary for the birth of relativity. For instance, non-Euclidean geometry emerged in 1870s as instantiated by models, but only in the 1920s did Klein articulate the boundary of velocity space as a Cayley absolute. See Felix Klein, M Ackerman translator (1979) Development of Mathematics in the 19th Century, p 138, Math Sci Press. Furthermore, linear algebra developed over the same period, and in our instance that includes the squeeze mapping. All fields converge in this encyclopedia project, so there is no separation between the Math dept and the Physics dept as on campuses. Rapidity is on the boundary of the two, so the challenge in this article is similar to others. Looking at the history of ideas clarifies essentials, and in this case of hyperbolic angle it is a transcendental function in the lens. Calculus teachers would do well to explain how these transcendental functions arise, and draw out the hyperbolic sector when they present the dented trapezoid as producing the natural logarithm. If these subjects were properly laid out the fantastical aspects of special relativity would be mundane.Rgdboer (talk) 02:28, 29 January 2017 (UTC)
 * Certainly, as you say, some of the relevant mathematical ideas preceded relativity but initially they were not applied, mainly because physicists were not familiar with them or that kind of mathematics. The attempt by Varicak and a few others to introduce Non-Euclidean geometry into Special Relativiy received little attention and was forgotten after General Relativity appeared. And, as you said, biquaternions (and also complex quaternions) were already used by Silberstein 1910, 1914 but no-one apparently followed up until recently. What you say about Klein and velocity space is very relevant to our present topic although I don't find it stated exactly as you say (it is not p.138 in my German reprint by Chelsea 1956). The ideas of hyperbolic angle and squeeze mapping look very interesting and relevant. They seem to have been reconstructed mainly by yourself from scattered sources. JFB80 (talk) 21:01, 29 January 2017 (UTC)

No, the flat spacetime arrangement was put together by the Swearingen Sisters in Austin Texas when Alexander Macfarlane joined G. B. Halsted at the University of Texas in 1885. Reference to Minkowski’s pompous pronouncement in 1908 is due to eminence of the German schools compared to American, Canadian, and English communities. Minkowski evidently caught wind of "The Great Vector Debate" in the pages of Nature (journal) and elsewhere, and his enunciation has been recalled as the watershed. — Rgdboer (talk) 19:35, 31 January 2017 (UTC)

Concerning the Klein reference, the Ackerman translation published by Robert Hermann in 1979 was cited for the page 138. Checking the bookshelf here at QA 26 K 6 there is the 1950 Chelsea reprint of the German original. The cited passage is from chapter four "Herausbeitung einer rein projektiven Geometrie" where he cites Cayley on general projective measure. So the page number is 150 in Chelsea, volume 1, Entwicklung der Mathematik. The equation given in dx, dy, dz, and dt is in homogeneous coordinates and is equivalent to v2 = 1, boundary of velocity space, and Cayley absolute in the passage. — Rgdboer (talk) 00:26, 2 February 2017 (UTC)


 * Yes it is on p.150 also in my 1956 reprint. JFB80 (talk) 16:36, 2 February 2017 (UTC)

I'm new so sorry if I don't do this right, but trying to read this article and the one for proper velocity, I got very confused until I realized they both use opposite variables for the two quantities. w and η. —Gravitative (talk) 12:07, 11 April 2017 (UTC)

Inconsistancy between pages
Rapidity and Celerity are closely linked, but according to the wiki pages for both, both use w to represent each other. The Celerity page refers to rapidity and eta so this page should use eta too.

Rapidity in terms of conservation of energy
@User:DVdm: thank you for your dedication to keeping Wikipedia top notch. In this case, you might have used   markup instead of the big undo axe, but I shall seek out one or more sources as you request. All the best — Q uantling (talk &#124; contribs) 17:33, 23 January 2019 (UTC)


 * no worries, when you find the sources, you can simply hit the undo-button and slide the citations in. Happy hunting! - DVdm (talk) 18:08, 23 January 2019 (UTC)

Extra dimensions
The following was removed as beyond the scope to Rapidity:

In more than one spatial dimension
The relativistic velocity $$\boldsymbol \beta$$ is associated to the rapidity $$\mathbf{w}$$ of an object via


 * $$\mathbf w = \boldsymbol{\hat \beta}\tanh^{-1}\beta, \quad \boldsymbol \beta = \boldsymbol \beta_1 \oplus \boldsymbol \beta_2,$$

where $$\boldsymbol \beta_1 \oplus \boldsymbol \beta_2$$ refers to relativistic velocity addition and $$\boldsymbol \hat \beta$$ is a unit vector in the direction of $$\boldsymbol \beta$$. This operation is not commutative nor associative. Rapidities $$\mathbf w_1, \mathbf w_2 $$ with directions inclined at an angle $$\theta$$ have a resultant norm $$w \equiv |\mathbf w|$$ (ordinary Euclidean length) given by the hyperbolic law of cosines,


 * $$\cosh w = \cosh w_1\cosh w_2 + \sinh w_1\sinh w_2 \cos \theta.$$

The geometry on rapidity space is inherited from the hyperbolic geometry on velocity space via the map stated. This geometry, in turn, can be inferred from the addition law of relativistic velocities. Rapidity in two dimensions can thus be usefully visualized using the Poincaré disk. Geodesics correspond to steady accelerations. Rapidity space in three dimensions can in the same way be put in isometry with the hyperboloid model (isometric to the $3$-dimensional Poincaré disk (or ball)). This is detailed in geometry of Minkowski space.

The addition of two rapidities results not only in a new rapidity; the resultant total transformation is the composition of the transformation corresponding to the rapidity given above and a rotation parametrized by the vector $$\boldsymbol \theta$$,


 * $$\Lambda = e^{-i\boldsymbol \theta \cdot \mathbf J}e^{-i\mathbf w \cdot \mathbf K},$$

where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule


 * $$[K_i, K_j] = -i\epsilon_{ijk}J_k,$$

where $$J_k, k = 1, 2, 3,$$ are the generators of rotation. This is related to the phenomenon of Thomas precession. For the computation of the parameter $$\boldsymbol \theta$$, the linked article is referred to. Rgdboer (talk) 04:38, 31 October 2022 (UTC)
 * Rgdboer (talk) 04:38, 31 October 2022 (UTC)