Talk:Velocity-addition formula

What a mess

I've taken SR and GR courses. My impression of this article is that it is a mess. Here's three examples: 1) the vectors u and v are used in a formula, but not defined!! 2) A ball dropping (vx=0) on a ship (vx=k) is used as the example of the addition of velocities! 3) A "fly" is mentioned and yet no example scenario using a fly is given. ... I suggest that first velocity (speed) in one dimension be handled (two parallel vectors), followed by the general case of three objects with different x,y&z motions. If three objects, A, B, & C are used in an example, why not determine the velocities of each of the other two relative to the third? (B,C) relative to A, (A,C) relative to B and (A,B) relative to C? I'd also suggest that both the case where B & C both have positive velocity relative to A, and the case where B's velocity is of opposite sign compared to C's for both the 1-D case and the 3-D case.71.29.173.173 (talk) 17:36, 12 July 2016 (UTC)

The article needs much reorganizing but is not a mess for the reasons you give. (1) Vectors u and v require an article on elementary physical vectors but there is no such article in WP. (2) Galileo's ball dropping is the correct historical origin of velocity addition. (3) I don't see any mention of a fly - it used to be in the article and the location of the fly was Galileo's ship. There are extended discussions on Bradley, Fizeau and Doppler which are well written but out of place here and are only incidental. There is no mention of the work of Einstein and Sommerfeld which was central. The derivations of the addition formula are well done but there could be a mention of their origins. The vector versions using the special addition notation which follow have not been well established in the literature and could even be wrong - who knows? There is a section on hyperbolic geometry but it does not connect up or explain anything. Hyperbolic geometry is highly relevant but should connect up directly with the simple relativistic addition law using rapidity and the triangle law of velocities in hyperbolic space. The article needs a lot of work but I am afraid I cannot volunteer. JFB80 (talk) 20:24, 13 July 2016 (UTC)

I swatted the fly. JFB80, why would you say applications are out of place here? YohanN7 (talk) 10:07, 15 July 2016 (UTC)

A lot of the clutter was my doing, thanks to Yohan for tidying up. M∧Ŝc²ħεИ_τlk 14:08, 15 July 2016 (UTC)

There are already good articles on Fizeau experiment, aberration and Doppler Effect. Duplicating them here is unnecessary and out of place. A brief reference to those articles is enough. It is also confusing because the reader gets the impression that the work of these people somehow led to the relativistic addition law whereas it was Einstein (1905) who did it all in the first place (his name hardly mentioned here). It was he not Doppler who derived the relativistic Doppler formula. Bradley had no connection with relativistic velocity addition and Fizeau's experiment only by hindsight. JFB80 (talk) 17:24, 15 July 2016 (UTC)

Not agreed. The application subsections in this article are short, and they balance other people's investigations on the subject. They don't have to actually derive the velocity addition law in general. If people want to trim the applications, fine, but preferably no complete deletion, replaced by links. The article edit history and this talk page shows you seem intent on biasing everything for Einstein. M∧Ŝc²ħεИ_τlk 06:45, 16 July 2016 (UTC)

There is no question of biasing in favour of Einstein, I am just saying you should give the correct origin of the formulae and not mislead your readers. If you have taken a normal course in Special Relativity you will know that the addition formula a well as the relativity Doppler formula were first derived by Einstein in his 1905 paper. But you, and others who constructed this article, were content to not even mention his name. It was I who put in a basic reference to him as you seem to have found out. The way you have presented it a reader would think that Doppler himself derived the relativistic Doppler formulae! JFB80 (talk) 18:22, 16 July 2016 (UTC)

I don't know how Einstein's paper got left out all together in the first place. But I do know why I didn't use it for inline citations (it is because it is a primary source). I see no harm however in having it alongside secondary sources. On the other hand (I learnt the hard way), it is dangerous to claim that any formula was first derived by Einstein (or anyone else for that matter). Such claims requires secondary sources, and those claims are also frequently incorrect. Several SR formulas were discovered before Einstein, but only with Einstein did they get their correct setting and interpretation. At any rate, the article is now oozing Einstein, put a picture in there as well. It would be nice if the general referencing style (citation templates) for this article were observed.YohanN7 (talk) 13:19, 18 July 2016 (UTC)

Ok that's a step in the right direction. But I don't know what you mean by saying that several SR formulas were discovered before Einstein - can you name one apart from Fizeau's result (which was only an approximation) and the Lorentz transformation (Voigt and Poincaré)? JFB80 (talk) 09:26, 19 July 2016 (UTC)

I don't know (and don't want to know) exactly which formulas, but people were aware of the Fizeau experiment and the Michelson–Morely experiment and physicists have a tendency to look for formulas that match the behavior of nature. Most definitely, the length contraction formula was derived from one of these early incorrect theories. I know there are others, but don't want to delve in to it. (Aether theories are wrong as we know, but they aren't entirely air, and the people working with them not stupid.) Like I said, these formulas were not obtained in their correct setting before Einstein, and probably incorrectly interpreted. Special relativity was at 1905 a ripe fruit, ready to be picked and enjoyed, but it was Einstein who did the picking, not anyone else. By contrast, general relativity was not a ripe fruit, and with that one Einstein was at least 50 years ahead of his time. (Yet, you'd be in trouble if you claim Einstein was first with the field equations for GR.) YohanN7 (talk) 09:43, 19 July 2016 (UTC)

I better clarify the issue. The issue (for wiki-lawyers) is not who was first, but whether there exists a secondary reference claiming that this or that dude was first. I (who IRL actually cares a little bit more about who actually was first, than who a secondary source claims was first) was together with another editor dragged to the ANI by a crackpot idiot for claiming Dirac was first with a certain result (which he was, everybody who should know knows) and citing the original Dirac paper. (We didn't like crackpot idiot splattering cn-templates en masse in mostly well-referenced articles.) Dirac didn't write in his paper I have discovered this equation and I am first to do so. Even if primamry sources were counted as reliable (they aren't really), it doesn't work. See the point?

For example, if you were to tell the truth about Hilbert's first problem (which is that it was composed of the two problems "Can the reals be wellordered?" and "Is the continnum hypothesis true"), then you will run into a million secondary sources saying it is composed of the continuum problem only. A primary source (e.g. transcript of Hilbert's actual speech, these exist) would tell you the truth, but doesn't count. YohanN7 (talk) 10:16, 19 July 2016 (UTC)

I think we just have to tell the truth as we see it and hope there are no crackpot idiots around. JFB80 (talk) 03:57, 20 July 2016 (UTC)

Confusing Symbols

In the section 'Standard configuration', I was confused by gamma sub v. Took me a while to realize it was not gamma times v.73.220.235.20 (talk) 02:46, 2 August 2016 (UTC)

I agree. I have made a little change: [1]. - DVdm (talk) 06:41, 2 August 2016 (UTC)

I don't. Every gamma refers to some v. The previous notation said which V. Besides, the rest of the article uses the old notation. YohanN7 (talk) 09:11, 2 August 2016 (UTC)

Oops, sorry. I had made a silly typo. Ok now. — Preceding unsigned comment added by DVdm (talk • contribs) 10:31, 2 August 2016 (UTC)

It is at any rate no big deal. I meant that since this article by definition involves gammas referring to different velocities, it should i m o be useful to have the notation indicate to which velocity the gamma in question refers. The rest of the article does this. But like I said - no big deal, latest edit works just as wellYohanN7 (talk) 11:02, 2 August 2016 (UTC)

Yes, I hadn't really looked at the remainder of the article yet. You make a valid point. Perhaps instead of what I did, we could lower the V-index and use $\gamma _{_{V}}$ as opposed to $\gamma _{V}$. Much less confusing. - DVdm (talk) 11:16, 2 August 2016 (UTC)

Wow, never thought of that. Looks really nice. YohanN7 (talk) 11:41, 2 August 2016 (UTC)

Ok, shall I make the change then? - DVdm (talk) 12:11, 2 August 2016 (UTC)

Please do! YohanN7 (talk) 12:19, 2 August 2016 (UTC)

 Done: [2]. I only did it with the capital V, no need with the lowercase u and v. I also kept the first explicit definition in the 'Standard configuration' section. Looks better indeed. - DVdm (talk) 12:24, 2 August 2016 (UTC)

Unsourced part removed

It looks like the following unsourced part (added here, here and here) was removed for the third time now ([3], [4], [5]):

In classical mechanics vectors :$\mathbf {v} ,\mathbf {u}$ may be referred (using equipollence) to .the same origin and usually are. But by definition they are relative velocities starting from different origins. This point is important in the generalization to Special Relativity where equipollence does not apply in general.

Rightly so—see edit summaries of removals. It is not just unsourced. It also suffers from typographical, grammatical and semantic errors. It reads like gibberish indeed. - DVdm (talk) 09:15, 16 September 2016 (UTC)

You might actually be correct and vector addition in SR be equipollent when looked at the right way but it is not looked at that way in this article and even you might find difficulty understanding how vectors adding noncommutatively could be equipollent. But there are apparently no sources on this subject. JFB80 (talk) 19:31, 16 September 2016 (UTC)

Indeed, and without sources, we don't take it. - DVdm (talk) 19:59, 16 September 2016 (UTC)

And now it is claimed by this this individual that something is equipollent "when looked at the right way but it is not looked at that way in this article". The same attitude (article bad) prevails in earlier posts by this individual. What does it mean? Perhaps this individual should refrain from editing SR related articles in the future. Misunderstanding the concept of velocity in SR and in Newtonian mechanics is not a good starting point. YohanN7 (talk) 07:17, 19 September 2016 (UTC)

First DVdm and especially YohanN7 please stop using disrespectful and abusive language as we are are requested to do in the guidelines. You seem to regard yourselves as some kind of Wiki-policemen who do not have to observe this rule. Secondly you don't have a profound knowledge of this subject and should only express an opinion not a judgement. Thirdly I suggest you make a start by reading the paper of Sommerfeld 1909 referred to at the end of the article. JFB80 (talk) 17:57, 19 September 2016 (UTC)

The sentence was removed because it is unsourced. It is wrong, it is badly formed, it reads like nonsense, but first of all, it is wp:unsourced, and, per wp:BURDEN, it is your job to find a proper source. So, unless you can come with a proper reliable source that completely backs what you were trying to add, you are disrupting this talk page per the wp:talk page guidelines. There is no need to call other editors "kind of Wiki-policemen". Other editors are the ones with whom you are supposed to collaborate and establish a wp:consensus about some change to the article. Consensus for the above sentence is unlikely to happen, so the best thing to do might be to leave it at that. - DVdm (talk) 18:30, 19 September 2016 (UTC)

DVdm perhaps it should be added that the sentence in question was only intended as an explanation, principally to yourself, and there was no proposal to put it in the article. Such remarks should not require a source. I see nothing in the guidelines that comments in talk pages require a source, nor should there be, in my opinion, so as to ensure free discussion. Otherwise it is like a medieval disputation using quotations from Aristotle. But I am willing to let the matter drop now.JFB80 (talk) 15:50, 22 September 2016 (UTC)

Indeed, comments on talk pages don't need sources. But the "the sentence in question" was put in the article by you. So it was removed. But then, continuing to argue on the talk page—against consensus—about putting something from the sentence in the article without being able to back it with a source, would be talk-page-disruptive, and therefore, thanks for dropping the matter. - DVdm (talk) 16:13, 22 September 2016 (UTC)

SOURCE ADDED:

Silberstein: The Theory of Relativity 1914 p.179 He says 'In connexion with this we have only the triangle rule and not the parallelogram rule as in Newtonian mechanics. There are no parallelograms in hyperbolic space... etc'

So now please stop your insulting 'gibberish' talk and learn more about what you saying.JFB80 (talk) 07:33, 19 January 2017 (UTC)

That does not support the sentence above at all. Bringing that as a source would be wp:original research by wp:synthesis. - DVdm (talk) 07:58, 19 January 2017 (UTC)

What do you understand by equipollence then? Isn't it the possibility in Euclidean geometry of freely moving the point of origin of vectors? It is this which justifies the parallelogram rule in Euclidean space and which does not apply in hyperbolic space. This is basic geometry but I tried to give a relativity source. I think there is no need for you to be always so legalistic quoting Wikilaw as this is a talk page where we try to get mathematical ideas clear. JFB80 (talk) 16:11, 19 January 2017 (UTC)

I know what equipollence is. Silberstein doesn't mention the term. He also does not mention vector origins, nor the difference between pairs of vectors with the same origin vs pairs with different origins. So he does not make the analysis that is made in the removed sentence. If that particular analysis would be worth being mentioned—as in wp:DUE—in Wikipedia, then surely a source that does indeed make the analysis must exist. It's just a matter of finding it, and you're in business. - DVdm (talk) 16:31, 19 January 2017 (UTC)

Non-standard notation for vector addition formulae - very confusing!

Normally (and consistently on Wikipedia everywhere else Special Relativity is discussed - see for example https://en.wiki.x.io/wiki/Special_relativity ), one starts in an unprimed reference frame S, in which the velocity of a body $\mathbf {u}$ is known, and one wishes to find the transformed velocity $\mathbf {u'}$ in a moving reference frame S' that is moving with velocity $\mathbf {v}$ relative to (and as measured from) the original reference frame. In the Standard Configuration, the co-ordinates are further selected so that the velocity $\mathbf {v}$ is along the x direction in S, and v is treated as a scalar speed, but in the general case $\mathbf {v}$ is a vector pointing anywhere.

That is always the Standard Notation convention; one starts in S and moves to S'. The symbols $\mathbf {u}$ and $\mathbf {v}$ alwaysrefer to "a body moving in S" and "the motion of S' relative to S" within the Standard Notation, respectively.

However, this page inexplicably moves away from this standard notation in the Standard Configuration section, using V for what is normally v, and even more confusingly, $v_{x}$, $v_{y}$ and $v_{z}$ for what are the components of $\mathbf {u}$ in the standard notation.

This then ends up with the Translation of velocity (Cartesian components) section being very confusing and non-standard, as it gives what would normally be the inverse translation (prime -> unprimed), without giving the normal (unprimed -> primed) transformation.

To conform with the standard notation used on this site and elsewhere, it should therefore look like this instead:

Transformation of velocity (Cartesian components)

$u_{x}={\frac {u_{x}'+v}{1+{\frac {v}{c^{2}}}u_{x}'}},\quad u_{x}'={\frac {u_{x}-v}{1-{\frac {v}{c^{2}}}u_{x}}}$

$u_{y}={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}u_{y}'}{1+{\frac {v}{c^{2}}}u_{x}'}},\quad u_{y}'={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}u_{y}}{1-{\frac {v}{c^{2}}}u_{x}}}$

$u_{z}={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}u_{z}'}{1+{\frac {v}{c^{2}}}u_{x}'}},\quad u_{z}'={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}u_{z}}{1-{\frac {v}{c^{2}}}u_{x}}}$

This is how it appears in A.P. French's book on Special Relativity, for example, which also uses the standard convention for the meaning of the symbols described above, and used everywhere else on Wikipedia.

This also then makes the formulae consistent with the one-dimensional special case formula shown in the master https://en.wiki.x.io/wiki/Special_relativity#Composition_of_velocities by which most people will find this page. — Preceding unsigned comment added by Kebl0155 (talk • contribs) 20:26, 8 January 2017 (UTC)

We get into further difficulty with the General Configuration section, again because of non-standard notation. The result at the end should look like:

$\mathbf {u} =\mathbf {u} _{\parallel }+\mathbf {u} _{\perp }={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\left[\alpha _{v}\mathbf {u} '+\mathbf {v} +(1-\alpha _{v}){\frac {(\mathbf {v} \cdot \mathbf {u} ')}{v^{2}}}\mathbf {v} \right]$

This is the backward transform. It should definitely be accompanied by the more useful forward transform:

$\mathbf {u'} =\mathbf {u} _{\parallel }'+\mathbf {u} _{\perp }'={\frac {1}{1-{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left[\alpha _{v}\mathbf {u} -\mathbf {v} +(1-\alpha _{v}){\frac {(\mathbf {v} \cdot \mathbf {u} )}{v^{2}}}\mathbf {v} \right]$

The page then suggests: "In order to facilitate generalization and to avoid proliferation of primes, change notation of V to u, and v′ to v"

No! That's a disaster! Now we have $\mathbf {u}$ and $\mathbf {v}$ meaning the EXACT OPPOSITE of the conventional uses of $\mathbf {u}$ and $\mathbf {v}$ everywhere else. In the current live version, $\mathbf {u} \oplus \mathbf {v}$ now means what is meant by $\mathbf {u}$ in the standard notation, $\mathbf {v}$ actually means $\mathbf {u} '$ in the standard notation, and $\mathbf {u}$ actually means $\mathbf {v}$ in the standard notation.

Ouch! Very ouch! No no no! DANGER DANGER DANGER!

The phrase starting with "In order to facilitate generalization..." needs to completely go, apart from the notes. It's WORSE than useless - it's positively harmful. There needs to be NO change of variables here, having used standard notation FROM THE START, and what should then be written in the equation box is:

${\begin{aligned}\mathbf {u} &={\frac {1}{1+{\frac {\mathbf {u} '\cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {v} +{\frac {\mathbf {u} '}{\gamma _{v}}}+{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}(\mathbf {u} '\cdot \mathbf {v} )\mathbf {v} \right]\\&={\frac {1}{1+{\frac {\mathbf {u} '\cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {v} +\mathbf {u} '+{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}\mathbf {v} \times (\mathbf {v} \times \mathbf {u} ')\right],\end{aligned}}$

and in the forwards direction

${\begin{aligned}\mathbf {u} '&={\frac {1}{1-{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[{\frac {\mathbf {u} }{\gamma _{v}}}-\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}(\mathbf {u} \cdot \mathbf {v} )\mathbf {v} \right]\\&={\frac {1}{1-{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {u} -\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}\mathbf {v} \times (\mathbf {v} \times \mathbf {u} )\right]\end{aligned}}$

The above could be further simplified by just writting $\gamma$ instead of $\gamma _{v}$, as this is the standard meaning of $\gamma$.

and all-important equations for $\gamma$ for added speeds become:

${\begin{aligned}\gamma _{u}&=\gamma _{u'}\gamma _{v}(1+{\frac {\mathbf {u} '\cdot \mathbf {v} }{c^{2}}})\end{aligned}}$

and the more useful

${\begin{aligned}\gamma _{u'}&=\gamma _{u}\gamma _{v}(1-{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}})\end{aligned}}$

This is the ONLY WAY that the equations on this page can be used interoperably with the equations that appear on all the other Special Relativity pages.

This then gives the same equations as are shown on the main page for Lorentz transformations, which links to this page in the hope that it will derive them - which it currently doesn't! See https://en.wiki.x.io/wiki/Lorentz_transformation — Preceding unsigned comment added by Kebl0155 (talk • contribs) 20:30, 8 January 2017 (UTC)

These results are important: it is difficult to develope special-relativistic conservation of momentum in a general way, beyond the specific cases of elastic and inelastic collisions between two particles that are usually offered in text books, without them.

The formulae for $\gamma$ also don't appear to be listed on Wikipedia anywhere else; I think we should therefore make the effort to keep the notation consistent with the standard that is used elsewhere on this site, as well as many reference books and other sites.

The Notational Conventions section should probably also be deleted, once the Standard Notation has been adopted - though perhaps a warning might be appropriate here instead?

Using Standard Notation is also the only way to get the formulae on this page to be consistent with the formulae given on the master Special Relativity page, which links to this one as a See Also in the Composition of Velocities section. For this reason, this page really should use the same Standard Notation as the main Special Relativity page; the formulae we currently show on this page are not the same as the ones shown on the master Special Relativity page.

Would you like me to have a go at revamping this section to use standard notation throughout?

Kebl0155 (talk) 19:57, 8 January 2017 (UTC)

Looking at the current text in the article and the source from which it is drawn, I see no accuracy problem with what's here. But in order to be more in line with our other articles, and no doubt with a larger and more modern part of the relevant literature (as opposed to the old—possibly notationally outdated—Landau and Lifshitz), I wouldn't mind an overhaul along the lines of the above. That is, provided you pull it from a good source and stay as close to it as possible.

References

Ping @YohanN7: what are your thoughts? - DVdm (talk) 21:47, 8 January 2017 (UTC)

Changing notation does not change physics, Kebl0155. Typos aside, the article is fine as is. If you could follow it through, and others can follow it through, then surely that's more important than fussing over notation. M∧Ŝc²ħεИ_τlk 08:18, 9 January 2017 (UTC)

It is not even notation that confuses, it is that we present the velocity as a stationary observer sees a velocity given in a moving frame, and that is the most straightforward question one might have, not how a moving observer sees a velocity we measure. Besides, it is referenced to the masterpiece L&L, and I am unaware of a "Standard Notation". The only semi-standard there is is that the moving frame is primed. So it is here too. If anything, is should be changed elsewhere because this is the main article on the subject. YohanN7 (talk) 08:42, 9 January 2017 (UTC)

Thank you all for your kind and considerate comments. In reply to M: Yes I completely agree that changing notation does not change physics or indeed mathematics; for instance when one writes a^2 + b^2 = c ^2 in regards to a right angled triangle, one could equally well write A^2 + B^2 = C ^2 and still be quoting Pythagorus, who did not in fact use any of these letters as they did not exist at the time and place of his writing. However, I dispute that I or others can/could follow the current article or found it easy; in actual fact this non-standard notation prevented me from developing what should have been an easy proof for several days, until I spotted that the notation had been reversed from the standard convention. I therefore believe your own argument implies that a certain amount of fussing over notation is therefore warranted.

In reply to DVdm: I wholeheartedly agree with your position. As per Maschen's comment, and the example allusion to Pythagorus, I would also take the position that resymbolising using Standard Notation would mean this section is still a direct quote from the original and respected source. Perhaps it might be appropriate to add a note to indicate that the quote has been resymbolised; what are your thoughts on this suggested way forwards?

In reply to YohanN7, I could not parse the first sentence of of your reply so I cannot reply to that. As regards the second sentence, even with resymbolising, the section will still be a direct quote of the original source, as per Maschen's reply. As regards the third sentence, while this page is the only dedicated article on velocity transformation, it is by no means the only article on Wikipedia that mentions it or prints formulae. Indeed, I politely and respectfully dispute your claim that that this page is the 'Main' page on velocity transformation, as both the Special Relativity page and the Lorentz Transformation page are objectively 'more main' than this one, in that they contain transformation formulae that are read much more frequently (104 hits per day this page, compared with 1,302 per day for Special Relativity, and 718 per day for Lorentz Transformations).

Indeed the combined visitation weight of those two pages makes them 20 times more 'main' than this page. Both these pages already use the Standard Notation described above, as does the Wikibook for Special Relativity at https://en.wikibooks.org/wiki/Special_Relativity/Principle_of_Relativity#Frames_of_reference.2C_events_and_transformations, which is the stated base of the Relativity Taskforce https://en.wiki.x.io/wiki/Wikipedia:WikiProject_Physics/Taskforces/Relativity under whose aegis this page is run (see link at top of this Talk page). I humbly submit that there are therefore multiple, strong and objective grounds to argue that the current symbolisation on this page is at fault, not in accordance with the stated aims of the Relativity Wikipedia project, and likely to cause confusion and strife in those who read it.

I therefore hopefully and respectfully would appreciate it if you could all please confirm that it is quite correct that the specified sections of this page be resymbolised in the manner suggested; my aim always is to create concord, delight in concord, and act in concord, regardless of reference frame. Kebl0155 (talk) 11:05, 9 January 2017 (UTC)

This is the main article on velocity addition. The formula is correct. It is well referenced. There is no "Standard Notation".

But, If you desperately wish to have the formula inverted, then invert the formula before you write yourself to death here over a non-issue. But make sure you understand what you are doing first. Since you became so very confused about the formulas being given on this form, it is good advice. YohanN7 (talk) 11:21, 9 January 2017 (UTC)

Thank you YohanN7. Subject to agreement from the other posters, I will begin to undertake the work. I hope that my own foolishness, which I freely acknowledge, will make me exceedingly careful in this endeavour. I do hope you might be able to spare the time to check my work when complete. Your advice is good indeed. Thank you again. Kebl0155 (talk) 12:09, 9 January 2017 (UTC)

You suggested to delete the notation/operand conventions section, which is bad because not everyone uses the same conventions and people may become confused when they read other books/papers on the topic. Using one convention in an article for internal consistency is fine, but writing as if it is the only convention is wrong. The list in this article is not exhaustive, and more could be added.

I don't object to which notation is used, as long as the derivation is clear and correct. The current notation is fine as is. Contrary to your claims, in the literature sometimes the primes in the Lorentz transformations are reversed from

${\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}$

to

${\begin{aligned}t&=\gamma \left(t'-{\frac {vx'}{c^{2}}}\right)\\x&=\gamma \left(x'-vt'\right)\\y&=y'\\z&=z'\end{aligned}}$

which both correspond to the same LT, in different notation. The second set of equations is not the inverse transformation of the first. There is no standard notation.

Also, take care when you edit. It is not up to other editors to check work for you. M∧Ŝc²ħεИ_τlk 08:43, 10 January 2017 (UTC)

My two cents on the whole issue: If you have the time an ability to improve articles on relativity, then it is a total waste of it to steer up a storm. You can find plenty of talk pages where highly qualified people have spent hours, even days, arguing over such things as units and conventions. Some people even believe that there is a right convention and a wrong convention (choice of units, etc), and that all choices but theirs actually are inconsistent and plainly wrong. They argue furiously time and time again that they are so OBVIOUSLY RIGHT. Such people are called crackpots. You have not gone that far, and it is perfectly okay to have a personally preferred set of units, conventions, etc. But, reading through your posts, you are dangerously close.

In the particular case at hand, yes, I do have a preferred presentation. Namely, the one in the article. It answers the, to me, primary question. In a frame moving with V, an object has velocity v′, which velocity v do we see?

The complementary question is, if we have an object moving with velocity v, which velocity v′ does an observer moving with V see, is to me the secondary question. Apparently, some distinguished authors feel the same way.

But I am not religious about it. Frankly, I don't care. The two presentations are mathematically dual in the usual prescription (reverse sign of velocity, swap primes) for relativistic formulas of this sort. It is just my two cents on which presentation is, by a nose, slightly preferable over the other. But you seem to be upset even over the choice of capital V to denote the velocity of the primed frame.

---

You are also implicitly suggesting that

• Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (4th ed.). Butterworth–Heinemann. ISBN 0 7506 2768 9. {{cite book}}: Invalid |ref=harv (help) (graduate level)

should perhaps be tossed. This is something I could actually consider worth spending time arguing over, and is the reason I spend time on writing this post. Make no mistake, there is a reason that an 80 years old textbook is still selling. It is not because it is bad. It is because it is good. Such books just don't come about every decade. As you might note, it is in its forth updated edition, and it has been revised thoroughly over the decades. The same goes for every title in Course of Theoretical Physics. If you ever venture to learn general relativity (perhaps you know it already), then this particular book will give the most bang for the time spent. Without comparison. I don't think I am (in fact I know I am not) the only fan of this book. YohanN7 (talk) 11:39, 10 January 2017 (UTC)

For the avoidance of doubt, I am certainly not suggesting that Landau & Lifschitz should be tossed! I apologise if I have given that impression, even implicitly. Indeed I believe the results presented on this page are important, crucial even.

Given your kind comments (and thank you again), I agree that we should indeed keep the Notation section. When I make the changes, very carefully, I will make it clear in this section that the derivation has been resymbolised from the original source.

Despite my foolishness, I was once considered very good indeed at Special Relativity. That was a long time ago. I am about to begin making the changes. I will update this page when complete. I hope the changes will help fools like me become less foolish in the future. Thank you again for your patience and your kind comments and attention on this matter. — Preceding unsigned comment added by Kebl0155 (talk • contribs) 12:27, 10 January 2017 (UTC)

OK it's done. It took all day. I was very careful. The only bit I couldn't do was the hidden section on 4-velocities; there are actual problems with this section, which is not internally self-consistent, and I cannot resymbolise it until they are fixed. I'm about to start a separate heading for this subsection. Interestingly, while checking the references, I noticed that in may cases the notation had already been resymbolised in some way, including Landau and Lifschitz, even before I started editing this page. I was not able to check two of the alternate notations presented as the references are missing.Kebl0155 (talk) 18:38, 10 January 2017 (UTC)

I think you have misunderstood General configuration. Primes are totally out of place there. There are not necessarily any frames there. YohanN7 (talk) 08:24, 11 January 2017 (UTC)

I am so sorry that you have written the above. I am sorry that I will now have to respond rather robustly:

No.

The section does say that "It deserves special mention that if u and v refer to boost velocities, then both u ⊕ v, and v ⊕ u are correct expressions for the combined boost velocity." Perhaps this is the basis of your assertion.

However, it is vital to understand that this only applies to the specific case where both u and v are boost velocities, by which is exclusively meant "the velocity of one frame relative to another frame". As soon as body velocities are involved, by which is meant "the velocity of a body within a particular frame" then whether or not one is talking about the primed or unprimed frame becomes very important, as if one is not paying very close attention to this, one ends up with sign errors in one's formulae.

If one is using those formulae with experimental data, one predicts an outcome that is not observed.

If one is using those formulae for proofs, the proofs are frustrated, with much crossing out and the needless destruction of trees. I am speaking from bitter, bitter experience of having made such mistakes many times, and would seek to save you and other readers of this page from similar anguish, frustration and grief. They're a devil to ferret out once they get in there; I have had to do this myself many times.

The General Configuration is explicitly a generalisation of the standard configuration, which again is explicitly the case where a body velocity and a frame velocity are relativistically added. The specific result for two frame velocities in the subsequent Properties section is a specific case derived from the General Configuration section (which would be more accurately labelled the 'Isotropic configuration', as the generalisation is in the allowable direction of v, which is no longer forced to be colinear with the x axis). the 'General Section' derivation is still specific to the addition of a body velocity and a frame velocity. Careful, mindful attention must therefore be paid to prime and unprime, and the sign of the relative frame velocity.

I'm sorry if that's uncomfortable reading for you. The importance of paying really close attention to the sign of relative frame velocity and which letters should be primed was something I had to learn from trial and error when I did all this at Oxford University, and I was reminded of the experience strongly in the process of sorting out the notation on this page. So much pain, so long ago. I would like to thank you for recommending Landau and Lifschitz; it looks very good. Perhaps one day I might even be as qualified to speak about General Relativity as I am to speak about Special Relativity. Thank you again.Kebl0155 (talk) 11:41, 11 January 2017 (UTC)

Primes aren't used in the literature. But since you know better, ... YohanN7 (talk) 07:32, 12 January 2017 (UTC)

I'm clicking on the links for the References in the Notational Conventions section, but apart from Ungar they're not actually going anywhere. I don't know how to fix this. I don't think it's my browser. I can't find places to link to, or even the original books, with Google, so I'm a bit at a loss. Can you help? I very much like the changes you have made since I did the massive resymbolisation by the way - thank you so much. Also having had my attention brought to an etiquette guide by another user, I thought I might apologise if I've been a bit brusque with all this. Sorry about that. The page is looking magnificent. Thank you. Kebl0155 (talk) 20:03, 11 January 2017 (UTC)

The templates "harvtxt" (used there) and "harvnb" (used elsewhere) are supposed to match with either of the templates "cite book", "cite journal" or "citation". In "harvtxt", the first arguments is the last name of the authors(s). It must match with parameter "last(n)" in "cite book" etc. The first numeric argument should match "year". Quirk: For this to work with "cite book" and "cite journal", the extra argument "ref=harv" must be supplied (works automatically for "citation"). YohanN7 (talk) 07:32, 12 January 2017 (UTC)

Ah thank you for the further information. I will try to remember. I went to fix them but found someone had fixed them already; I didn't have time to check the history for whom, but my thanks to whoever that was.

The applications section still uses the old notation. YohanN7 (talk) 11:10, 12 January 2017 (UTC)

Problems with subsection: A proof using 4-vectors and Lorentz transformation matrices

I was unable to resymbolise this section because it was not sufficiently well defined for me to penetrate its meaning, and I sensed either an internal contradiction or sign error, possibly both. Without the proper definition of terms it's hard to tell. Possibly impossible to tell.

Because there are no references or citations for this subsection, I was also unable to check it against any original source. All quotations from this subsection that I tried searching for online turned out to be quotes from this page.

Specifically, the page starts by asserting in the second paragraph that V is a fly's four velocity as seen by a ship, and that V is roughly in the same direction as the x axis. The wording with planes and axes is setting us up for a transform of velocity of some body in the ship's frame into the fly's frame. So far so good.

Next a Lorentz Transformation matrix is given in which the V1 components have opposite sign to the ones in the example matrix in the Proper Transformations section of the Lorenz transformation page. I can just about cope with that, but it would be non-obvious to a lay reader: that indicates it's going to be a reverse transform (either that or a sign error). So we're transforming the velocity of something moving in the fly's frame to that which would be measured in the ship's frame. OK that's confusing, but OK. The section asserts that this 'boosts the rest frame', however it is impossible to boost a frame, as in Special Relativity all frames are inertial, meaning having constant velocity. That's the Special bit. So, I don't know what that actually means. One could legitimately talk about boosting velocities measured within a particular frame; perhaps that is what is meant. Perhaps.

Usually it's best to entirely avoid talking about rest frames (and particularly one should avoid asserting the existence of 'the rest frame') when discussing Special Relativity, as no frame is privileged to be at rest; that's kind of the point, but perhaps that's just my opinion.

The main problem with the terminology/notation at this point is that 'the rest frame' is not actually defined anywhere in this subsection. I can say for sure that if you were to actually use the matrix to multiply a velocity - any velocity - it would transform a velocity having this value measured in the fly's frame to the ship's frame, though.

Personally, I couldn't evaluate the next sentence about 'this matrix rotates the a pure time-axis vector' as either true or false, but again perhaps that's just me.

Here's where things get really tricky. The next bit starts 'If a fly is moving with four-velocity U in the rest frame'. Well now. What are we to do with this. We already have the velocity of the fly as V within the ship's frame. So now we're either changing letters to represent the fly's velocity (Bad), or we're about to attempt to do a Lorentz transformation on this fly with its own velocity (also Bad), or we're talking about a new, second fly (please make it stop).

If this new velocity U (whatever thing that is referring to. The first fly? Some other fly? Some other thing?) in the rest frame (whatever that is) is 'boosted by multiplying by the matrix' (yes it's OK to use the word boost this time because we're talking about a velocity), the new four velocity S will be a velocity that has been transformed from the original fly's frame into the ship's frame (definitely because of the signs of V1 in the matrix).

I just don't know the velocity of what. It certainly can't be the original fly.

So then we have three equations which, from the distribution of pluses and minus signs, have the same form as a transform from a primed frame (right hand side) to an unprimed frame (left hand side). Unprimed normally means rest frame, and would be consistent with the use of an unprimed S on the left hand side. But that then contradicts the assertion that U is in the rest frame. Doesn't it?

The honest answer is that the terms in this subsection are so poorly defined that I cannot tell you for sure what velocity is actually being transformed, from which frame it is being transformed and to what frame, to what the transformed velocity refers, or to what this 'rest frame' refers, or even how many flies there are.

I also tried to start from the end and work backwards, to see if I could get it to make sense that way, but ended up writing 'If a fly is moving with four velocity U′ in the rest frame', and could go no further. I've never seen anyone use primes to describe a velocity in a rest frame before.

Until someone can explain what this subsection actually means, or even what it is trying to say, or even how it is trying to say it, I cannot understand it, let alone find the error with certainty (assuming there is one), let alone resymbolise it for consistency. My apologies if I am being in any way foolish with this; I frequently am. At the moment I couldn't even tell you for sure that this subsection is actually saying anything at all.

I really can't see how anyone with less than a doctorate would be able to follow this subsection as it stands, apart perhaps from the original author, whoever that is, to whom I am sincerely appealing for help, and apologising for any foolishness on my part. I almost certainly have been stupid somewhere.

The subsection has been restored to its original state; it has NOT been resymbolised as part of today's work for this reason.

I'm now going to lie down in a darkened room and try not to think of relativity for a little while. — Preceding unsigned comment added by Kebl0155 (talk • contribs) 20:38, 10 January 2017 (UTC)

I managed to trace back the addition of this subsection to 15:34 on July 9th 2015. At that time there were already multiple inconsistencies in notation within the rest of the page that makes determining the original intent of the notation for this subsection difficult. That is probably not the fault of the subsection's original author. I have been able to ascertain that we are talking about a ship in the sailing sense rather than the star trek sense, that the rest frame is meant to be the shore, and I'm pretty sure we're down to just one fly. It would be helpful to know from where the addition was sourced, YohanN7 this looks like this subsection was your addition in the first place, so far as I can tell; where did you get it? Kebl0155 (talk) 23:09, 10 January 2017 (UTC)

No. It is not difficult using the article history to see that it is not. YohanN7 (talk) 08:02, 11 January 2017 (UTC)

My humble and sincere apologies. I am new here. I guess I must have misread or misinterpreted the history in some way. Does anyone know where this section has come from, then?Kebl0155 (talk) 11:41, 11 January 2017 (UTC)

OK I managed to get this subsection to be internally self consistent, and mathematically and physically correct, so far as I can tell from the corresponding section(s) of Lorentz transformations. This required CLARIFICATION of notation in some cases, and unfortunately, the COMPLETE REDEFINITION of several terms, which means the MEANING HAS SUBSTANTIALLY CHANGED, unlike the resymbolisation effort for the other sections. I do not know whether these new meanings are in line with the original author's intent. Before and After changes are highlighted in the version history. I am loathe to resymbolise until I have confirmation that the changes I have made are in fact correct. I think the section warning should stay in place until someone else can check this; preferably until several people have checked this. YohanN7 please advise. Hopefully I will be able to sleep now! Kebl0155 (talk) 00:23, 11 January 2017 (UTC)

A patroller Ping @Wbm1058: has removed the warning so I have resymbolised the section as earlier discussed, to ensure promise made in Notational Conventions is fulfilled. Kebl0155 (talk) 19:32, 11 January 2017 (UTC)

The patroller probably removed it for reasons other that that the content is problem free. Suggestion: Put that whole hidden section in the dumpster and write a new fresh 4-vector section. I put that old one in a hide box because I weren't able to parse it in reasonable time. Thus I suggest this:

• Given 3-velocity, manufacture 4-velocity (e.g. divide $dx=(cdt,dx,dy,dy)$ by proper time $d\tau$).
• 4-velocities transform just as coordinates
• Done

Then one might add in that new section the most general case $u'^{\mu }={\Lambda ^{\mu }}_{\nu }u^{\nu }$ with $\Lambda =\exp({\boldsymbol {\xi }}\cdot \mathbf {K} )$, where $\mathbf {K}$ is the vector of boost generators and ${\boldsymbol {\xi }}$ are boost parameters (related to ${\boldsymbol {\beta }}$), and perhaps also how the matrix $\Lambda$ appears in all generality (think it is given in Jackson's EM book) for pure boosts. YohanN7 (talk) 07:50, 12 January 2017 (UTC)

Actually, the formula is in Lorentz transformation#Proper transformations. YohanN7 (talk) 09:18, 12 January 2017 (UTC)

I'd be quite happy to see the back of that section too. Do you happen to know any reference books we could quote here? I'm happy to write a new section from scratch, but I'd rather be quoting something in a journal or textbook; I think part of the problem with the current section is it's unreferenced. Do you think you might be able to point me in a helpful direction? Respectfully, Kebl0155 (talk) 10:14, 12 January 2017 (UTC)

Four-velocity construction should be anywhere and everywhere. It is definitely in L&L (they divide by $ds=cd\tau$). The velocity addition formula is derived using 4-velocity (one case only) in

• Kleppner, D.; Kolenkow, R. J. (1978) [1973]. An Introduction to Mechanics. London: McGraw-Hill. ISBN 0-07-035048-5. {{cite book}}: Invalid |ref=harv (help) (introductory level),

chapter 14. YohanN7 (talk) 10:26, 12 January 2017 (UTC)

Thank you. I should be able to get to this in the next couple of days. The section in its current state is at the very least self consistent and has consistent notation with the rest of the page; it should hold til then, I feeel. Respectfully, Kebl0155 (talk) 12:30, 12 January 2017 (UTC)

Hyperbolic problems

I am trying to verify the statement

$dl_{\boldsymbol {\beta }}^{2}={\frac {d{\boldsymbol {\beta }}^{2}-({\boldsymbol {\beta }}\times d{\boldsymbol {\beta }})^{2}}{(1-\beta ^{2})^{2}}}={\frac {d\beta ^{2}}{(1-\beta ^{2})^{2}}}+{\frac {\beta ^{2}}{1-\beta ^{2}}}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}),$

referenced to Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (4th ed.). Butterworth–Heinemann. p. 36. ISBN 0 7506 2768 9. {{cite book}}: Invalid |ref=harv (help)

For full reference, this is stated in a problem as such:

Problem: Find the "element of length" in relativistic "velocity space".

Solution: The required line element $dl_{v}$ is the relative velocity of two points with velocities $\mathbf {v}$ and $\mathbf {v+dv}$. We therefore find from (12.6),

$dl_{\mathbf {v} }^{2}={\frac {d\mathbf {v} ^{2}-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {dv^{2}}{(1-v^{2})^{2}}}+{\frac {v^{2}}{1-v^{2}}}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}),$

where $\theta ,\phi$ are the polar angle and azimuth of the direction of $\mathbf {v}$. If in place of $v$ we introduce the new variable $\chi$ through the equation $v=\tanh \chi$, the line element is expressed as:

$dl_{\boldsymbol {\chi }}^{2}=d\chi ^{2}+\sinh ^{2}\chi (d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}).$

I cannot verify the first formula. The first equality is no problem. It follows from the expression for relativistic relative velocity (which is (12.6) in ref). But then I have to make a couple of intentional errors to get to the right hand side.

Starting point of my failed calculation — maybe I go wrong already at this point

I take $\mathbf {v} =(0,0,v)$ and $d\mathbf {v} =(dx,dy,dz)$ and use

${\begin{aligned}x&=v\,\sin \theta \,\cos \varphi \\y&=v\,\sin \theta \,\sin \varphi \\z&=v\,\cos \theta \end{aligned}}$

Note $v$ in place of $r$ Note $x$, etc in place of more cumbersome $v_{x}$, etc.

I'd highly appreciate if someone could attack this. I have starred myself blind on my calculation. YohanN7 (talk) 12:37, 17 January 2017 (UTC)

As a suggestion... If you have simplified the calculation by taking

$\mathbf {v} =(0,0,v_{z})$

then split dv into components parallel and perpendicular to v

$d\mathbf {v} =d\mathbf {v} _{\parallel }+d\mathbf {v} _{\perp }$

where

$d\mathbf {v} _{\perp }=(dv_{x},dv_{y},0)$

then

$|\mathbf {v} \times d\mathbf {v} |^{2}=|\mathbf {v} \times d\mathbf {v} _{\perp }|^{2}=v_{z}^{2}(dv_{x}^{2}+dv_{y}^{2})=v_{z}^{2}dv_{\perp }^{2}=v_{z}^{2}(dv^{2}-dv_{\parallel }^{2})=v_{z}^{2}(dv^{2}-dv_{z}^{2})$

then it should be a case of finding dv² and dv_z² and converting back to vector form. What are your intentional errors? M∧Ŝc²ħεИ_τlk 19:47, 17 January 2017 (UTC)

I do get $|\mathbf {v} \times d\mathbf {v} |^{2}=v_{z}^{2}(dv^{2}-dv_{z}^{2})$. I know of two ways of making intentional errors to get at the result, but both require some typing and I don't have the calcs here a t m. I'll get back on that later. YohanN7 (talk) 08:53, 18 January 2017 (UTC)

I know you knew that, just summarizing for myself (and anyone). M∧Ŝc²ħεИ_τlk 11:11, 18 January 2017 (UTC)

No, I appreciate all attempts to find bloopers in my calculation, and could not possibly be offended, since I am asking for help. After all I must screw up somewhere. YohanN7 (talk) 11:35, 18 January 2017 (UTC)

Okay, with the ansatz I get

$dl_{v}^{2}={\frac {(d\mathbb {v} \cdot d\mathbb {v} )^{2}-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {dv_{x}^{2}+dv_{y}^{2}+dv_{z}^{2}-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {dv^{2}+v^{2}(d\theta ^{2}+sin^{2}\theta d\varphi ^{2})-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}$

Now argue that in the last term, $dv$ should be taken as zero. (I believe this is incorrect.) Then

$dl_{v}^{2}={\frac {(d\mathbb {v} \cdot d\mathbb {v} )^{2}-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {dv^{2}+v^{2}(d\theta ^{2}+sin^{2}\theta d\varphi ^{2})-v^{4}(cos^{2}\theta d\theta ^{2}+sin^{2}\theta d\varphi ^{2})}{(1-v^{2})^{2}}}.$

Now accidentally drop a factor $cos^{2}\theta$ in the second to last term. (This is decidedly incorrect.) Then

$dl_{v}^{2}={\frac {(d\mathbb {v} \cdot d\mathbb {v} )^{2}-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {dv^{2}+v^{2}(1-v^{2})(d\theta ^{2}+sin^{2}\theta d\varphi ^{2})}{(1-v^{2})^{2}}},$

which is the L & L formula. I know of a second incorrect way of reaching the equation, but this would hardly be enlightening since it is just composed of two computational errors like the last error above.

One might try to calculate without assumptions about $\mathbf {v}$. The computations will be from hell due to all squaring of differentials.

It is interesting to note that we do explicitly (second equality, first equation this post) have the Poincaré ball model if the cross product term is dropped altogether. YohanN7 (talk) 11:25, 18 January 2017 (UTC)

I think I am close, but stuck. Starting from

$dl_{v}^{2}={\frac {dv_{x}^{2}+dv_{y}^{2}+dv_{z}^{2}-v_{z}^{2}(dv^{2}-dv_{z}^{2})}{(1-v^{2})^{2}}}={\frac {dv_{x}^{2}+dv_{y}^{2}+(1+v_{z}^{2})dv_{z}^{2}-v_{z}^{2}dv^{2}}{(1-v^{2})^{2}}}$

then since v_z = v (but not dv_z = dv)

$dl_{v}^{2}={\frac {dv_{x}^{2}+dv_{y}^{2}+dv_{z}^{2}+(1+v^{2})dv_{z}^{2}-v^{2}dv^{2}-dv_{z}^{2}}{(1-v^{2})^{2}}}$

$dl_{v}^{2}={\frac {dv^{2}+v^{2}d\theta ^{2}+v^{2}\sin ^{2}\theta d\phi ^{2}+v^{2}dv_{z}^{2}-v^{2}dv^{2}}{(1-v^{2})^{2}}}$

$dl_{v}^{2}={\frac {(1-v^{2})dv^{2}+v^{2}d\theta ^{2}+v^{2}\sin ^{2}\theta d\phi ^{2}+v^{2}dv_{z}^{2}}{(1-v^{2})^{2}}}$

The stumbling block for me is dv_z²:

$dv_{z}=dv\cos \theta -v\sin \theta d\theta$

$dv_{z}^{2}=dv^{2}\cos ^{2}\theta +v^{2}\sin ^{2}\theta d\theta ^{2}-2v\cos \theta \sin \theta dvd\theta$

which involves dvdθ. M∧Ŝc²ħεИ_τlk 11:59, 18 January 2017 (UTC)

Maybe we should take this to the reference desk... YohanN7 (talk) 12:20, 18 January 2017 (UTC)

The result follows from the initial equation on using the vector Lagrange formula

$(\mathbf {v} \times d\mathbf {v} )^{2}=(\mathbf {v} .\mathbf {v} )(d\mathbf {v} .d\mathbf {v} )-(\mathbf {v} .d\mathbf {v} )^{2}$

JFB80 (talk) 23:14, 10 March 2018 (UTC)JFB80 (talk) 19:39, 15 March 2018 (UTC)

Here are the details which may be of general interest.

${\frac {d\mathbf {v} ^{2}-(\mathbf {v} \times d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {(1-v^{2})d\mathbf {v} ^{2}+(\mathbf {v} .d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}={\frac {d\mathbf {v} ^{2}}{(1-v^{2})}}+{\frac {(\mathbf {v} .d\mathbf {v} )^{2}}{(1-v^{2})^{2}}}$

On representing v in spherical coordinates in the convention used this is

${\frac {dv^{2}}{(1-v^{2})^{2}}}+{\frac {v^{2}}{(1-v^{2})}}(d\theta ^{2}+\sin ^{2}\theta .d\varphi ^{2})$

Finally, putting v = $\tanh \chi$ it becomes

$d\chi ^{2}+\sinh ^{2}\chi (d\theta ^{2}+\sin ^{2}\theta .d\varphi ^{2})$

JFB80 (talk) 20

13, 15 March 2018 (UTC)

Speed of light in fluid, not in aether

In the section "History", it says:

the speed is c + (1 − ​1⁄n^2)V instead of c + V, where c is the speed of light in the aether

But we are talking from light traveling in a fluid, so c should be replaced with c/n and "aether" should be replaced with "fluid".

I will wait a day before posting an edit, to give chance for a justification of the current version. — Preceding unsigned comment added by George Albert Lee (talk • contribs) 20:15, 10 March 2021 (UTC)

JFB80 (talk) 19:34, 17 March 2022 (UTC)== Velocity composition not linear ==

Regarding my undo of user Randallbsmith's edit, I think the original formulation with two constants was OK, so I restored it, but do we have a proper inline textbook source to settle it? - DVdm (talk) 10:16, 11 February 2022 (UTC)

Let me explain my edit: consider routine real number addition: f(X,Y) = X+Y. Would you expect f(aX, bY) = ab f(X,Y)? It is almost never true, and even if it were true, it is certainly not what is meant by linearity: for an appropriate definition of linearity, I'd appeal to the link listed which then leads us to https://en.wiki.x.io/wiki/Linear_map, in which our function's domain is the direct product space of the two input vectors. The definition refers to a single scalar c operating on the domain, and transliterates in our case to f(cX, cY) = c f(X,Y). This IS true for simple addition, even when X and Y are vectors, matrices, or complex numbers. One might wonder (as I actually did for my application of velocity addition) if I could count on f(aX, aY) = a f(X,Y). It does not hold for relativistic velocity addition, and I gently suggest (as per Wikipedia) the equality is a routine meaning for linearity AND the fact that is does not hold is a useful and interesting observation. To summarize, the equation as it stood was not a test for linearity, and is of little use since no one would expect f(aX, bY) = a b f(X,Y) anyway. My edit shows the routine meaning of linearity does not hold for relativistic velocity addition, and is both useful and, IMHO, kind of interesting! I trust that was a clear enough explanation to warrant replacing my edit? Anyway, stuck it back in there. Randallbsmith (talk) 17:49, 11 February 2022 (UTC)

Still, I think we really need a solid, proper, inline, textbook source for this.I spent some significant time looking for one, and failed to find one. ANYONE? - DVdm (talk) 18:58, 12 February 2022 (UTC)

To my taste, because the ordinary 3-vector addition function f() can be considered a mapping over a six dimensional space with the property f(aX) = af(X) makes the use of the term "linear" so obvious as to obviate the need for a non-Wikipedia reference. However, if this is truly a stretch, perhaps we could get right to the point without the clutter of explaining why some term ("lineraity") might be used here. Rather, simply observe "For the usual addition of 3-vectors $\mathbf {v}$ and $\mathbf {u}$, we know that $\lambda \mathbf {v} +\lambda \mathbf {u} =\lambda (\mathbf {v} +\mathbf {u} )$ for real $\lambda$. However, for relativistic velocity addition, in general $\lambda \mathbf {v} \oplus \lambda \mathbf {u} \neq \lambda (\mathbf {v} \oplus \mathbf {u} )$." Bottom line: the fact that equality doesn't hold is an important feature, worthy of being pointed out. Randallbsmith (talk) 17:30, 14 February 2022 (UTC)

The formula has a simple geometrical interpretation which makes it quite clear it is not linear. It only becomes linear when v/c -> 0 when it reduces to ordinary vector addition. But I think the article is based on a misunderstanding. This oplus formula does not represent relativistic addition of velocities because it omits the Thomas precession. This is what resulted in the "Mocanu paradox" and the non-associativity of the oplus formula. Regarding DVdm's remark I don't believe there exists any clear textbook explanation and, from what I have seen, in the literature. Ungar and Mocanu wrote about it but what they say is not very easy to follow. JFB80 (talk) 07:47, 15 February 2022 (UTC)

It sounds like you are saying the formula using $\oplus$ is simply incorrect? But I just went through my own derivation from scratch using a Lorentz transformation on spacetime intervals for the simpler case of 2+1 dimensions, and it appears to replicate exactly the equation at the end of the "General Configuration" section. Perhaps it is not clear: I assumed all 3 frames start at mutual rest with 2 spatial axes parallel, then take off in arbitrary directions at constant velocity, with no rotation. (Rotation can be physically verified as absent, using a mass on a string or whatever.) Is the error only present in 3 dimensions, or in the case the frames also rotate? Randallbsmith (talk) 19:28, 16 February 2022 (UTC)

No it is not incorrect and represents part of the answer. See p.6 of Thomas' 1927 paper where he says:"A Lorentz transformation with velocity -v followed by one with velocity v+δv where δv is infinitesimal, resolves into a Lorentz transformation with infinitesimal velocity γ {δv + (γ-1)(v.δv)v/v²} together with an infinitesimal rotation (γ-1)(vxδv)/v²". The` first term can be shown to be what is given by the oplus (or ominus) formula. (sorry I don't use the markup - it is too slow, I am not very used to it).JFB80 (talk) 17:09, 17 February 2022 (UTC)

You have my sympathies regarding the markup! Well, I am going to come to peace with things was they are then. I can't understand the distinction between "not incorrect" and being just "part of the answer." I would like to use the full answer, not just part, but since I also derived the result in the article, I will go with that. Also the rotation term in the formula you quoted from Thomas has a single cross product and thus has a handiness sensitivity, so it can't be simply an additional term for the resultant, on physical grounds.(If δv is to the left the resultant velocity would point slightly up, if it is to the right, the resultant points slightly down. The universe doesn't work like that, the resultant must be in the plane of the two vectors). — Preceding unsigned comment added by Randallbsmith (talk • contribs) 20:55, 17 February 2022 (UTC)

The paper I quoted is everywhere accepted. Check up yourself if you like on p.6. The link is http://www.clifford.org/drbill/csueb/4250/topics/thomas_papers/Thomas1927.pdf. You might also like to see Wiki 'Wigner rotation'(but very difficult to understand). The mystery is explained by the fact that the Lorentz transformation is really a transformation in hyperbolic space which gives a rotation by 'parallel transport' which is what Thomas' paper was about to explain electron precession.(see also Borel 1913 CR Paris)JFB80 (talk) 16:47, 18 February 2022 (UTC)

But surely you accept the statement: product of 2 boosts = boost x rotation matrix. A boost gives a translation in hyperbolic space. So is'nt it the same sort of thing? JFB80 (talk) 13:04, 19 February 2022 (UTC)

Absolutely agree. And directly related important to note that for Lorentz Boosts $B$, $B(U)B(V)\neq B(U\oplus V)$. As to the correctness of the addition formula given here (definition of $\oplus$) I believe it is correct and is used in many papers discussing Thomas rotation and the Mocanu paradox. The trickiness of the formula is as follows: The formula is correct in giving a resultant vector aiming in the proper direction with the proper length in space. You can literally watch the moving object slide along the resulting velocity vector as time passes. But to read off the coordinate values of this vector, one needs to account for the relative orientation of the frame axes you are using. This is clearly explained in a short and accessible paper intended for undergraduate level physics by Silagadze "Thomas rotation and Mocanu paradox -- not at all paradoxical." Lat. Am. J. Phys. Educ. 6 (2012), 67-71 arXiv:0907.4482 [physics.class-ph]. Frames may start out parallel in which case coordinates are easily shared, but after two boosts to a frame moving with $U\oplus V$ the frames are no longer parallel thanks to the Thomas rotation. Randallbsmith (talk) 18:24, 26 February 2022 (UTC)

I'm glad you agree. The Silagadze paper was earlier in arXiv at Silagadze 2009 https://arxiv.org/abs/0907.4482v1.JFB80 (talk) 16:16, 27 February 2022 (UTC)

It is interesting to note that the oplus formula for addition was originally that stated by Einstein (1905 section 5) and is sometimes correctly called Einstein addition. But the formula omits Thomas rotation and so people, naturally believing in the Einstein result, make a mistake.JFB80 (talk) 19:34, 17 March 2022 (UTC)

Quick lookup for velocity addition formula

As a quick reference, readers need a definition of u' right below the velocity addition formula. The formula is good, but hunting around for u' did not give me an unambiguous definition. Right below the equation, add "where u' is the relative velocity of the systems."97.73.100.22 (talk) 21:53, 31 August 2023 (UTC)quemadojournal.org

Broken English or otherwise unmotivated phrasing in Galilean relativity section

This section starts with the sentence "It was observed by Galileo that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward". As a native English speaker (and incidentally a physicist), this sentence is either broken or written in a way that suggests prior context about a heavy body, why it might falling, etc. It should be edited accordingly, although I will refrain since I am not sure if the original author had other intentions. — Preceding unsigned comment added by Mrkinzie (talk • contribs) 13:37, 27 January 2024 (UTC)