Talk:Velocity-addition formula/Archive 1

http://upload.wikimedia.org/math/6/d/c/6dc88e88a74eac0290cb0d575aa7a147.png There is an algebraic error in the above equation. The rightmost term should read: (c v1 + c v2)/(c + [v1 v2]/c). Basic unit analysis shows that the original quantity is incorrect.

Sorry for the botched image include. But the URL is there for reference. Cheers, Justin.


 * Right you are. I fixed it. —Keenan Pepper 02:52, 2 July 2006 (UTC)

Problem of symmetry
Sorry folks, but are you sure that this formula is correct: $$\mathbf{v_1} \oplus \mathbf{v_2}=\frac{\mathbf{v_1}+\mathbf{v_2}}{1+ \frac{\mathbf{v_1}\cdot\mathbf{v_2}}{c^2}} + \frac{1}{c^2} \cdot \frac{1}{1+\sqrt{1-\frac{v_1^2}{c^2}}} \cdot \frac{\mathbf{v_1}\times(\mathbf{v_1}\times\mathbf{v_2})}{1+\frac{\mathbf{v_1}\cdot\mathbf{v_2}}{c^2}}$$

I have my doubts about that, since $$\oplus$$ is not symmetric. The result of this formula seems to be correct only, if $$v_1$$ and $$v_2$$ are collinear.

Can anyone confirm this? --141.33.44.201 10:20, 14 December 2006 (UTC)


 * The correct formula is only symmetric in the collinear case, as others have said below. The formula above is an attempt to correct for the fact that there is an extra time dilation factor of $$\sqrt{1-v_1^2}$$ for the perpendicular component in the case of non-collinear motions. The triple cross product is a three-dimensional way of selecting the component of $$v_2$$ perpendicular to $$v_1$$.


 * The triple cross product evaluates to $$|v_1|^2$$ times the desired component of $$v_2$$. The fraction in front is supposed to add with the $$v_2$$ from the first term, to produce the required $$\sqrt{1-|v|^2}$$. Since the first term contains the component perpendicular multiplied by 1, the second factor needs to equal $$\sqrt{1-v_1^2}-1 \over |v_1|^2$$ to work. It does, as can be seen by simplifying this fraction by multiplying top and bottom by $$\sqrt{1-v_1^2} +1 $$. But because the algebraic manipulations were designed to work with the cross product, this is a confusing formula. Selecting perpendicular and parallel components should be done with the dot product, as in the rewrite.  —Preceding unsigned comment added by 71.127.173.103 (talk) 02:35, 3 September 2007 (UTC)

Velocity units
"When Velocity is expressed in metres per second, instead of as a fraction of the speed of light the equation becomes..."

Of course, this statement is true, but it would also be true if velocity is expressed in kilometers per hour, miles per hour, or knots too. The main thing is unit distance and time rather than fraction of C. Perhaps the statement should reflect this fact.

ANSWER: Problem of symmetry
The formula IS NOT and DOES NOT HAVE TO BE symmetric in the component vectors. The problem is that the article does not explain the physical meaning of these components. The explanation should be added because many misunderstand the relativistic addition of velocities. (see p. ex. the award winning article "Speed of light". A bitter fight is going on because of the author's stubbornness. And guess what, they refer to THIS article as the source of their interpretation. Since there is no interpretatiton here the whole fight is futile.)

Here is the interpretatiton

Let us picture 3 observers 1, 2, and 3. Let v1 denote the volocity vector of 2 as observed by 1. Also, let v2 denote the velocity vector of 3 as observed by 2. Then if 1 observes 3, too, then they will see the velocity vector given by the formula. Is it clear now that the formula does not have to be symmetrical? The point is thet the roles of the observers are not symmetrical.

The rather suprising thing is that if 1, 2, 3, move along the same straight line then the formula becomes symmetrical.

IMPORTANT: SOMEBODY SHOULD INCLUDE THIS EXPLANATION ABOUT THE PHYSICAL MEANING OF THE COMPONENTS. I repeat, many misunderstand the Einstein formula. This is very dangerous. Relativity theory is already the hot bed of of stupidity and misunderstanding.

zgyorfi

Answer for the Answer
Thanks. Meanwhile I looked the general lorentz-boost up in one of my physics books. And I (surprisingly) realised, that the formula indeed is not symmetric. --141.33.44.201 09:50, 29 March 2007 (UTC)

cleanup tag or expert needed tag or what
the people here and a physicist elsewhere say this article needs work. I'm putting an expert needed tag since it seems likely to need smarties to clean it up.Rich 19:10, 28 April 2007 (UTC)


 * I did a complete rewrite. Likebox 02:45, 3 September 2007 (UTC)

Two successive dot products in the first equation?
Why are ther two successive dot products in the first equation? Errors as simple as this give great cause for concern about the accuracy of the rest of the program. —The preceding unsigned comment was added by Special:Contributions/ (talk)

Those dots can be removed as what it on the lhs of each one is a scalar and not a vector. --EMS | Talk 19:40, 23 August 2007 (UTC)

Doppler Stuff is Not Very Physical
I only kept it because the previous version had it as a reasonable addition formula. Maybe it should be erased entirely. At least the relativistic and nonrelativistic versions are correct now. Likebox 03:42, 3 September 2007 (UTC)

I figured out why it's a velocity addition law. But it is very peculiar--- its for the Doppler effects in a one-dimensional right-moving wave. Only then is there a group which gives this addition law. This has no relativistic analog. I don't know the context in which this addition law is used, if any. Likebox 23:43, 4 September 2007 (UTC)

What about c-c?
I want to determine what we get by adding $$c$$ and $$-c$$ together. Because we are adding two relativistic speeds we should use the velocity addition formula, right? But it is undefined for $$v_1 = -v_2 = c$$. And things don't get any clearer if we consider the two limits $$\lim_{v \rightarrow c} \frac{c - v}{1 - \frac{cv}{c^2}} = c$$ and $$\lim_{v \rightarrow c} \frac{v - c}{1 - \frac{cv}{c^2}} = -c$$. It obviously is not a revokable singularity, so what is really going on here?

And if we set $$c \oplus -c = 0$$ we loose the associativity, since then we have $$(v \oplus c) \oplus -c = c \oplus -c = 0$$, and $$v \oplus (c \oplus -c) = v \oplus 0 = v$$. Can someone please explain what all the trouble is about? --Rndusr (talk) 00:02, 21 December 2007 (UTC)


 * The trouble is made clearer by taking a limit. If you have a particle moving with c-e where e is very small compared to c, and you add the velocity -(c-e) you get a particle at rest. If you add -(c-e/2) you get a particle moving with a speed c/3. If you boost by c-ae where a is an arbitrary number, you get an arbitrary result. There is no unique answer to adding c and -c because for speeds very close to light, the result of subtraction is any answer you want.Likebox (talk) 02:06, 21 December 2007 (UTC)

Rndusr, good question, yet it doesn't even matter. In the composition formula one of the "velocities to compose" is the velocity of one observer (object) w.r.t the other. This velocity cannot be c or -c. Light signals cannot be treated as observers, so your case (c,c) is out scope for the formula. DVdm (talk) 09:18, 21 December 2007 (UTC)

Suggested change of name
I suggest that the name of the page is changed to 'velocity-composition formula' with a redirect from the original name. As has already been pointed out, relativity is a hotbed of misunderstanding and stupidity and use of the term 'addition' causes much confusion in that some people seem to think that in that relativity 2 + 2 can equal 3. Also, velocities can be added normally in some circumstances such as closing speeds. Martin Hogbin (talk) 12:29, 23 March 2008 (UTC)

too complicated
The previous comment about renaming the article 'velocity-composition formula' makes sense, since it would not imply a simple 'addition' method. I was checking to see if the previous incorrect formula had been changed, and obviously it has. Now the article seems way too complicated with the 'long' discussion of doppler effects.Phyti (talk) 15:38, 17 May 2008 (UTC)phyti


 * I agree, the Doppler shift bit has no real relevance to the main subject and should be deleted from this article.Martin Hogbin (talk) 22:00, 1 July 2008 (UTC)

Citation should be fine
First formula is the same as in "Zur Elektrodynamik der bewegter Koerper". Ann. Phys., 17, 891-921.

paragraph 5.

This should be appropriate reference;) AlexShkotin (talk) 15:43, 18 October 2008 (UTC)

Gyrovector space??
Anyone know anything? --unsigned comment--


 * I added the link. I am currently constructing that page. Only colinear velocites are commutative and associative, but in general, addition of non-colinear velocities is non-associative and non-commutative. The set of admissable velocities forms a hyperbolic space. There have been various attempts in the past to use hyperbolic geometry to study special relativity but these have been unpopular due to lack of new results and hampered by the non-associativity, however gyrovectors changes all this. It "repairs" the situation with gyroassociativity and gyrocommutativity.

A book review is given here: http://www.springerlink.com/content/h7u2nq7mp8apunx9/fulltext.pdf Delaszk (talk) 21:04, 21 November 2008 (UTC)

Concerning gyrovectors someone might like to ponder the following problem:
 * It is known that mathematically $$\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w})$$ is not identically equal to $$(\mathbf{u} \oplus \mathbf{v})\oplus \mathbf{w}$$. So that at most one expression can be correct. Which one is it and why?

JFB80 (talk) 20:46, 27 October 2010 (UTC)


 * This kind of pondering is not really on topic on a article talk page and should really be put forward at, for instance, wp:Reference desk/Science, but anyway... I don't think that "correctness" is a relevant expression property here, but surely, when u is combined with the result of the combination of v and w, then the former is "correct", whereas when the combination of u and v is combined with w then the latter is "correct". DVdm (talk) 21:00, 27 October 2010 (UTC)


 * Since there is talk about non-associativity is it not good to be clear? Your reply avoids the problem which is obviously this: if there are points A,B,C,D having relative velocities u,v,w, which (if either) of the two formulae expresses the velocity of D relative to A? At most one can, so maybe the talk about non-associativity is meaningless JFB80 (talk) 08:28, 28 October 2010 (UTC)


 * It's worse than that - since it isn't commutative either, then all the following expressions are different: u+(v+w), (v+w)+u, (w+v)+u, u+(w+v), (u+v)+w, w+(u+v), w+(v+u), (v+u)+w
 * and it's even worse than that still - if you add more points E,F,... with relative velocites x,y,... then there are even more combinations of expressions every time you add a new point. 89.241.236.121 (talk) 11:30, 28 October 2010 (UTC)
 * The noncommutativity and nonassociativity mean that you have to be careful to match the expression to the physical meaning. If the velocity of D from A's perspective is given by u+(v+w) then you must be careful to use that expression in your calculations and not accidentally use (u+v)+w. But is the velocity given by u+(v+w) ? 89.241.236.121 (talk) 11:38, 28 October 2010 (UTC)
 * I don't think we can currently say that any of these expressions has physical meaning. How do we know the velocity composition law matches reality? There haven't been any experiments done to test the validity of the velocity addition law except when the velocites are colinear. Ungar says stellar aberration shows that the addition law is not a composition law and instead the addition should be given by what he calls the co-addition to the usual addition law. The co-addition law is symmetric (commutative) but it is still nonassociative. 89.241.236.121 (talk) 11:57, 28 October 2010 (UTC)
 * (ec) Ah yes, I see what you mean now. Tricky, although of course the magnitudes of the vectors are equal. Some of this is related to the fact that even $$\mathbf{v} \oplus \mathbf{u} \neq \mathbf{u} \oplus \mathbf{v}$$, so we don' even know the "real" relative velocity between A and C. The question is mentioned and treated in the cited source here but the essential pages (7 and 8) seem to be left out from the preview. If anyone has full access to it, they can add a remark in the text. DVdm (talk) 12:14, 28 October 2010 (UTC)
 * The content of those pages also appear in the paper The Relativistic Composite Velocity Reciprocity Principle, however in later works Ungar came to the conclusion that $$\oplus$$, should be replaced with the co-addition which is denoted $$\mathbf{u} \boxplus \mathbf{v}$$, for which we do have $$\mathbf{u}\boxplus \mathbf{v} = \mathbf{v} \boxplus \mathbf{u}$$. Also the associativity problem of choosing where to put the brackets can be solved in the case of the co-addition by, instead of composing $$\boxplus$$, deriving a formula for $$\mathbf{u} \boxplus_3 \mathbf{v} \boxplus_3 \mathbf{w}$$ which is symmetric in all 3 velocities, so co-addition $$\boxplus$$ is denoted $$\boxplus_2$$ to distinguish it from $$\boxplus_3$$, and this can be extended to derive a formula for $$\boxplus_n$$ which appears under the name gyroparallelepiped law in the books 1, 2. 89.241.230.124 (talk) 15:31, 28 October 2010 (UTC)


 * That's nice. Thanks for the link. As far as I'm concerned, please feel free to amend the article. DVdm (talk) 15:39, 28 October 2010 (UTC)


 * Are you suggesting that from now on we should use this new idea instead of the familiar one due to Einstein?  Do you consider it established mathematics? JFB80 (talk) 21:39, 13 November 2010 (UTC)


 * I'm not sure what you mean with "we should use", but the thing is properly sourced and it does solve a problem in an obviously established mathematical way, so I don't see why it should not be briefly mentioned, as it is now. DVdm (talk) 21:33, 13 November 2010 (UTC)

During an overhaul of the article, the gyrovectors were left out entirely. This is because they are related to a treatment of a topic much more advanced than that of this article, namely Thomas precession. It is doubtful whether they belong there either, because they are based on an abstraction and further generalization of the Thomas rotation formalism. At any rate, the "which is correct" question is treated in the article, in the case of $u ⊕ v ≠ v ⊕ u$. The expressions are all correct, they just refer to different coordinate systems, related by rotations. Since these coordinate systems are non-parallel, direct parallel transport is not possible, unlike in the case of a single boost or collinear boosts. YohanN7 (talk) 23:06, 16 July 2015 (UTC)

Euclidean rotation through a purely imaginary angle
The section discussing rapidity is obscure. While it is true that some authors introduce hyperbolic functions by using the imaginary variable in the classical circular functions sine and cosine, this approach makes the subject difficult for novices. Infinite series or reference to the length of sides of a right triangle on y=x from a point (u, 1/u) are more direct. Certainly the idea of Euclidean rotation is incorrect here.
 * Some modifications to the section using rapidity have been made.Rgdboer (talk) 00:41, 13 January 2009 (UTC)
 * Rapidity is not an angle, it is a relativistic velocity (scaled by the factor c). It did originate with Minkowski's (and Sommerfeld's) interpretation of the Lorentz transformation as a Euclidean rotation through an imaginary angle but then it was reinterpreted by Varicak in hyperbolic geometry as a velocity.JFB80 (talk) 21:11, 17 October 2010 (UTC)

Constant accelerations
Frequently for space travel considerations, it is useful to know what the speed (relative to a stationary observer) over time is of a spacecraft undergoing a constant acceleration (in its reference frame, for instance if it were accelerating at 1g to simulate Earth gravity). This can be derived (below) from the velocity addition formula, and is v(t)=c tanh(at/c) (for a constant acceleration a). Is this worth including in the article?

Derivation: The relativistic addition of velocities formula is
 * $$s = \frac{v+u}{1+(uv/c^2)},$$

which means that if you are already traveling at a speed of v relative to an outside observer, and accelerate at a rate of a for a time interval $$\Delta t$$, your new speed is
 * $$v+\Delta v = \frac{v+a\Delta t}{1+(va\Delta t/c^2)},$$

so
 * $$\Delta v = \frac{v+a\Delta t}{1+(va\Delta t/c^2)}-v=\frac{v+a\Delta t}{1+(va\Delta t/c^2)}-\frac{v+(v^2/c^2)(a\Delta t)}{1+(va\Delta t/c^2)}=a\Delta t\frac{1-(v^2/c^2)}{1+(va\Delta t/c^2)}.$$

So we have
 * $$\frac{\Delta v}{\Delta t} =a \frac{1-(v^2/c^2)}{1+(va\Delta t/c^2)}.$$

Letting $$\Delta t \to 0$$, we obtain
 * $$\frac{dv}{dt}=a(1-v^2/c^2)$$.

This is a first order separable differential equation, so we can solve:
 * $$\frac{c^2}{c^2-v^2}\, dv=a dt$$
 * $$\int \frac{c^2}{c^2-v^2}\, dv=\int a dt$$
 * $$c \tanh^{-1}(v/c)=at+C\,$$
 * $$v(t)=c\tanh(at/c)\,.$$

skeptical scientist (talk) 02:48, 8 August 2009 (UTC)


 * Yes, this is a nice application of the composition formula. (See also my version - it is essentially the same). But unless we find an acceptable source for this, we couldn't really add it here.
 * By the way, I added one step and did some reformatting - hope you don't mind...
 * DVdm (talk) 20:03, 8 August 2009 (UTC)


 * A similar result was shown in the book: S.J.Prokhovnik The Logic of Special Relativity Cambridge U.P. 1967, Appendix 3, equation (3.8.6). More recently I have generalized it. JFB80 (talk) 20:01, 9 October 2010 (UTC)
 * A further comment: this problem goes right back to the beginning of the special theory with experiments of Kaufmann el al on the motion of an electron under constant acceleration in an electric field. These experiments first gave clear confirmation of the special theory (Bucherer Phys.Z 1908) So there is plenty of background if the topic is to be introduced into the article. The problem was discussed in Minkowski Raum und Zeit 1908 and Max Born Ann d Physik 1909 who gave the motion the name 'hyperbolische bewegung'. It would also fit in well with introductory comments on Galileo because it was he who first clarified the idea of constant acceleration with experiments on balls rolling down inclined planes. JFB80 (talk) 09:13, 11 October 2010 (UTC)


 * Yes, as I said it is a nice application of the formula. We could introduce the derivation in the article provided (1) the derivation can be found in these sources, and (2) other contributes agree that it is sufficiently important to be included. DVdm (talk) 15:32, 11 October 2010 (UTC)


 * This is also presented as problem 12.11 in Kleppner & Kolenkow (the formula for the acceleration, not its integration). It fits nicely in the article imo (provided correct ). YohanN7 (talk) 23:22, 16 July 2015 (UTC)
 * I think that relativistic acceleration is usually defined by differentiation with respect to proper time τ which puts in an extra γ factor. You may like to check with Rindler's book. Another point is that the final result is nicely interpreted in terms of rapidity. JFB80 (talk) 09:12, 17 July 2015 (UTC)
 * Yes, $a = du⁄ds$ is $4$-acceleration according to Landau and Lifshitz. This is (if I get things right) the proper acceleration in the rest frame of the thing being accelerated. But for this article, the quantity of interest is the one in the formulae above. YohanN7 (talk) 11:33, 17 July 2015 (UTC)
 * If I remember correctly, Rindler says (proper) acceleration a = γ^3 dv/dt which does not agree with your calculation which would give a = γ^2 dv/dt. Can you relate your calculation to your quote from Landau and Lifschitz? JFB80 (talk) 20:14, 17 July 2015 (UTC)
 * I have not made any calculation. To what do you refer? I quote K&K, not L&L. YohanN7 (talk) 15:33, 22 July 2015 (UTC)

See four vector for the components of the basic four vectors including acceleration. We could put acceleration the article (even if just constant), but four acceleration would be a more relevant home and this article could link to there. M&and;Ŝc2ħεИτlk 07:25, 24 July 2015 (UTC)

W. Kantor's 1972 paper on non-parallel velocities
I added the sentence "Experimental results have been reported that conflict with the formula's prediction for non-parallel velocities" about this paper: Nonparallel convention of light but it was reverted. Are you so sure that the paper is crackpot ? Charvest (talk) 19:46, 25 September 2009 (UTC)

Yes. That journal has published papers by Jack Sarfatti. Martin Hogbin (talk) 19:50, 25 September 2009 (UTC)


 * I haven't heard of him, but crackpot by association doesn't seem to be a watertight argument. Charvest (talk) 19:54, 25 September 2009 (UTC)


 * In this case it does. If I were you, I'd stay away from it afap :-) - DVdm (talk) 20:00, 25 September 2009 (UTC)


 * I've reverted my edits to other articles that included this ref. Upon closer reading, the paper even says that experiments contradict the co-linear case of the formula as well. Charvest (talk)

Sommerfeld's paper: On the Composition of Velocities in the Theory of Relativity
I've removed the source because the accompanying commentary was wrong. The source does not say velocities are not cartesian. The source says "it apparently better corresponds to the meaning of the theory of relativity to calculate and (by consideration of the reality relations) to construct with rotation angles, instead of only using its tangents, the velocities." 89.241.225.124 (talk) 08:55, 12 December 2010 (UTC)
 * You were very quick to remove it. You should have thought about it a little because you are quite wrong. If you look at fig.2 you will see that Sommerfeld is combining arcs on the surface of a sphere. fig.1 is the Cartesian (Euclidean) figure. Just after fig.2 Sommerfeld said:
 * In summarizing we can say: For the composition of velocities in the theory of relativity, not the formulas of the plane, but the formulas of the spherical trigonometry (with imaginary sides) are valid. By this remark the complicated transformation calculus becomes dispensable, and can be replaced by a lucid construction on a sphere. Is not this clear enough? Please replyJFB80 (talk) 17:03, 12 December 2010 (UTC)
 * The adjective "cartesian" describes coordinate systems. It does not describe objects. For example complex numbers can be given in either cartesian or polar form. The word "cartesian" does not refer to a plane vs a sphere: Even a sphere can be described with cartesian coordinates. The equation of a sphere of unit radius at the origin in a cartesian coordinate system is x^2 + y^2 + z^2 = 1, but the equation of the same sphere in a spherical coordinate system is just r=1.
 * Fig 2 in 1 does combine arcs on a sphere but the arcs don't represent velocities, they represent angles. The source says "If we namely compose (Fig. 2) the rotation angles $$\varphi_{1},\varphi_{2}$$ as arcs on a unit sphere". Angles used in that paper are related to velocities by the equation near the beginning of the paper $$\cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}}$$ where β=v/c.


 * The phrase "not the formulas of the plane, but the formulas of the spherical trigonometry (with imaginary sides) are valid." is pure rhetoric. Things can be represented all sorts of ways and as long as they give the right answers then no representation is more or less valid than any other. For example equations with trig functions can be rewritten with hyperbolic functions using the identities $$\cosh x = \cos ix \,$$ and $$\sinh x = -i \sin ix \,$$.
 * The set of admissable velocities is described by the cartesian coordinates { v=(x,y,z) such that |v|<c }. The addition of velocities is of course not the addition of R^3 but the velocity-addition formula. 89.241.239.164 (talk) 19:08, 12 December 2010 (UTC)
 * (1) Your remark about the use of 'Cartesian' is a quibble. Read 'Euclidean' for 'Cartesian' if you like. What are ordinary vectors, Euclidean or Cartesian? Anyone who wants to understand, can understand. (2) Spherical arcs correspond to Minkowski imaginary angles which then define rapidities. (3) Sommerfeld was an outstanding authority on special relativity. He did not indulge in rhetoric because he knew what he was talking about - refer to p.14 of S Walter's article [] In my opinion Sommerfeld's paper was a fundamental contribution which, already 100 years ago, indicated a correct scientific solution to velocity composition currently discussed by the theory of gyro-vectors which is invading Wiki articles and monopolizing discussion. I would like to bring to notice the existence of an alternative - a good one. Can you object to that?JFB80 (talk) 17:35, 13 December 2010 (UTC)
 * You added to the article the Sommerfeld ref, and said "An alternative way to explain the effect known as gyro-rotation" with the note: "This approach recognizes that velocities are not Cartesian." The sentence begins with "This approach recognizes" which implies that what is being recognized is not recognized by the other approach. You say "Read 'Euclidean' for 'Cartesian' if you like.". In that case the sentence would become "This approach recognizes that velocities are not Euclidean." implying that the other approach (gyrovectors) treats velocities as Euclidean vectors, which is not true, gyrovectors don't treat velocities as Euclidean vectors. Perhaps you meant to say "This approach recognizes that velocities are not ordinary vectors." but again nobody is saying that they are ordinary vectors, so it is incorrect to write "This approach recognizes" as if the other approach doesn't. 89.241.237.134 (talk) 12:07, 14 December 2010 (UTC)
 * The gyrovector theory started off from Einstein's 1905 derivation generalized to 3 dimensions as described earlier in this velocity-addition article. The operation v(+)u is the velocity observed in a frame S of a point moving with velocity u in another frame S' which is moving relative to S with velocity v. So initially at least what you said (gyrovectors don't treat velocities as Euclidean vectors) was not true. Subsequently Ungar made an algebra out of this operation introducing the gyr angle and it became more and more abstract. But it is unclear whether these abstractions have any significant application other than the initial one for which ordinary mathematics should suffice (and is necessary anyway because it is difficult to locate and understand proofs of the gyro-formulae - for example, where is the formula for calculating gyr?). Hyperbolic velocities make their appearance via unproved statements about the Beltrami representation which in the article is not clearly related to the initial abstract definition of gyrovector.JFB80 (talk) 18:42, 15 December 2010 (UTC)JFB80 (talk) 07:05, 16 December 2010 (UTC)
 * The addition of Euclidean vectors is commutative. Relativistic velocity addition is not commutative in the 2d case discussed by Einstein, therefore it is nonsense to suggest the gyrovector approach treats velocities as Euclidean vectors. 89.241.234.151 (talk) 07:43, 16 December 2010 (UTC)
 * Please read more carefully - I said initially they were Euclidean vectors. In fact in his 1989 paper Ungar defined his variables u, v as belonging to R^3, the set of all 3-vectors in the Euclidean 3-space R^3 with magnitude smaller than the speed of light c in empty space. And in the 2 dimensional case, was'nt Einstein using ordinary Euclidean vectors? Who is talking nonsense? Reference: A.A.Ungar The relativistic velocity composition paradox and the Thomas rotation. Foundations of Physics vol.19, no.11, 1989 p.1386JFB80 (talk) 16:27, 16 December 2010 (UTC)JFB80 (talk) 16:35, 16 December 2010 (UTC)
 * Well the definition of a vector space (or gyrovector space) includes not just the set but also the addition operation. The set of relativistic velocities is a subset of euclidean vectors but they have different addition operations therefore they are not the same.


 * You make various remarks about unproved statements but Wikipedia is an encyclopedia not a textbook so proofs are usually not going to be included. There are plenty things the gyrovector space article could explain more fully but Wikipedia articles are works in progress. gyr[u,v] is a spatial rotation and its effect on a vector w is a new vector denoted by gyr[u,v]w the formula for which is given in this article. As a rotation gyr[u,v] can be represented as a 3x3 matrix, which can be calculated by finding the angle and plane between w and gyr[u,v]w and writing a rotation matrix for that angle and plane. 89.241.228.234 (talk) 18:21, 16 December 2010 (UTC)
 * Naturally complete proofs cannot be included in the Wiki articles (although references would be useful). But the gyro-speak would be clearer if more related to the mathematics we know. JFB80 (talk) 22:10, 18 December 2010 (UTC)

velocity addition formula with velocity expressed as gamma
the velocity addition formula with velocity expressed as gamma is:
 * z = x y + sqrt(x^2 - 1) sqrt(y^2 - 1)
 * http://www.physicsforums.com/showthread.php?p=3036746#post3036746
 * Just granpa (talk) 17:35, 14 December 2010 (UTC)

Reverted edits by 96.237.182.21
I've reverted the changes made today. Note that this article writes the formula as v+u to match up with the rest of this article, but ungar writes the formula as u+v, therefore the formula as it appears in this article swaps u and v wherever they appear in Ungar. Also note that the coordinates of v were already in the next formula, but they were factored, i.e. v+(stuff)v = { 1 + (stuff) }v. 92.41.6.146 (talk) 20:26, 20 June 2011 (UTC)

Small typo in 'Special theory of relativity' section
When it talks of the hyperbolic tangents it says
 * $$ s = {v+u \over 1+(vu/c^2)} . $$

is the same as

\tanh(\alpha + \beta) = {\tanh(\alpha) + \tanh(\beta) \over 1+ \tanh(\alpha) \tanh(\beta) } $$

where

{v\over c} = \tanh(\alpha) \,\,\,\, {u \over c}=\tanh(\beta) \,\,\,\,\, {s\over c}=\tanh(\alpha +\beta) ,$$

But unless I am missing something very obvious, the last term should just be $$ s =\tanh(\alpha +\beta)$$. Washyleopard (talk) 15:17, 8 August 2012 (UTC)


 * Looks ok to me. - DVdm (talk) 15:35, 8 August 2012 (UTC)

Nevermind, I see it now. Forgot about the c's in the numerator. As I thought, obvious mistake on my part.Washyleopard (talk) 16:22, 8 August 2012 (UTC)

Typo in the final expression for orthogonal velocities
The expression $$S = \sqrt{V^2 + U^2 - V^2 U^2}$$ should be as follows $$S = \sqrt{V^2 + U^2 - V^2 U^2 / c^2}$$

You can easily check it out. If inertial reference frame K' is moving with the speed V in the positive x direction relative to the frame K then
 * $$v_x = {v'_x + V \over 1 + v'_x V / c^2 } \, \quad v_y = {v'_y \sqrt{1 - V^2 / c^2} \over 1 + v'_x V / c^2 }$$

For the case of orthogonal velocities:
 * $$v'_x = 0 \, \quad v'_y = U \ , \quad S = \sqrt{v_x^2 + v_y^2} = \sqrt{V^2 + U^2 (1 - V^2 / c^2)} = \sqrt{V^2 + U^2 - V^2 U^2 / c^2}$$

MrPlough (talk) 08:36, 30 June 2013 (UTC)


 * In that context "units where c=1" are used. I.o.w. V=v/c and U=u/c. See the opening remark in the next subsection where "engineering units" are used. -DVdm (talk) 11:14, 1 July 2013 (UTC)

Non-relativistic Doppler shift
A notion of velocity addition can also be formulated in the theory of the nonrelativistic, one-dimensional Doppler shift. When the source of a wave is moving with non-relativistic velocity $s$ toward the receiver, the frequency of the waves is increased by a factor of $1/(1 − s/c)$. If the receiver is moving with velocity $v$, the frequency of the waves detected is decreased by a factor of $(1 − v/c)$. When both the source and the receiver are moving, the frequency measured is given by:



f' = f { 1- v/c \over 1- s/c } \, .$$

If a receiver measures velocities using Doppler shifts, and it determines that an object coming towards it is moving with velocity $u$, it is actually determining the shift in frequency, from which it calculates the velocity. Suppose that the receiver itself is moving with velocity $v$, but it does not take this into account in the calculation. It calculates the value $u$ falsely assuming that it is at rest. The velocity $u$ can then be thought of as the inferred velocity relative to the ship from Doppler shifts alone. What, then, is the actual velocity of the object relative to the medium?

Since the ship determined $u$ from the frequency, the frequency shift factor relative to the ship is



{1 \over 1-u/c} \, .$$

But this factor is not the frequency shift relative to a stationary receiver. For a stationary observer, it must be corrected by dividing by the frequency shift of the ship:



f' = f {1 \over (1-u/c)(1-v/c)} \, .$$

The velocity of the object relative to the medium is then given by


 * $$ s = v + u - {v u \over c} \, .$$

This is the true velocity of the object. Unlike the relativistic addition formula, the velocity $u$ is not the physical velocity of the object.

There is a group of transformations in one space and one time dimension for which this operation forms the addition law. The group is defined by all matrices:



\begin{pmatrix} 1 & 0 \\ {-v \over 1-v/c} & {1\over 1-v/c} \end{pmatrix} \,$$

When they act on $$\scriptstyle(t,x)$$, they produce the transformations


 * $$\, t'=t \qquad x'={x-vt\over 1-v/c}

\,$$

which is a Galilean boost accompanied by a rescaling of the $x$ coordinate. When two of these matrices are multiplied, the quantity $v$ (the velocity of the frame), combines according to the Doppler addition law.

The physical meaning can be extracted from the transformation. Time is the same for both frames, but the rescaling of the $x$ axis keeps the right-moving speed of sound fixed in the moving frame. This means that if the ship uses this transformation to define its frame, the ruler that it uses is the distance that the waves move to the right in one unit of time. The velocity $u$ can now be given a physical interpretation, although an unusual one. It is the velocity of the object as measured from the ship using a Doppler contracted ruler.

I am presently vandalizing the article from beginning to end. I feel that this section doesn't belong at all (and that the section on relativistic Doppler shift needs work). YohanN7 (talk) 00:08, 11 July 2015 (UTC)
 * A good piece of vandalism so far. Martin Hogbin (talk) 18:24, 11 July 2015 (UTC)
 * Thank you

Relativistic Doppler shift


In the theory of the relativistic Doppler shift, the case where the speed of the wave is equal to the speed of light is special, because then there is no preferred rest-frame. In this case the frequency of the received waves can only depend on the relativistic sum of the velocities of the emitter and the receiver. But when the speed of the wave $c ≠ 1$, meaning that the phase velocity of the wave is different from that of light, the relativistic Doppler shift formula does not depend only on the relative velocities of the emitter and receiver, but on their velocities with respect to the medium.

In the rest frame of the medium, the frequency emitted by a relativistic source moving with velocity $v$ is decreased by the time dilation of the source:


 * $$ f'= {f \sqrt{1-v^2/c^2} \over 1-v/c}.$$

If the receiver is moving with a velocity $u$ through the fluid perpendicular to the wave fronts, the frequency received is determined by the proper time between the events where the receiver crosses crests. The fluid frame time between crest-crossings does not require changing frames and is the same as in the non-relativistic case:


 * $$ \Delta t' = {\Delta t\over 1-u/c }.$$

In this time, the receiver has moved (in the fluid frame) an amount


 * $$ \Delta x' = u \Delta t'. \,$$

And the proper time between the two crest crossing is



\Delta \tau' = \Delta t'^2 - \Delta x'^2 =\Delta t{\sqrt{1-u^2/c^2}\over 1-u/c}. \,$$

And this is the time between crest-crossings as measured by the receiver. From this, the received frequency can be read off:
 * $$ f' = f \frac{1-u/c}{\sqrt{1-u^2/c^2}}=f \frac{\sqrt{1-u/c}}{\sqrt{1+u/c}} \,$$

Multiplying the two factors for the emitter and receiver gives the relativistic Doppler shift:


 * $$ f' = f {\sqrt{1-v^2/c^2}\over (1-v/c)}{(1-u/c)\over \sqrt{1-u^2/c^2}}.

\,$$

When $c = 1$, it simplifies:


 * $$ f' = f \sqrt{1+v \over 1-v}\sqrt{1-u \over 1+u} \,$$

and then



\sqrt{(1+v)(1-u) \over (1-v)(1+u)} = \sqrt{ 1+ (v-u)/(1-vu) \over 1 - (v-u)/(1-vu) } \, , $$

so that the relativistic Doppler shift of light is determined by the relativistic difference of the two velocities.

It is also possible to determine, in the relativistic case, the actual velocity of a source, when a moving ship falsely determines it from a Doppler shift without taking its own motion into account. Just as in the non-relativistic case, this is the velocity at which a source would have to be moving in order to make the Doppler shift factor for a moving receiver equal to the Doppler shift factor for the velocity $u$. It is the solution of the equation:



{\sqrt{1- s^2/c^2} \over (1- s/c)} = { \sqrt{1-v^2/c^2} \sqrt{1-u^2/c^2} \over (1- v/c)( 1-u/c)} $$

This is the relativistic analog of the Doppler velocity addition formula. When $c$ is not the speed of light, the velocity $u$ is not the velocity of anything, just a false inferred velocity from the point of view of the moving ship. In the relativistic case, there is no group of transformations for which this is the velocity addition law, since it is impossible to independently rescale time and distance measurements.

I feel that this section doesn't belong either. The purpose of the applications of the article should illustrate use of the velocity addition formulas. This is not done in the above. I have replaced both. YohanN7 (talk) 17:21, 14 July 2015 (UTC)

I should mention that I don't think the above is bad, it is just not spot on for the article. Perhaps it can be merged into Doppler effect and Relativistic Doppler effect respectively. YohanN7 (talk) 19:19, 14 July 2015 (UTC)


 * Good work, agreed with what you're saying. Will look at this more tomorrow falling asleep now. M&and;Ŝc2ħεИτlk 23:15, 15 July 2015 (UTC)


 * Later I'll add more refs and a diagram (same for other sections). It may be instructive to generalize to relative velocities in arbitrary directions, but this last bit is not essential.
 * About this edit, the time dilation does apply when considering time intervals at the same point in space, but I mixed the observer/emitter the wrong way round... M&and;Ŝc2ħεИτlk 07:45, 19 July 2015 (UTC)


 * Sure, but $T = γT&prime;$ isn't applicable since the time duration occurs at the same spatial point in the observer's frame (in fact it doesn't if $(x&prime;_{1}, t&prime;_{2})$ and $(x&prime;_{1}, t&prime;_{1})$ (with $t&prime;_{2} − t&prime;_{1} = T&prime;$) are Lorentz transformed. If you require the same spatial point in the observers frame, you end up with the quantity $τ$. YohanN7 (talk) 10:45, 19 July 2015 (UTC)


 * What do you mean by the formula "isn't applicable" when it is being used? M&and;Ŝc2ħεИτlk 15:54, 19 July 2015 (UTC)


 * You misread it (or I miswrite it, either way). It is a quote from the previous version. The reason that it holds is not because "the time duration occurs at the same spatial point in the observer's frame". It just holds. Hence writing "...is applicable since the time duration occurs at the same spatial point in the observer's frame" is misleading bordering to downright wrong. YohanN7 (talk) 16:07, 19 July 2015 (UTC)


 * OK. Diagram is in preparation now. M&and;Ŝc2ħεИτlk 16:10, 19 July 2015 (UTC)

The two last added references seem to be the same, and the generalization (to non-collinear motion) of the formula is identical to the original. This doesn't seem to be correct. YohanN7 (talk) 12:14, 20 July 2015 (UTC)


 * Oops that formula and description was pasted in by mistake and not meant to be there in that edit, but the references are not the same. Anyway I'll correct the formula, because the transverse motion has a shift by the Lorentz factor. (Sorry about the diagrams I know I said they are in prep and will be done but my laptop is being shared with someone else for now, only time for comments, will get them done ASAP). M&and;Ŝc2ħεИτlk 13:01, 20 July 2015 (UTC)


 * No hurry at all. B t w, do the new refs have the original formula? It is w/o reference as it stands. It is in the main article, likewise there w/o ref. YohanN7 (talk) 13:16, 20 July 2015 (UTC)


 * Oh, I'd very much like to see the modification due to non-collinearity coming last (and for both light waves and bullets). Now it disrupts the flow. YohanN7 (talk) 13:25, 20 July 2015 (UTC)


 * Those physics encyclopedias do not actually have the collinear formula, they just have the general result when the velocities are non collinear. I placed the new formula at the end of the light description because it is for light, but agree it can be moved down. I'll add more refs for the original formula (which can be transferred to the main article). M&and;Ŝc2ħεИτlk 18:58, 20 July 2015 (UTC)

For this edit, I'm sure it's a plus sign since those refs denote the source and emitter frequency emitters the opposite way round, so the relative velocity vector is reversed. M&and;Ŝc2ħεИτlk 00:13, 21 July 2015 (UTC)

No, it should be a minus sign, thanks YN7. M&and;Ŝc2ħεИτlk 00:14, 21 July 2015 (UTC)

Diagrams


Finally done, after some external good faith life interruptions but also my own blunders. I have redrawn three times because of how cluttered it became each time. The part which slowed the process down was actually defining clearly what the refs (specifically the two physics encyclopedias) meant by "the direction of wave propagation"... The wave propagates throughout space, in all directions. Concerning the Doppler effect, the important thing is the direction it is measured relative to the relative velocity. It is still cluttered, suggestions to simplify are welcome. M&and;Ŝc2ħεИτlk 00:00, 21 July 2015 (UTC)


 * The diagrams are fine if the emitter is stationary with respect to the medium. If not, they are not really correct unless my intuition is playing tricks with me. YohanN7 (talk) 01:32, 21 July 2015 (UTC)


 * Yes the medium is stationary with respect to the emitter. M&and;Ŝc2ħεИτlk 08:33, 21 July 2015 (UTC)


 * We could possibly include the angle $θ$ in the already in the "geometrical observation". Then we could begin with the general case. That would be the shortest and crispest presentation of the general case relativistic Doppler shift available anywhere, but still verifiable. YohanN7 (talk) 12:26, 21 July 2015 (UTC)


 * Even crisper, since this article is about the relativistic velocity addition where the emphasis should be, just start from the general case of relativistic velocity addition, including the theta angle. Then state the result for light (still including the theta angle), which can be cited by refs. What's the point of doing the light case before the mechanical case, only to return back to the light case again after? This is not the article on the relativistic Doppler effect itself so no need to elaborate on it. I'll try it later today. M&and;Ŝc2ħεИτlk 09:29, 22 July 2015 (UTC)

section: General configuration
This should be much crisper, shorter, more encyclopediac, and the proof should not be so messy. I am going through this to streamline and make it easier to follow while providing sufficient background and trying to maintain mathematical correctness, previously bits were introduced ad-hoc. The same manipulations for the parallel/perp split can be applied to other LTs in standard config and this should be in the text (no, this doesn't make it a textbook, it will inform the reader of a general and useful procedure that is easy to follow). I also added another reference. More to follow. M&and;Ŝc2ħεИτlk 21:31, 15 July 2015 (UTC)


 * While I like the edits, in the previous, more terse, version, things were certainly not introduced ad hoc. Introducing something ad hoc means introducing something unexplained in order to fit the correct answer. We do not need to glean at the correct answer to decompose vectors into parallel and perpendicular components with respect to another vector.


 * The problem with the current version is that it presents a wall of formulas, suggesting that this is difficult, while it is not. I suggest putting the middle part (essentially decomposition of a vector in parallel and perpendicular components) in a hide box. The rationale is that this is not even "one level down", it is two levels down, and rightly belonging in an article about elementary vector analysis. YohanN7 (talk) 14:15, 16 July 2015 (UTC)


 * Maybe "ad-hoc" was not the right word and if it sounded rude I apologize, but in the previous version, some parts were not introduced as clearly as they could have because of the terseness.
 * Yes, I don't particularly like the "wall of formulae" either, but disagree it looks "difficult". Since the diagram is there, let's cut the explanation on the parallel and perpendicular components and just refer to the figure. Also, since we agree not to use gamma and alpha, some of the formulae can be deleted (alternatively - the gamma versions can be placed in an nb with a note on how to get to them). M&and;Ŝc2ħεИτlk 15:37, 16 July 2015 (UTC)


 * Rude? Ha! If that was rude by you, then all discussion due to some perceived politeness code is made entirely impossible . No rudeness intended by you and no rudeness perceived by me. I was just nitpicking on choice of words, because the phrase ad hoc is misused often, and is not to be confused with hand-waving, or leaving out details, all three are different.
 * You can even actually be rude without me taking offense. I truly miss Incnis! A little mutual rudeness livens up gloomy days. Just don't, like someone did, report me to the police for doing OR on talk pages, because the next time that happens, I'll really go ballistic. YohanN7 (talk) 16:01, 16 July 2015 (UTC)


 * How about this keeping the figure, making a hide box for the middle block of text? Just cutting out the "wall" leaves the section without full proof, which the original actually had. We both agree that the wall isn't difficult, but it will have undesirable effects on readers. As for me, all equations I understand beforehand are easy and formulas I don't understand (yet or ever, doesn't matter) imply, if not difficulties, at the very least an effort. YohanN7 (talk) 16:27, 16 July 2015 (UTC)


 * The original proof had loads of tedious algebra and it was not done as efficiently as it could have been. I'm not sure where you want to draw the start and end of the "middle block" of text - please feel free to add the show/hide box yourself, or edit the section altogether. Thanks, M&and;Ŝc2ħεИτlk 16:45, 16 July 2015 (UTC)


 * Ok, I will give it a try a little later. YohanN7 (talk) 16:56, 16 July 2015 (UTC)


 * Done. But the new proof leaves out what was in the old proof entirely. The new proof just states that
 * Substituting the parallel component into...
 * and the formula results. (This observation has nothing to do with my last edit.) The old proof was a full proof, this isn't. But don't misunderstand me now, I think this is probably better, and leaves a little homework for the reader. YohanN7 (talk) 18:41, 16 July 2015 (UTC)


 * I like your new series of edits and arrangement of the text. And yes, leaving some homework for the reader will actually deepen their understanding with a little of their own self-experimentation. They will be guided but thinking for themselves. Thanks! M&and;Ŝc2ħεИτlk 19:42, 16 July 2015 (UTC)

Why switch between gamma and alpha??
Why can't we just stick with one or the other, whichever makes the expressions simpler?? The velocity addition equation in general configuration is simpler with alpha, not gamma, and alpha is just as easy to calculate limits with as gamma. Compare which side of the identity is simpler:


 * $$\frac{1}{u^2}(1-\alpha_u) = \frac{1}{c^2}\dfrac{\gamma_u}{1+\gamma_u}$$

... M&and;Ŝc2ħεИτlk 23:13, 15 July 2015 (UTC)

Put more explicitly:


 * $$ \mathbf{v} =\frac{1}{1+\frac{\mathbf{V}\cdot\mathbf{v}'}{c^{2}}}\left[\left(1+\frac{\mathbf{V}\cdot\mathbf{v}'}{V^{2}}\left(1-\alpha_V\right)\right)\mathbf{V}+\alpha_V\mathbf{v}'\right] $$

or


 * $$ \mathbf{v} =\frac{1}{1+\frac{\mathbf{V}\cdot\mathbf{v}'}{c^{2}}}\left[\left(1+\frac{\mathbf{V}\cdot\mathbf{v}'}{c^{2}}\frac{\gamma_V}{1+\gamma_V}\right)\mathbf{V}+\alpha_V\mathbf{v}'\right] $$

Similarly


 * $$\mathbf v = \frac{\mathbf u + \mathbf v}{1 + \frac{\mathbf u \cdot \mathbf v}{c^2}} + \frac{1}{u^2}(1-\alpha_u)\frac{\mathbf u \times(\mathbf u \times \mathbf v)} {1 + \frac{\mathbf u \cdot \mathbf v}{c^2}} $$

or


 * $$\mathbf v = \frac{\mathbf u + \mathbf v}{1 + \frac{\mathbf u \cdot \mathbf v}{c^2}} + \frac{1}{c^2}\frac{\gamma_u}{1+\gamma_u}\frac{\mathbf u \times(\mathbf u \times \mathbf v)} {1 + \frac{\mathbf u \cdot \mathbf v}{c^2}} $$

M&and;Ŝc2ħεИτlk 23:41, 15 July 2015 (UTC)


 * Yes, stick to one in final expressions. Direct verifiability (← that word English? Google:No!) perhaps goes out the window, but so what? YohanN7 (talk) 14:37, 16 July 2015 (UTC)

Why the reverts?
I haven't a clue what your are complaining about. I think it is the blank lines in the list of citations at the end. Again, per WP:Listgap, there can be no blank lines in a list. Also, I'm not a bot. You are not making sense in your edit summaries. Bgwhite (talk) 20:10, 17 July 2015 (UTC)


 * It is not lists. They are footnotes. Blank lines are used to render large blocks of text readable. But ordinary blank lines don't work in footnotes. This is a bug in the handling of footnotes. You can check the appearance yourself, I'll not revert for a few hours. YohanN7 (talk) 20:16, 17 July 2015 (UTC)


 * It would help if you tell me exactly where in the article your issue is located. Bgwhite (talk) 20:26, 17 July 2015 (UTC) Bgwhite (talk) 20:26, 17 July 2015 (UTC)


 * It is footnotes (or you can call them pop-ups if you want to) with the labels [nb 1] and [nb 2]. YohanN7 (talk) 20:32, 17 July 2015 (UTC)


 * The two in a row?  I just Replaced them with a .  HTML specs say not to use  as blank space, that is what  is for.  Bgwhite (talk) 20:53, 17 July 2015 (UTC)


 * Yes, that does it. Thank you. I never did read the HTML spec, and probably never will YohanN7 (talk) 21:00, 17 July 2015 (UTC)


 * It is a very useful read. One will easily fall asleep while reading it.   Next time, could you only revert to what you object to.  It will in knowing what exactly you have a problem with, especially when dealing with "slow" people like me. Bgwhite (talk) 21:18, 17 July 2015 (UTC)


 * I will. Perhaps I should explain another revert. I had placed identical references in a number of places with the intention of adding page numbers a little later. These were merged into (a named) reference. I had to revert that to save time (so I'm not waging war). YohanN7 (talk) 21:25, 17 July 2015 (UTC)


 * Just explain it in the edit summary. Hopefully I can remember. Bgwhite (talk) 22:57, 17 July 2015 (UTC)