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level-5 vital article is rated C-class on Wikipedia's
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As presented today, the matrix has minus signs in front of two of its components. These minus signs disappear when the parameter φ is made negative because sinh is odd and cosh is even. I realize that elsewhere this minus sign may be used but that is an insufficient reason to include it here. The use of any unnecessary minus signs tends to lead to confusion when one compares the matrix operator with a comparable matrix that represents ordinary rotation in a plane by ordinary complex multiplication. In that case there is one minus sign on the counter-diagonal. Using two minus signs does not improve the analogy. There should be no minus signs when none are necessary. Rgdboer ( talk) 22:44, 14 November 2008 (UTC)
The minus sign is not found in the use of the rapidity parameter in E.T. Whittaker's History of Electricity, neither in the 1910 edition nor in the 1953 edition. The superfluous symbols have then an evidence of needlessness. Rgdboer ( talk) 23:20, 14 July 2009 (UTC)
The problem of sign can be looked at in this way: Einstein 1905 had two reference frames S and S' with S' moving away from S along the x axis with uniform velocity v. He called S the stationary frame. Looking at frame S from S' (as Einstein did) we get the minus sign and looking at frame S' from S we get a plus sign. The books tend to follow Einstein but is it not more natural to view from the stationary system S and get the plus sign? (PS Notation: Einstein had frames K and k for S and S') JFB80 ( talk) 20:02, 11 October 2010 (UTC)
The issue here is that of active and passive transformations. As that article says, the terms "alias" and "alibi" have been adapted from detective fiction. The use of the minus sign means that "another place" is the frame of reference being brought to rest. Use of a plus sign refers to a new identity of a moving frame compared to the one at rest. As also mentioned in that article, the two points of view often arise from two different disciplines of study. Preference for one over the other then is a matter of taste. However, the issue has implications for the article on hyperbolic angle as the use of the minus sign here confuses the mathematical idea. See Talk:Hyperbolic angle#Sense of angle. Comments to improve this article are sought. Rgdboer ( talk) 20:55, 8 April 2014 (UTC)
As the article stands today, events are represented by column vectors and the transformation matrix operates on the left of this column vector. Successive transformations are then built up by placing transformation matrices to the left. On the other hand, as originally posted 11 November 2008, events were represented by row vectors so that transformation matrices may be written in the ordinary left-to-right direction. This stylistic convention of having further developments arise to the right in text is a strong tradition in European languages. Since the matrices for various φ form a multiplicative group, it seems more natural to apply them in an ordering like group multiplication. For these reasons, in this fundamental article in relativity theory, I recommend the left-to-right notation be adopted (restored). From the greater transparency here, editors may adapt these reasonable measures elsewhere. Rgdboer ( talk) 21:28, 24 November 2008 (UTC)
I've removed the last line, as I can't see that "constrains the future to a quadrant of a spacetime plane" actually have any meaning in this context. Apologies if I'm missing the point. The future is constrained to a light-cone in a normal spacetime plane - an isosceles triangle for one spatial dimension. Are we now envisioning a new measure of spatial extent defined as rapidity × speed of light × time? If so, and if it's relevant, it probably warrants some explanation. As it stands, it seems a bizarre, glib, confusing line; whereas the previous line seems to be a fine conclusion. Thoughts welcome - Bobathon ( talk) 20:51, 12 January 2009 (UTC)
Rgdboer, I am a little perplexed by your comment.
Ever since Einstein first published his special theory of relativity, people have been looking for appropriate examples, situations, and thought experiments to illustrate the unusual consequences of this theory. To my understand, the effort to come up with better examples of an existing theory is a pedagogical exercise and not original research.
Why do you consider this thought experiment to be original research? The consequences described in the story are precisely those predicted by special relativity, and special relativity is over 100 years old and is almost universally accepted. —Preceding unsigned comment added by Unitfreak ( talk • contribs) 05:14, 12 July 2009 (UTC)
In particle physics rapidity is usually defined as
,
where z is the beam axis. See for example eq 38.39 in
http://pdg.lbl.gov/2009/reviews/rpp2009-rev-kinematics.pdf
Maybe this definition should be presented as an alternative, especially since it is never called "longitudinal rapidity" or something else that it would make it distinguishable from the rapidity defined in the current article version. —Preceding unsigned comment added by 128.141.103.146 ( talk) 14:50, 5 November 2009 (UTC)
The article says "possible values of relativistic velocity form a manifold, where the metric tensor corresponds to the proper acceleration (see above)." I don't find anything "above" in the article to explain this correspondence. What does it mean for the metric tensor to "correspond to" the proper acceleration? Flau98bert ( talk) 14:33, 17 September 2012 (UTC)
At present the article is quite confused over this difference. It needs correction. JFB80 ( talk) 19:19, 18 January 2013 (UTC)
The following text was removed:
The introduction of new terms hyperbolic velocity and relativistic velocity in this article without reference and without contribution to understanding of the topic calls for discussion here. — Rgdboer ( talk) 02:47, 6 March 2016 (UTC)
You asked who gets to name things. The subject title is nomenclature, in this case scientific terminology as exemplified by the international scientific vocabulary. New terms are subject to naming conventions as in a systematic name. The encyclopedia only reports on usage as found in primary and secondary reliable sources. Notability of terminology is determined by standard metrics on educational and scientific texts. — Rgdboer ( talk) 03:28, 11 March 2016 (UTC)
Thank you for your thoughts. On new terms see neologism which is the target of the redirect at coining (linguistics). As for hyperbolic angle and its applications, references have been given and the material is classical though neglected in some study sequences. The application in this article provides a motive to investigate the concept more fully, such as in the area of a hyperbolic sector. But the fact that your textbooks missed some applications does not mean they are not standard. Your idea to accelerate the use of neologisms in this encyclopedia is contrary to policy and would tend to discredit the project as going off in its own direction rather than reporting on usage as is the policy now. — Rgdboer ( talk) 23:26, 12 March 2016 (UTC)
is uncommon. Usually the hyperbolic arc tangent is either or . Please provide a reference where is commonly used. iou ( talk) 18:59, 16 January 2017 (UTC)
I deleted a section beginning with
Rapidities aren't confined to a space of unit radius. Velocities are (with ). One has . The rest, copyedited for errors, might belong in Velocity-addition formula, but the relevant material is there already.
It was all promptly reverted with motivation that I should provide a reference claiming the incorrectness of the content. To be honest, finding reliable sources confirming that is wrong are hard to come by. You might find one saying is wrong with a bit of luck. No, I rely mostly on references giving what is right and take anything else as false. This should suffice.
As for the sentence in the lead deleted by me, a thread just above highlights what is going on. Even if someone 107 years ago wrote , and even if the contributor wrote a conference paper 1992, it is not notable and to my knowledge not used to the extent that it warrants mention in the lead of this article. YohanN7 ( talk) 10:07, 18 January 2017 (UTC)
I have not examined the dispute in detail, but in all fairness it would be interesting to discuss rapidity for boosts in different directions somewhere on WP, provided it is clear and correct. Stated as a problem: if you have
then how do ζ1 and ζ2 combine into ζ? Alternatively, is there a function z such that
I am guessing the dispute is related to the content of this:
but we need modern books with references to the original sources. That would be stronger for WP. M ∧Ŝ c2ħε Иτlk 11:34, 20 January 2017 (UTC)
The section In more than one space dimension has versions done by two editors. The one by JFB80 includes the link hyperboloid model which is pertinent. The version by YohanN7 introduces the Lie algebra of the Lorentz group with no mention of bivector (complex). Though the Lorentz group is a Lie group, presuming the reader can follow Lie theory is a stretch. Therefore I prefer the JFB80 version. Furthermore, YohanN7 asserted on January 25 that I removed something which I did not touch. Derogatory comments in an edit description are offside, and a falsehood is easily confirmed by History comparisons. We work here with just ourselves and software, with hope for uplifting but sometimes subject to degradation. Such is the state of man on earth. Rgdboer ( talk) 01:05, 27 January 2017 (UTC)
Thank you both for responding. Look at that History comparison YohanN7 since it shows more than a reversion, there was an insertion you made, and I did not remove it. Please stop perpetuating the falsehood.
It should be recalled that hyperbolic geometry was developed in the nineteenth century and that it is a metric geometry not applicable to spacetime. But A Treatise on Electricity and Magnetism was using t,x,y,z variables so some manifold was needed to make sense of it. Also from the nineteenth century came quaternions and biquaternions of Hamilton. In the struggle to achieve mathematical physics another contribution was hyperbolic quaternions. The stampede in the twentieth century was led by Ludwik Silberstein using biquaternions. Admittedly, the contributions of Sophus Lie and Elie Cartan are huge, but this article should be pegged to the likely reader, and he or she needs to learn examples of continuous groups. Advanced terminology is not helpful to the novice reader. Furthermore, disdain for mathematics of the past is unjustified. Rgdboer ( talk) 00:28, 28 January 2017 (UTC)
Yes, my point is that hyperbolic geometry and quaternions preceded spacetime relativity and enabled the physical theory. They amount to "off-the-shelf" technology that the conceptual engineers apply to physics. While there is no doubt that rapidity is a Lie group parameter, that feature does not define it. As we discuss a particular section aiming to expand the concept, some historic aspects have been brought up to highlight several decades of development that were necessary for the birth of relativity. For instance, non-Euclidean geometry emerged in 1870s as instantiated by models, but only in the 1920s did Klein articulate the boundary of velocity space as a Cayley absolute. See Felix Klein, M Ackerman translator (1979) Development of Mathematics in the 19th Century, p 138, Math Sci Press. Furthermore, linear algebra developed over the same period, and in our instance that includes the squeeze mapping. All fields converge in this encyclopedia project, so there is no separation between the Math dept and the Physics dept as on campuses. Rapidity is on the boundary of the two, so the challenge in this article is similar to others. Looking at the history of ideas clarifies essentials, and in this case of hyperbolic angle it is a transcendental function in the lens. Calculus teachers would do well to explain how these transcendental functions arise, and draw out the hyperbolic sector when they present the dented trapezoid as producing the natural logarithm. If these subjects were properly laid out the fantastical aspects of special relativity would be mundane. Rgdboer ( talk) 02:28, 29 January 2017 (UTC)
No, the flat spacetime arrangement was put together by the Swearingen Sisters in Austin Texas when Alexander Macfarlane joined G. B. Halsted at the University of Texas in 1885. Reference to Minkowski’s pompous pronouncement in 1908 is due to eminence of the German schools compared to American, Canadian, and English communities. Minkowski evidently caught wind of "The Great Vector Debate" in the pages of Nature (journal) and elsewhere, and his enunciation has been recalled as the watershed. — Rgdboer ( talk) 19:35, 31 January 2017 (UTC)
Concerning the Klein reference, the Ackerman translation published by Robert Hermann in 1979 was cited for the page 138. Checking the bookshelf here at QA 26 K 6 there is the 1950 Chelsea reprint of the German original. The cited passage is from chapter four "Herausbeitung einer rein projektiven Geometrie" where he cites Cayley on general projective measure. So the page number is 150 in Chelsea, volume 1, Entwicklung der Mathematik. The equation given in dx, dy, dz, and dt is in homogeneous coordinates and is equivalent to v2 = 1, boundary of velocity space, and Cayley absolute in the passage. — Rgdboer ( talk) 00:26, 2 February 2017 (UTC)
I'm new so sorry if I don't do this right, but trying to read this article and the one for proper velocity, I got very confused until I realized they both use opposite variables for the two quantities. w and η. —
Gravitative (
talk) 12:07, 11 April 2017 (UTC)
Rapidity and Celerity are closely linked, but according to the wiki pages for both, both use w to represent each other. The Celerity page refers to rapidity and eta so this page should use eta too.
@ User:DVdm: thank you for your dedication to keeping Wikipedia top notch. In this case, you might have used {{citation needed}} markup instead of the big undo axe, but I shall seek out one or more sources as you request. All the best — Quantling ( talk | contribs) 17:33, 23 January 2019 (UTC)
The following was removed as beyond the scope to Rapidity:
The relativistic velocity is associated to the rapidity of an object viaCite error: A <ref>
tag is missing the closing </ref>
(see the
help page).
[nb 1]
where refers to
relativistic velocity addition and is a unit vector in the direction of . This operation is not commutative nor associative. Rapidities with directions inclined at an angle have a resultant norm (ordinary Euclidean length) given by the
hyperbolic law of cosines,Cite error: The <ref>
tag has too many names (see the
help page). Rapidity in two dimensions can thus be usefully visualized using the
Poincaré disk.<ref {{harvnb|Rhodes|Semon|2003}} ref> Geodesics correspond to steady accelerations. Rapidity space in three dimensions can in the same way be put in
isometry with the
hyperboloid model (isometric to the 3-dimensional Poincaré disk (or ball)). This is detailed in
geometry of Minkowski space.
The addition of two rapidities results not only in a new rapidity; the resultant total transformation is the composition of the transformation corresponding to the rapidity given above and a rotation parametrized by the vector ,
where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule
where are the generators of rotation. This is related to the phenomenon of Thomas precession. For the computation of the parameter , the linked article is referred to.
Rgdboer (
talk) 04:38, 31 October 2022 (UTC)
Cite error: There are <ref group=nb>
tags on this page, but the references will not show without a {{reflist|group=nb}}
template (see the
help page).
This
level-5 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||
|
As presented today, the matrix has minus signs in front of two of its components. These minus signs disappear when the parameter φ is made negative because sinh is odd and cosh is even. I realize that elsewhere this minus sign may be used but that is an insufficient reason to include it here. The use of any unnecessary minus signs tends to lead to confusion when one compares the matrix operator with a comparable matrix that represents ordinary rotation in a plane by ordinary complex multiplication. In that case there is one minus sign on the counter-diagonal. Using two minus signs does not improve the analogy. There should be no minus signs when none are necessary. Rgdboer ( talk) 22:44, 14 November 2008 (UTC)
The minus sign is not found in the use of the rapidity parameter in E.T. Whittaker's History of Electricity, neither in the 1910 edition nor in the 1953 edition. The superfluous symbols have then an evidence of needlessness. Rgdboer ( talk) 23:20, 14 July 2009 (UTC)
The problem of sign can be looked at in this way: Einstein 1905 had two reference frames S and S' with S' moving away from S along the x axis with uniform velocity v. He called S the stationary frame. Looking at frame S from S' (as Einstein did) we get the minus sign and looking at frame S' from S we get a plus sign. The books tend to follow Einstein but is it not more natural to view from the stationary system S and get the plus sign? (PS Notation: Einstein had frames K and k for S and S') JFB80 ( talk) 20:02, 11 October 2010 (UTC)
The issue here is that of active and passive transformations. As that article says, the terms "alias" and "alibi" have been adapted from detective fiction. The use of the minus sign means that "another place" is the frame of reference being brought to rest. Use of a plus sign refers to a new identity of a moving frame compared to the one at rest. As also mentioned in that article, the two points of view often arise from two different disciplines of study. Preference for one over the other then is a matter of taste. However, the issue has implications for the article on hyperbolic angle as the use of the minus sign here confuses the mathematical idea. See Talk:Hyperbolic angle#Sense of angle. Comments to improve this article are sought. Rgdboer ( talk) 20:55, 8 April 2014 (UTC)
As the article stands today, events are represented by column vectors and the transformation matrix operates on the left of this column vector. Successive transformations are then built up by placing transformation matrices to the left. On the other hand, as originally posted 11 November 2008, events were represented by row vectors so that transformation matrices may be written in the ordinary left-to-right direction. This stylistic convention of having further developments arise to the right in text is a strong tradition in European languages. Since the matrices for various φ form a multiplicative group, it seems more natural to apply them in an ordering like group multiplication. For these reasons, in this fundamental article in relativity theory, I recommend the left-to-right notation be adopted (restored). From the greater transparency here, editors may adapt these reasonable measures elsewhere. Rgdboer ( talk) 21:28, 24 November 2008 (UTC)
I've removed the last line, as I can't see that "constrains the future to a quadrant of a spacetime plane" actually have any meaning in this context. Apologies if I'm missing the point. The future is constrained to a light-cone in a normal spacetime plane - an isosceles triangle for one spatial dimension. Are we now envisioning a new measure of spatial extent defined as rapidity × speed of light × time? If so, and if it's relevant, it probably warrants some explanation. As it stands, it seems a bizarre, glib, confusing line; whereas the previous line seems to be a fine conclusion. Thoughts welcome - Bobathon ( talk) 20:51, 12 January 2009 (UTC)
Rgdboer, I am a little perplexed by your comment.
Ever since Einstein first published his special theory of relativity, people have been looking for appropriate examples, situations, and thought experiments to illustrate the unusual consequences of this theory. To my understand, the effort to come up with better examples of an existing theory is a pedagogical exercise and not original research.
Why do you consider this thought experiment to be original research? The consequences described in the story are precisely those predicted by special relativity, and special relativity is over 100 years old and is almost universally accepted. —Preceding unsigned comment added by Unitfreak ( talk • contribs) 05:14, 12 July 2009 (UTC)
In particle physics rapidity is usually defined as
,
where z is the beam axis. See for example eq 38.39 in
http://pdg.lbl.gov/2009/reviews/rpp2009-rev-kinematics.pdf
Maybe this definition should be presented as an alternative, especially since it is never called "longitudinal rapidity" or something else that it would make it distinguishable from the rapidity defined in the current article version. —Preceding unsigned comment added by 128.141.103.146 ( talk) 14:50, 5 November 2009 (UTC)
The article says "possible values of relativistic velocity form a manifold, where the metric tensor corresponds to the proper acceleration (see above)." I don't find anything "above" in the article to explain this correspondence. What does it mean for the metric tensor to "correspond to" the proper acceleration? Flau98bert ( talk) 14:33, 17 September 2012 (UTC)
At present the article is quite confused over this difference. It needs correction. JFB80 ( talk) 19:19, 18 January 2013 (UTC)
The following text was removed:
The introduction of new terms hyperbolic velocity and relativistic velocity in this article without reference and without contribution to understanding of the topic calls for discussion here. — Rgdboer ( talk) 02:47, 6 March 2016 (UTC)
You asked who gets to name things. The subject title is nomenclature, in this case scientific terminology as exemplified by the international scientific vocabulary. New terms are subject to naming conventions as in a systematic name. The encyclopedia only reports on usage as found in primary and secondary reliable sources. Notability of terminology is determined by standard metrics on educational and scientific texts. — Rgdboer ( talk) 03:28, 11 March 2016 (UTC)
Thank you for your thoughts. On new terms see neologism which is the target of the redirect at coining (linguistics). As for hyperbolic angle and its applications, references have been given and the material is classical though neglected in some study sequences. The application in this article provides a motive to investigate the concept more fully, such as in the area of a hyperbolic sector. But the fact that your textbooks missed some applications does not mean they are not standard. Your idea to accelerate the use of neologisms in this encyclopedia is contrary to policy and would tend to discredit the project as going off in its own direction rather than reporting on usage as is the policy now. — Rgdboer ( talk) 23:26, 12 March 2016 (UTC)
is uncommon. Usually the hyperbolic arc tangent is either or . Please provide a reference where is commonly used. iou ( talk) 18:59, 16 January 2017 (UTC)
I deleted a section beginning with
Rapidities aren't confined to a space of unit radius. Velocities are (with ). One has . The rest, copyedited for errors, might belong in Velocity-addition formula, but the relevant material is there already.
It was all promptly reverted with motivation that I should provide a reference claiming the incorrectness of the content. To be honest, finding reliable sources confirming that is wrong are hard to come by. You might find one saying is wrong with a bit of luck. No, I rely mostly on references giving what is right and take anything else as false. This should suffice.
As for the sentence in the lead deleted by me, a thread just above highlights what is going on. Even if someone 107 years ago wrote , and even if the contributor wrote a conference paper 1992, it is not notable and to my knowledge not used to the extent that it warrants mention in the lead of this article. YohanN7 ( talk) 10:07, 18 January 2017 (UTC)
I have not examined the dispute in detail, but in all fairness it would be interesting to discuss rapidity for boosts in different directions somewhere on WP, provided it is clear and correct. Stated as a problem: if you have
then how do ζ1 and ζ2 combine into ζ? Alternatively, is there a function z such that
I am guessing the dispute is related to the content of this:
but we need modern books with references to the original sources. That would be stronger for WP. M ∧Ŝ c2ħε Иτlk 11:34, 20 January 2017 (UTC)
The section In more than one space dimension has versions done by two editors. The one by JFB80 includes the link hyperboloid model which is pertinent. The version by YohanN7 introduces the Lie algebra of the Lorentz group with no mention of bivector (complex). Though the Lorentz group is a Lie group, presuming the reader can follow Lie theory is a stretch. Therefore I prefer the JFB80 version. Furthermore, YohanN7 asserted on January 25 that I removed something which I did not touch. Derogatory comments in an edit description are offside, and a falsehood is easily confirmed by History comparisons. We work here with just ourselves and software, with hope for uplifting but sometimes subject to degradation. Such is the state of man on earth. Rgdboer ( talk) 01:05, 27 January 2017 (UTC)
Thank you both for responding. Look at that History comparison YohanN7 since it shows more than a reversion, there was an insertion you made, and I did not remove it. Please stop perpetuating the falsehood.
It should be recalled that hyperbolic geometry was developed in the nineteenth century and that it is a metric geometry not applicable to spacetime. But A Treatise on Electricity and Magnetism was using t,x,y,z variables so some manifold was needed to make sense of it. Also from the nineteenth century came quaternions and biquaternions of Hamilton. In the struggle to achieve mathematical physics another contribution was hyperbolic quaternions. The stampede in the twentieth century was led by Ludwik Silberstein using biquaternions. Admittedly, the contributions of Sophus Lie and Elie Cartan are huge, but this article should be pegged to the likely reader, and he or she needs to learn examples of continuous groups. Advanced terminology is not helpful to the novice reader. Furthermore, disdain for mathematics of the past is unjustified. Rgdboer ( talk) 00:28, 28 January 2017 (UTC)
Yes, my point is that hyperbolic geometry and quaternions preceded spacetime relativity and enabled the physical theory. They amount to "off-the-shelf" technology that the conceptual engineers apply to physics. While there is no doubt that rapidity is a Lie group parameter, that feature does not define it. As we discuss a particular section aiming to expand the concept, some historic aspects have been brought up to highlight several decades of development that were necessary for the birth of relativity. For instance, non-Euclidean geometry emerged in 1870s as instantiated by models, but only in the 1920s did Klein articulate the boundary of velocity space as a Cayley absolute. See Felix Klein, M Ackerman translator (1979) Development of Mathematics in the 19th Century, p 138, Math Sci Press. Furthermore, linear algebra developed over the same period, and in our instance that includes the squeeze mapping. All fields converge in this encyclopedia project, so there is no separation between the Math dept and the Physics dept as on campuses. Rapidity is on the boundary of the two, so the challenge in this article is similar to others. Looking at the history of ideas clarifies essentials, and in this case of hyperbolic angle it is a transcendental function in the lens. Calculus teachers would do well to explain how these transcendental functions arise, and draw out the hyperbolic sector when they present the dented trapezoid as producing the natural logarithm. If these subjects were properly laid out the fantastical aspects of special relativity would be mundane. Rgdboer ( talk) 02:28, 29 January 2017 (UTC)
No, the flat spacetime arrangement was put together by the Swearingen Sisters in Austin Texas when Alexander Macfarlane joined G. B. Halsted at the University of Texas in 1885. Reference to Minkowski’s pompous pronouncement in 1908 is due to eminence of the German schools compared to American, Canadian, and English communities. Minkowski evidently caught wind of "The Great Vector Debate" in the pages of Nature (journal) and elsewhere, and his enunciation has been recalled as the watershed. — Rgdboer ( talk) 19:35, 31 January 2017 (UTC)
Concerning the Klein reference, the Ackerman translation published by Robert Hermann in 1979 was cited for the page 138. Checking the bookshelf here at QA 26 K 6 there is the 1950 Chelsea reprint of the German original. The cited passage is from chapter four "Herausbeitung einer rein projektiven Geometrie" where he cites Cayley on general projective measure. So the page number is 150 in Chelsea, volume 1, Entwicklung der Mathematik. The equation given in dx, dy, dz, and dt is in homogeneous coordinates and is equivalent to v2 = 1, boundary of velocity space, and Cayley absolute in the passage. — Rgdboer ( talk) 00:26, 2 February 2017 (UTC)
I'm new so sorry if I don't do this right, but trying to read this article and the one for proper velocity, I got very confused until I realized they both use opposite variables for the two quantities. w and η. —
Gravitative (
talk) 12:07, 11 April 2017 (UTC)
Rapidity and Celerity are closely linked, but according to the wiki pages for both, both use w to represent each other. The Celerity page refers to rapidity and eta so this page should use eta too.
@ User:DVdm: thank you for your dedication to keeping Wikipedia top notch. In this case, you might have used {{citation needed}} markup instead of the big undo axe, but I shall seek out one or more sources as you request. All the best — Quantling ( talk | contribs) 17:33, 23 January 2019 (UTC)
The following was removed as beyond the scope to Rapidity:
The relativistic velocity is associated to the rapidity of an object viaCite error: A <ref>
tag is missing the closing </ref>
(see the
help page).
[nb 1]
where refers to
relativistic velocity addition and is a unit vector in the direction of . This operation is not commutative nor associative. Rapidities with directions inclined at an angle have a resultant norm (ordinary Euclidean length) given by the
hyperbolic law of cosines,Cite error: The <ref>
tag has too many names (see the
help page). Rapidity in two dimensions can thus be usefully visualized using the
Poincaré disk.<ref {{harvnb|Rhodes|Semon|2003}} ref> Geodesics correspond to steady accelerations. Rapidity space in three dimensions can in the same way be put in
isometry with the
hyperboloid model (isometric to the 3-dimensional Poincaré disk (or ball)). This is detailed in
geometry of Minkowski space.
The addition of two rapidities results not only in a new rapidity; the resultant total transformation is the composition of the transformation corresponding to the rapidity given above and a rotation parametrized by the vector ,
where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule
where are the generators of rotation. This is related to the phenomenon of Thomas precession. For the computation of the parameter , the linked article is referred to.
Rgdboer (
talk) 04:38, 31 October 2022 (UTC)
Cite error: There are <ref group=nb>
tags on this page, but the references will not show without a {{reflist|group=nb}}
template (see the
help page).