Talk:Lorentz transformation/Archive 3

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

→

Archive 8

Error in section Lorentz transformation for frames in standard configuration

It is written that "Also assume that the origins of both coordinate systems are the same." However, origins of both coordinate system are the same at t=t'=0, this statement should be corrected. --144.122.128.169 (talk) 21:48, 24 October 2011 (UTC)

The origins of the systems include the time coordinate. This is mentioned in the text as "... their own Cartesian coordinate system to measure space and time intervals." So we have one event with coordinates (t,x,y,z)=(0,0,0,0) in one system and (t',x',y',z')=(0,0,0,0) in the other system. So the origins are indeed the same and the text is correct. DVdm (talk) 07:45, 25 October 2011 (UTC)

ok I got it. Thank you --144.122.128.169 (talk) 00:18, 26 October 2011 (UTC)

Errors in matrix

user:DVdm: Can you explain where the errors are in the second simplified matrix below?

__________________________________________

${\begin{bmatrix}c\,t'\\x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &-\beta _{x}\,\gamma &-\beta _{y}\,\gamma &-\beta _{z}\,\gamma \\-\beta _{x}\,\gamma &1+(\gamma -1){\dfrac {\beta _{x}^{2}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{x}\beta _{y}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{x}\beta _{z}}{\beta ^{2}}}\\-\beta _{y}\,\gamma &(\gamma -1){\dfrac {\beta _{y}\beta _{x}}{\beta ^{2}}}&1+(\gamma -1){\dfrac {\beta _{y}^{2}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{y}\beta _{z}}{\beta ^{2}}}\\-\beta _{z}\,\gamma &(\gamma -1){\dfrac {\beta _{z}\beta _{x}}{\beta ^{2}}}&(\gamma -1){\dfrac {\beta _{z}\beta _{y}}{\beta ^{2}}}&1+(\gamma -1){\dfrac {\beta _{z}^{2}}{\beta ^{2}}}\\\end{bmatrix}}{\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}\ ,$

where $\beta _{x}={\frac {v_{x}}{c}},$ $\beta _{y}={\frac {v_{y}}{c}},$ $\beta _{z}={\frac {v_{z}}{c}},$ $\beta ={\frac {v}{c}}={\frac {|\mathbf {v} |}{c}},$, and $\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}.$

.....

introduce another factor capital beta Β:

$\mathrm {B} _{ij}=(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}+\delta _{ij}=(\gamma -1){\dfrac {v_{i}v_{j}}{v^{2}}}+\delta _{ij},\,\!$

in which:

• i and j can each be set to x, y or z, or using index notation, 1 replaces subscript x, similarly 2 replaces subscript y and 3 replaces subscript z),
• $\delta _{ij}\,\!$ is the kronecker delta (in numeric form - not as a tensor),

hence:

${\begin{bmatrix}c\,t'\\x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{x}&-\gamma \beta _{y}&-\gamma \beta _{z}\\-\gamma \beta _{x}&\mathrm {B} _{xx}&\mathrm {B} _{xy}&\mathrm {B} _{xz}\\-\gamma \beta _{y}&\mathrm {B} _{xy}&\mathrm {B} _{yy}&\mathrm {B} _{yz}\\-\gamma \beta _{z}&\mathrm {B} _{xz}&\mathrm {B} _{yz}&\mathrm {B} _{zz}\\\end{bmatrix}}{\begin{bmatrix}c\,t\\x\\y\\z\end{bmatrix}}.$

__________________________________________

If you don't like mixing x, y, z with i, j (probably fair eneogh), why not add the statement somewhare:

".......The x, y, z components of velocity (using cartesian coords) correspond to the index labels 1, 2, 3, (more general notation, for 4-vectors and tensors). That is: (v_x, v_y, v_z) = (v₁, v₂, v₃). Letting i, j be dummy indicies which take index values 1, 2, 3......."

I simply fail to see any mistake...

I also absolutley fail to see how the inclusion of the kronecker delta "induces errors"...

When there are products of perpedicular velocites (non-diagonal elements), the components of the velocities are non-equal, zero is added to the fraction

$(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}.$

For the diagonal elements, where velocity components are equal, 1 is added to that fraction. The kronecker delta easily relates to this:

${\begin{aligned}&i\neq j:&(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}+0\\&i=j:&(\gamma -1){\dfrac {\beta _{i}\beta _{i}}{\beta ^{2}}}+1\\\end{aligned}}$

${\begin{aligned}\delta _{ij}=\,&0:&i\neq j\\&1:&i=j\\\end{aligned}}$

It obviously a number in this context - not a matrix/2-rank tensor.

--F=q(E+v^B) (talk) 09:13, 26 November 2011 (UTC)

Since no-one has replied, i'm going to revert the matrix back with a referance, it's in the almighty Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne...Though I suppose people will disagree with that as well and merrily delete.--F=q(E+v^B) (talk) 09:13, 26 November 2011 (UTC)

Now that I checked it, there is indeed no error. I reverted because the previous version had been there (unchallenged, unsourced, correct) for quite a while and you didn't provide an edit summary for your edit. So one glance at your edit was sufficient to make me decide to undo it. If you had provided that source and a short edit summary, we would have saved ourselves some valuable time. I had a quick look at my copy of MTW, but I couldn't find the expressions. On which page do we find them? - DVdm (talk) 11:21, 26 November 2011 (UTC)

No worries! In MTW its on page 69 (exercise 2.7 "Boost in an arbitary direction", chapter 2 foundations of special relativity). Again you are correct about recources - that referance will be added, and i'll add more elsewhare. I might re-write some of the initial formalism so the referances fit in, and make the notation used in the literature (both cartesian and index-component) clearer. Apologies for getting angry and thanks --F=q(E+v^B) (talk) 12:12, 26 November 2011 (UTC) — Preceding unsigned comment added by F=q(E+v^B) (talk • contribs)

(Please sign your talk message with four tildes (~~~~). Thanks.)

Ok, I missed them in the source because the notation is different. Was looking for a matrix-picture. It might be a good idea to use the same notation here with capital lambda (Λ) etc. as opposed to beta (Β). Sticking as close as possible to the sources tends to avoid undos and reverts due to wp:original research. - DVdm (talk) 14:42, 26 November 2011 (UTC)

Fair point, i'll change B to Λ (by the way its a capital B, not capitial Beta). B was used lower down the page when explaining compositions of transformations, and was just quicker to type. I assumed B could be used too. It doesn't matter, notation is notation.

BTW I don't understand how compositions of boosts work, or the generalizations incorperating rotations, so I left those sections alone - including the notation.--F=q(E+v^B) (talk) 16:06, 26 November 2011 (UTC)

Merge, December 2011

The Lorentz factor article does not seem to offer anything on its own. It presents the LF as a definition and then has to explain what v is. But what is v, and where did v come from? It isn't clear until we get to the "applications" section, where the Lorentz transformation is given as an "application". As far as I know, the LF is meaningless (and underivable) without the context of the LT. Basically, I think that the LF article is of a very poor standard (take a look at it), the LT is quite a nice article, and we should get rid of the LF article, link it to the LT article, and then add a LF section to the LT article. — Fly by Night (talk) 01:13, 28 December 2011 (UTC)

But LF is long. We need to shorten it before mmerging, either without losing content, or declaring some content non-notable...--Dhatsavan 03:31, 29 December 2011 (UTC)

Most of the content would become redundant once merged. I'm not suggesting we cut-and-paste the LF content into the LT article; I'm suggest that we define

$LT_{\text{new}}:=LF\cap LT_{\text{old}}\,.$

The concept of notability refers primarily to the subject of an article, and not so much its content. Some level of notability is required for content, but that is much less than that required for the subject. — Fly by Night (talk) 22:49, 29 December 2011 (UTC)

• Merge request removed due to lack on involvement. — Fly by Night (talk) 02:32, 15 January 2012 (UTC)

Standard Configuration

The description of the Standard Configuration is not correct. The x and x' axes to not coincide in 4D spacetime. The x axis has a component in the t' direction, and the x' axis has a component in the t direction. — Preceding unsigned comment added by 173.59.121.56 (talk) 14:27, 4 January 2012 (UTC)

Please sign your talk page messages with four tildes (~~~~)? Thanks.

The text doesn't really say that they coincide in 4D spacetime, but that they are collinear, and this is implicitly assumed to be in 3D space. I have made the 3D explicit now. Looks better indeed. - DVdm (talk) 14:41, 4 January 2012 (UTC)

Diagrams

Shikes... all this time I haven’t even looked at the page and realized there was an error correction in the image I produced, so thanks very much to DVdm and the IP. Incredibly embarrassing... By the same editor who corrected the image, it seems to be wrong - so please tell me what the problem is. I can take it if you say its no use.--Maschen (talk) 11:40, 26 January 2012 (UTC)

Actually, please still speak up, though I intend to replace it anyway with another one, this can be moved somewhere else or deleted. It will incorporate the concept of rulers and clocks placed through out the frame, and how they would appear to the observers in their respective frames, which I can find a reference for: University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, ISBN (10-) 0-321-50130-6, ISBN (13-) 978-0-321-50130-1.--Maschen (talk) 11:43, 26 January 2012 (UTC)

Although I like the current diagram (File:Lorentz transforms.svg), if you have an alternative, feel free to upload with another name, so we can discuss it before we show it in the article. Good work. - DVdm (talk) 12:04, 26 January 2012 (UTC)

That was a quick response, here it is,

The space-time coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles.

Top: frame F' moves at velocity v along the x-axis of frame F.

Bottom: frame F moves at velocity −v along the x'-axis of frame F'.

Also created an extra one (optional) to illusrate the terminolgy hopefully a bit clearer.

An observer O, situated at the origin of a local set of coordinates - a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to detect a spacetime event, shown as a star.

Opinions? Thanks again for your previous appreciation. This article really lacks diagrams, the two (none of mine) at the very beginning are good, but perhaps insufficient to show they apply directly to a situation geometrically.--Maschen (talk) 15:12, 26 January 2012 (UTC)

BTW, the current one I added really isn't that good, so I honestly don't mind how people criticize it. --Maschen (talk) 15:25, 26 January 2012 (UTC)

Sorry - I forgot to mention the perhaps obvious: could someone who knows how archive talk pages please do so for this one? If not, I don't mind finding out how to then doing it myself. Its becoming very long...--Maschen (talk) 15:29, 26 January 2012 (UTC)

Brilliant. Just one quibble about the caption. I wouldn't write:

Bottom: F moves at velocity v in the -x direction relative to F'

but rather:

Bottom: F moves at velocity -v in the x' direction relative to F'

(note the prime at x' !), and in the figure I'd mark the bottom velocity as -v, or perhaps let's mark them even as vectors ${\vec {V}}$ and $-{\vec {V}}$.

I have set archive params. - DVdm (talk) 15:45, 26 January 2012 (UTC)

Crikey, another speed of light response! I'll make those changes. About the vector notation, arrow or bold, it may be redundant since its only in one dimension, but bold will not matter so i'll do it. You are clearly monitoring this page well - i'll be sure to keep track of it too so you're not waiting too long! And thanks for the archive setup =). --Maschen (talk) 17:10, 26 January 2012 (UTC)

Done. If you didn't mind, I cut and pasted your version of the statement directly - it was very short and effective, and couldn't explain it better myself!--Maschen (talk) 17:20, 26 January 2012 (UTC)

I made a slight tweak to make it langauge independent using speech bubbles, maybe with an infinitesimal pinch of "comicness" in it? Also the 4-positions had to be aligned to the same point - they wern't before. (while at it cleared up my rudeness above)--Maschen (talk) 17:56, 26 January 2012 (UTC)

It looks very good to me. I made little tweak to the wording of the caption from

Top: frame F' moves at velocity v in the +x direction relative to F.

Bottom: F moves at velocity −v in the x' direction relative to F'

to

Top: frame F' moves at velocity v along the x-axis of frame F.

Bottom: frame F moves at velocity −v along the x'-axis of frame F'

and I removed the "textbooky" notice about the symmetry and the apparently contradicting situations. I'd say, go for it :-) - DVdm (talk) 08:59, 27 January 2012 (UTC)

Xcellent - thank you very much for your support in all this =) Here it goes... --Maschen (talk) 09:06, 27 January 2012 (UTC)

I moved it a bit. I think it looks better on the right. I also removed the old figure.

Little note: the filename of the small figure has a little spelling error: "referance" should be "reference". I know, it's hard to fix ;-) - DVdm (talk) 09:47, 27 January 2012 (UTC)

Oops... oh well. I'll add the reference (!) above, should have done it before. Also I removed some text-booky character in the first line of the x-boost section by changing "these forms are most usually introduced" to "these are the simplist forms". =)--Maschen (talk) 10:42, 27 January 2012 (UTC)

Make that "simplest", and you're in biznez ;-) - DVdm (talk) 10:48, 27 January 2012 (UTC)

Yeah..... That was funny actually... As you can tell I am very bad at spelling: normally when checking masses of words for spelling errors I cut-and-paste into a word processor, fix, then paste back. In this instance I didn't because it was only one phrase, so thanks for fixing that. =)

Just to let you know, I intend to add another for the electromagnetic fields E and B, but it may take a while. It’s not because the fields are hard to draw - rather the diagrams become very cluttered... I'll see what I can do anyway. =)--Maschen (talk) 20:16, 27 January 2012 (UTC)

I'll hold off for a while on transformations of EM fields for a while, its not as easy as I thought... =(

Anyway, here is a possibly clearer one for the transformation in one-spatial dimension + time using the Minkowski diagram of a lightcone;

Lorentz transformations on the Minkowski light cone spacetime diagram, for one spatial dimension. The transformations correspond to hyperbolic rotations, the hyperbolic angle ϕ is called rapidity. The greater the relative speed between the inertial frames, the more "warped" the axes become. The red diagonal lines are worldlines for light - the relative velocity cannot exceed c.

Opinions? Maybe Minkowski spacetime diagram itself?--Maschen (talk) 01:02, 30 January 2012 (UTC)

I'm not enthusiastic about this one. For instance, I see 3 distinct events here ( (Dx,Dt), (Dx',Dt') and (Dx",Dt") —the blue, green and orange (missing) dots on the diagonal— ) but these events are not related. A Lorentz transformation diagram should show one single event with different sets of coordinates, projected on the axes, like for instance in Spacetime diagram:

Furthemore, the caption text doesn't make sense. "The red line is the speed of light" should be "The red line is the worldline of a light signal", or "The red line corresponds to the speed of light." The sentence "These transformations are all time-like intervals" doesn't make sense either. Where did you find this diagram? - DVdm (talk) 09:48, 30 January 2012 (UTC)

I (Maschen) drew it of course. The different axes were supposed to correspond to the Lorentz transformations at different velocities. I did include statements about world lines but then the caption became too long. The dots were only supposed to be points on the axes as they warp along the hyperbolae. If it’s wrong - by all means leave it.. guess I tried including too much detail in one diagram. The spacetime diagrams are much simpler.--Maschen (talk) 12:29, 30 January 2012 (UTC)

The dots on one hyperbolae do make sense of course, as they represent sets of events that, according to the primed observers, on the "vertical" hyperbolae, are simultaneous with the origin, or, on the "horizontal" hyperbolae, are colocal with it. It's the dots on the light cone diagonal, together with their projection lines to the axes and the delta markings that don't make sense. The graphics are nice though :-) — much nicer than the i.m.o. rather disturbing animated gif image in the lead. - DVdm (talk) 12:43, 30 January 2012 (UTC)

It has been repaired, thanks for complimenting the graphic SVG quality =).--Maschen (talk) 22:20, 30 January 2012 (UTC)

Much better. I have tweaked the caption a bit. I would't object to replacing the current top image with this one. Others? - DVdm (talk) 22:35, 30 January 2012 (UTC)

Please add as you see fit, I think the current one (Minkowski diagram) is ok, but may as well be replaced by this one since its essentially the same, though with more info. There aren’t any other images - for now (best not to overdo it!) --Maschen (talk) 22:59, 30 January 2012 (UTC)

I redrew it to be clearer and less cluttered. The actual drawing should be:

Lorentz transformations on the Minkowski light cone spacetime diagram, for one spatial dimension. The transformations correspond to hyperbolic rotations, the hyperbolic angle ϕ is called rapidity. The greater the relative speed between the inertial frames, the more "warped" the axes become. The darker regions of the cone indicate low relative velocity, increasing white tone indicates greater velocity. The red diagonal lines are worldlines for light - the relative velocity cannot exceed c.

but, as typical, WikiMedia didn't upload the image properly in SVG so this had to be PNG rendered instead. =( Anyway, for now i'll add the PNG to the article, it looks nicer and WikiPedia stupidly rasterizes everything anyway - the SVG quality is almost redundant in this respect. As a result the SVG version became simpler and less eye-catchy... --Maschen (talk) 13:08, 2 February 2012 (UTC)

Hm... I really prefer the above svg-version, the one with the uniformly grey past and future. I think the tone/velocity relation kind of overloads the picture again. To me svg is clearer, and of course, it's much easier to maintain. - DVdm (talk) 17:34, 2 February 2012 (UTC)

Sorry for the late response. I did anticipate this suggestion would be made, as it raises valid points. I don't think the tone/velocity relation overloads the picture, but the SVG properties are important. I'll change it to the SVG version.--Maschen (talk) 18:13, 5 February 2012 (UTC)

It wasn't such a big deal, but thanks for allowing future SVG-tweakers to contribute :-) - Good work. DVdm (talk) 18:38, 6 February 2012 (UTC)

No problem, and thank you. =) --Maschen (talk) 12:22, 7 February 2012 (UTC)

Article class?

The start class at the top should be changed to something better, say C or even B? NOT just becuase of my drawings! - because of the quality of the REST of the article. =) --Maschen (talk) 12:22, 7 February 2012 (UTC)

I'll just change it to C myself. If anyone disagrees change as seen fit.--Maschen (talk) 22:24, 8 February 2012 (UTC)

Transformation animation?

Any chance the very first image in the article could be animated (not linked to the animation file)?-- F = q(E + v × B) 21:10, 22 February 2012 (UTC)

I personally don't like the image there, and I think that replacing it with its animation, would even be more annoying—sort of, let's scroll it out of sight ASAP. - DVdm (talk) 21:20, 22 February 2012 (UTC)

So delete then? I really don't mind either way - just asking.-- F = q(E + v × B) 22:47, 22 February 2012 (UTC)

Afaiac, it can be removed, but it can of course stay if other contributors really like it—as long as it's kept frozen, that is :-) - DVdm (talk) 08:24, 23 February 2012 (UTC)

Actually its relativley redundant compared to Maschen's awesome, clearer and more meaningful image further down the article on hyperbolic transformations. He uses far less words in the caption and blends it very neatly with the local text. I would also favour deletion of the image, but raised the issue of animation becuase thats one feature Maschen's didn't have. -- F = q(E + v × B) 13:50, 23 February 2012 (UTC)

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

→

Archive 8