Talk:Lorentz transformation/Archive 6

Transformation of the electromagnetic field

I think the EM section should be given after the specification of how tensors transform. Actually, deriving the transformation laws using 3 + 1 spacetime (i.e. using E and B) is difficult. It is rather easy (no bag of tricks needed) using the field tensor F. YohanN7 (talk) 15:54, 15 December 2015 (UTC)

Yes it is tedious to derive the transformations of the E and B fields, but current section is the vector statement which simply mentions that they can be derived from the transformations of velocity and the Lorentz force (only the statement they can be derived is needed, not the actual derivation). It fits in well where it is. In the tensor section, we can still mention the angular momentum and EM field tensors, illustrating how these tensors transform. The reader can make up their own mind which formulation (vector or tensor) is easier. M∧Ŝc²ħεИ_τlk 16:29, 15 December 2015 (UTC)

I don't agree. I think this article should rely on special relativity (with manifest covariance) as much as possible. YohanN7 (talk) 11:08, 16 December 2015 (UTC)

I much prefer for this entire article if the vector formulation of the examples are given first, then the tensor formulation as a second treatment. Cartesian components (including on generators and parameters), vectors, and matrices are more accessible than using index notation and tensors for an introductory article like this.

Maybe we could create a new article Covariant formulation of Lorentz transformations (not set on title), which contains everything in this article in tensor form, using 4-vectors at the outset, and manifest covariance throughout. The tensor section in this article would be moved over there. The group theory can be cast in index form also (using the parameter matrix ω and generator matrix M, commutation relations can be given in index form, with a mention of structure constants). Transformations of spinors could also be given in that article. Do you agree with this?

Otherwise, I'm not keen on what you're saying, but on the other hand I don't want to own the article either. Feel free to move and rewrite the section. M∧Ŝc²ħεИ_τlk 12:20, 16 December 2015 (UTC)

Also, to further reduce this article size, maybe another article just on the problem of general compositions of Lorentz transformations in arbitrary directions could be created (not sure on the title, maybe "Compositions of Lorentz transformations" or something). The parts about hyperbolic triangles and spacetime diagrams for non-collinear boosts could be in there also. It is not in elementary textbooks (obviously), but the topic is a research area in its own right, and included in more advanced books, so it's notable. This article could just link to that article for the results. Would you agree with this also? M∧Ŝc²ħεИ_τlk 12:20, 16 December 2015 (UTC)

On the last paragraph: Transformation of velocities could be reduced to

Main article: Velocity addition formula

+ the formula. Most stuff on combining boosts could be (is already) in Thomas precession. It is what those articles are for.

Then, if you'd rather have this article as a reference (can answer what the formula for transformation of EM fields/3-velocity/momentum/angular momentum, spinors), then I'd agree. Just splash up the formulas. If you want to have this as learning material, then the tensor formulation should go first (after coordinate transformations). It provides the rather simple framework in which all explicit formulas on whatever form can be derived with comparative ease. I'll definitely not insist on this, but it is my POV. YohanN7 (talk) 13:05, 16 December 2015 (UTC)

Agreed on the velocity addition formula (statement and link, if the reader cannot take differentials, will not understand the section anyway so the x-boost is redundant).

For the composition of boosts and equality rotations and boosts, the formulation I've written gives systematically all the cases for two boosts and a boost then rotation, and not all of it is in Thomas precession. The material in this article could be merged into that article, or overwrite the redirect that is Thomas rotation. Didn't you mention that Thomas rotation and Thomas precession could be separate articles here?

"I suggest we collect the advanced stuff somewhere, perhaps Thomas precession, perhaps a new article Thomas rotation, which would make sense because Thomas precession really is a physical phenomenon with mathematical root Thomas rotation. Lorentz transformation would make sense too, but I don't know whether people want to allow for that article to swell much more. Meanwhile, Velocity-addition formula should be reduced to the basics as given in textbooks. YohanN7 (talk) 14:50, 5 July 2015 (UTC)"

If you agree, the compositions as given in this article could go in Thomas rotation, and Compositions of Lorentz transformations could redirect to Thomas rotation.

No - I don't want the tensor formulation throughout this article straight after the coordinates in place of vectors, but suggested another article for a completely covariant formulation all the way through. Yes, yes, yes, tensors are more efficient than vectors. So are differential forms. But if a reader doesn't have the background in tensor algebra what's the point in writing the entire article as manifestly covariant as possible, when vector and matrix algebra in Cartesian coordinates suffices? It will obfuscate what could be expressed in a lower-level language, if you know tensor algebra you also know vector and matrix algebra, but the converse is not necessarily true. M∧Ŝc²ħεИ_τlk 14:43, 16 December 2015 (UTC)

Fair enough. But since you ask, a manifestly covariant approach does make for a logically coherent and elegant, comprehensive, and easily understood (whether new to the reader or not) framework. Full-blown knowledge of tensors per se isn't needed, just index gymnastics. That is what is offered anyway in the physics texts. They don't even explain what covariant and contravariant vectors are, just how to manipulate them. There is nothing that says that we must present thing in the exact order that most (far from all, the really good ones don't (L&L and also Jackson's (non-introductory) text)) introductory textbooks do.

As for the actual topic of this thread, I don't know the order in which subjects are usually taught, but my introductory EM course came after introductory mechanics (including SR). The treatment of how the Lorentz transformation affects the EM field most certainly is introduced after SR has been introduced. Misner, Thorne and Wheeler devotes in Gravitation a complete section to demonstrate the superiority of the covariant approach, and use the transformation of the EM field as an example. They outline how it is done in the non-covariant approach (including the use of a dose of magic), but don't present the actual calculations because of their great length. I think we should follow this approach and present the EM formulas after tensors, and then derive them (the 3+1 formulas). It Is a piece of cake. The reader should not be left with the impression that relativity is harder than it really is. At any rate, this is toward the end of the article, and there is no guideline saying that everything towards the end should be accessible to the lay reader. YohanN7 (talk) 12:06, 17 December 2015 (UTC)

For now, let's follow your idea of moving and rewriting the EM field in the tensor section and see how it looks. My original plan of stating the vector transformations then tensor ones will take up too much space anyway.

(As an aside, for now, let's leave the group theory section in matrix form please. I find it much easier to follow this way. Maybe it can be rewritten back in index notation later). M∧Ŝc²ħεИ_τlk 15:12, 17 December 2015 (UTC)

Okay. I'll give it a try in a few days. It is actually the vector version I'll derive from the tensor version. So there is no change except for move and proof. On your last point, I don't really understand what you mean. It is now in index notation, but a display of the matrix version can be arranged for. This is a 6 × 6 matrix multiplying a 6 × 1 column containing E₁, E₂, E₃, B₁, B₂, B₃ in the EM case. This is (to me) highly desirable to display to make connection to the irreducilbe representations of the Lorentz group. (As it stands, the tensor representations are not irreducible, see posts above.) YohanN7 (talk) 12:39, 18 December 2015 (UTC)

About the aside, I meant this section Lorentz transformation#Introduction to the Lorentz group. Everything is in matrices, not indices. It is easier to follow since matrix multiplications can be done immediately. In particular,

$\langle A,B\rangle =A^{T}\eta B$

is more familiar and accessible (and all that is required) than

$\langle A,B\rangle =A_{\mu }\eta ^{\mu \nu }B_{\nu }=A^{\mu }\eta _{\mu \nu }B^{\nu }\,,$

it is trivial to extract

$\eta =\Lambda ^{T}\eta \Lambda$

from

$\langle A,B\rangle =\langle \Lambda A,\Lambda B\rangle =A^{T}\eta B=A^{T}\Lambda ^{T}\eta \Lambda B\,,$

taking the determinant of this

$\eta =\Lambda ^{T}\eta \Lambda$

is immediate and trivial compared to

$\eta _{\mu \nu }{\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }=\eta _{\alpha \beta }\,,$

and why use the clumsy Λ⁰₀ when a single simple letter Γ (or anything else) suffices?

Don't get me wrong throughout this thread. I do like tensor algebra and find it elegant and powerful, but it is strange and unfamiliar unless you know it, and don't agree it's "comprehensive, and easily understood" to any reader. Anyway, I'll not push this point further. M∧Ŝc²ħεИ_τlk 13:19, 18 December 2015 (UTC)

Concepts such as "easily understood" are relative. I still maintain that the tensor formulation is superior when dealing with tensors. I did put in a derivation of the transformation law of the EM field tensor in all painful detail. This can be understood by more people than any 3+1 proof. YohanN7 (talk) 12:54, 22 January 2016 (UTC)

I put in the whole story (added spinors and general fields). It will need some explanation, but that will have to wait for a few days. YohanN7 (talk) 12:54, 22 January 2016 (UTC)

Nice work adding a spinors section. The EM field section is good too, but the equations could each be put onto at least a extra line (they stretch past the page on my screen), also the EM field section earlier up is no longer relevant. I'll take the liberty of deleting Lorentz transformation#Transformation of the electromagnetic field and moving the current transformations and diagram lower down. M∧Ŝc²ħεИ_τlk 15:45, 22 January 2016 (UTC)

More planned rewriting

I am planning a rewrite of the vector transformations which could be made more general (examples can be tabulated), and lead onto the tensor analysis quickly. M∧Ŝc²ħεИ_τlk 17:22, 17 January 2016 (UTC)

I'll comment only on the "Tensor transformations" section in your link (which may make me sound overly negative, the rest looks good at first sight); I am not a huge fan of this. It pulls thing out of the hat by listing examples and relies entirely on index gymnastics. It may teach people how to grind the wheels but explains little. The present version of tensor formulation is actually not that bad when it comes to actual content. It has much more than usually found any introductory physics text, and explains what lies behind the gymnastics–even if it does so tersely. YohanN7 (talk) 11:02, 18 January 2016 (UTC)

Your'e not being negative, none of the planned rewrite in my sandbox is guaranteed to happen, its only a provisional draft and may be scrapped altogether if it doesn't work (right now it's probably fairly confusing for newbies to SR as is).

I started off my own tensor section to follow from the vector transformations, and expected blend in your section, or maybe my version could be replaced by your version entirely. If it was in the article, then we would have the basic x-boost covered everywhere, followed by a general vector transformation and speedily onto four vectors and tensors would be much sooner (then the group theory would follow after). I would prefer to have some transition from the basics to tensors (or just index notation), at the same time motivating the use of them, rather than diving straight into them. M∧Ŝc²ħεИ_τlk 12:02, 18 January 2016 (UTC)

Thomas rotation and precession

Also, "Thomas precession" is not the place for the composition of two boosts, which leads to a static rotation. That article should be about the coordinate frame rotating with an angular velocity, with physical implications (yes, some of this is included, but most of that article is just about two boosts equaling a rotation and boost, it doesn't even have the formula for the Thomas precession, nor its occurrence in the Bargmann-Michel-Telegdi equation, see Jackson's Electrodynamics).

All the content of combining two or more Lorentz boosts (in this and Thomas precession) should be in its own article. My preference is Compositions of Lorentz transformations with Thomas rotation as a section in the article, and the link redirecting to that section. Or at the very least, the composition stuff should all just be in Thomas rotation (not Thomas precession), whether or not the article is rewritten in tensor language. M∧Ŝc²ħεИ_τlk 17:54, 16 December 2015 (UTC)

I agree, Thomas precession should be split up. But this is another big article with problems, and we can in the meanwhile collect related things there. A better name for a new article is Wigner rotation (now a redirect). It is notable, while Thomas rotation is not. Anything else than combining two boosts is simple, and can be kept in this article. YohanN7 (talk) 12:14, 17 December 2015 (UTC)

OK, let's put it in Wigner rotation and redirect Thomas rotation to there. I am drafting a merge in my sandbox, which is provisional. M∧Ŝc²ħεИ_τlk 15:12, 17 December 2015 (UTC)

The Wigner rotation, b t w, can be found in chapter 2 in Weinberg. and in Wigner's original (massive and worthwhile!) paper, referenced in the rep theory article. YohanN7 (talk) 12:39, 18 December 2015 (UTC)

I am doing the best to preserve what both of us have written, and will soon overwrite Wigner rotation, so this article and Thomas precession can be immediately reduced. M∧Ŝc²ħεИ_τlk 13:19, 18 December 2015 (UTC)

Some deletion of what both of us have written has been necessary, but now it is almost ready to at least overwrite the Wigner rotation redirect. I'm not sure how or why this section you wrote is a clear and efficient way to obtain the composite velocity, and rotation matrix. (I know it's in Goldstein and likely elsewhere, but I don't see what the point of the manipulations are). You can do it just using block matrix multiplication, and find the composite velocities, axis, and angle. M∧Ŝc²ħεИ_τlk 20:48, 18 December 2015 (UTC)

It is a very clear but horribly inefficient way of finding boost + rotation from boost + boost. The point (see the Goldstein quote) is that there is no way to do this easily. People have tried. (Maybe Ungar did succeed (using a computer algebra system, but never published the full proof), his claimed formula is on that talk page somewhere. Never got further with that before Q stepped in and cooled off the action.) Remember that what one is really after is a formal solution in terms of parameters (depending on the original two boost parameters). It is not a matter of taking a 4×4-matrix of numbers and spitting out a new set of numbers. That is easy (by the same process as described). YohanN7 (talk) 15:08, 28 December 2015 (UTC)

OK. Yes, whichever method is used the composition is very tedious. Its just my POV that block matrices make things less tedious than full matrices, and the useful relations constructed from the given relative velocities automatically follow (I'm not denying how tedious this still is). M∧Ŝc²ħεИ_τlk 20:12, 29 December 2015 (UTC)

It is not only "tedious", it is hard. What you are trying to do is to solve

$e^{{\boldsymbol {\zeta _{1}}}\cdot \mathbf {K} }e^{{\boldsymbol {\zeta _{2}}}\cdot \mathbf {K} }=e^{{\boldsymbol {\zeta }}\cdot \mathbf {K} }e^{{\boldsymbol {\theta }}\cdot \mathbf {J} }$

for ς and θ given ς₁ and ς₂. The ς is relatively easy to get to. The θ = θ(ς₁, ς₂) is the Thomas rotation, which is much harder to find. This is what Goldstein alludes to, and also what has been the subject of a bunch of research papers over 80 years. YohanN7 (talk) 11:12, 8 January 2016 (UTC)

I see now that the article contains explicit formulae. (I haven't dealt with this in a long time.) Do these formulae agree with what is found here (the Ungar ones)? If these are correct, then the "Goldstein-passage" has no relevance. YohanN7 (talk) 12:26, 8 January 2016 (UTC)

I understand exactly what the problem was (finding the composite boost, and the axis-angle vector (not sure what its common name is)).

Yes, the formulae here are correct: the cosine of the angle from the trace of M is an established result given in the citations, the cross product of a and b is definitely proportional to the cross product of velocities (WP:CALC).

I'll have to double check how these formulae agree with Ungar's formulae, at one time (unearthed here) I naively substituted the cross products of a×b and u×v in terms of magnitudes and angles to obtain the relation between the angles as Ungar does, and the resultant formula agreed with one of Ungar's formulae. But since u and v apply in different frames, is it meaningful to define the angle between them? Sexl and Urbantke say no, Ungar and Mocanu say yes. However, S&U just reverse one of the velocities to obtain the perceived velocity from one of the frames. For example, in Σ′, the frame Σ moves with velocity -u (not u) and Σ ′′ moves with velocity v so -u×v defines the axis in this frame. In Σ the cross product of u (velocity of Σ′) with u⊕v (velocity of Σ′′) is proportional to -u×v, in Σ′′ the cross product of -v with (-u)⊕(-v) (or whatever) is again proportional to -u×v. So in all frames, the rotation axis is in the same direction (but the sense of rotation may be reversed), the angle between any pair of 3-vectors is only defined if they are perceived from the same frame. This is something else to clear up in Wigner rotation.

By the way you seem to interchange the rapidity vector ζ and the velocity vector v. Landau and Lifshitz's formula for the line element in velocity space uses velocities, not rapidity vectors. M∧Ŝc²ħεИ_τlk 14:38, 8 January 2016 (UTC)

It is not the case that I distrust you. The thing is this: When in a graduate level textbook (and also elsewhere) you find the statement

It can easily be shown that...,

then you and I as experienced students of science should infer that some very good mathematician/physicist active in the area actually can show the statement (perhaps even with, but usually not with, "ease"). Now, if a textbook (and several papers) say

it is damned difficult to show that...,

then at least my immediate reaction is that there is no way in hell it can be shown by me. YohanN7 (talk) 10:44, 14 January 2016 (UTC)

OK. It would be nice, now that Wigner rotation has its own article (hence more room), to summarize the various axis and angle formulae from Ungar, Mocanu, S&U, (others?), and clarify how the perceived relative velocities relate to the axis in each frame. On google books Gourgoulhon gives a variety of sine and cosine formulae for the Thomas angle, as well as "half-(Thomas) angle" formulae. M∧Ŝc²ħεИ_τlk 12:50, 14 January 2016 (UTC)

Done, now see Wigner rotation. M∧Ŝc²ħεИ_τlk 11:25, 25 December 2015 (UTC)

See also Thomas precession. M∧Ŝc²ħεИ_τlk 20:12, 29 December 2015 (UTC)