Talk:Lorentz transformation/Archive 5

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A question

The article says "Consider two observers O and O′, each using their own Cartesian coordinate system to measure space and time intervals. O uses (t, x, y, z) and O′ uses (t′, x′, y′, z′)." It seems to me that there is in fact only one cartesian coordinate system, and both observers are using it. Coordinates of an event as measured by observer 1 are expressed without primes, i.e. (t, x, y, z). Coordinates of the same event as measured by observer 2 are expressed with primes, i.e. (x', y', z', t'). Is that wrong?

Consider that one set of coordinates might be with rotated axes relative to the other. Since the coordinates for measuring any given point are not the same in the two systems, this makes it two separate coordinate systems. —Quondum 00:14, 21 June 2015 (UTC)

Apologies for conflicts

I'll not edit the page for the rest of the day (really, should not for the rest of the week after so many minor edits, but that may not happen). It was important to emphasize that Lorentz transformations are passive transformations so I did that, but it took a long time and I edit conflicted. Anyway others are welcome to edit. M∧Ŝc²ħεИ_τlk 13:52, 16 November 2015 (UTC)

Hmm... Lorentz transformations can be both active and passive passive transformations. Anyway, I am done for the day (I was happily unaware of edit conflicts for once - usually I am the one getting the bad news). No need to apologize for anything. YohanN7 (talk) 14:31, 16 November 2015 (UTC)

Active and passive transformations, transformations of other quantities

OK, I always thought of Lorentz transformations as passive and find them easier to understand that way, possibly typical readers may find it easier also, (and I haven't seen them viewed as active transformations), although with a change of interpretation its an active transformation. In time (probably later today) I'll mention (briefly) how to view the Lorentz transformation as active and passive, side by side. M∧Ŝc²ħεИ_τlk 09:32, 17 November 2015 (UTC)

It is much more common to use them in the passive sense. I can't even straight away give a reference for the "active view" except for perhaps for stuff (like an occasional proof of Noether's theorem) that deals with general symmetries. Mention that Lorentz transformations are usually used in the passive sense should suffice, and then the active view could perhaps be avoided altogether? YohanN7 (talk) 10:24, 17 November 2015 (UTC)

To prevent any WP:POV, WP:THIS, WP:THAT coming from other editors (fortunately you are not one of them), I mentioned both on equal footing. If we don't mention that the LTs can be active the reader may wonder why they can't be active. Maybe just briefly emphasize the passive version, with a quick mention they can also be active (only a few words will be needed, the paragraph I wrote should be trimmed a lot). M∧Ŝc²ħεИ_τlk 12:58, 17 November 2015 (UTC)

The following may or may not be helpful. I don't think it is wrong to have a bias for passive transformations. Whenever you involve

$x^{\prime }=\Lambda x,$

you involve a passive transformation. It is only slightly different if you, say, apply an active Lorentz transformation to a 4-vector like in

$p^{\nu }\rightarrow p_{\mathrm {boosted} }^{\nu }={\Lambda ^{\nu }}_{\mu }p^{\mu },$

(no primes) transforming a momentum 4-vector of (to be specific) zero spatial momentum in order to boost it in the positive x-direction, while maintaining the same coordinate system, because then the corresponding passive transformation is given by

$x^{\prime }=\Lambda x,$

with the same Λ. Here it is seen that the primed frame needs to be boosted in the negative x-direction in order to have spatial momentum pointing in the positive direction in the primed system. This determines Λ. Of course,

$p^{\nu \prime }={\Lambda ^{\nu }}_{\mu }p^{\mu }$

holds for the momentum in the primed system for the momentum that is (spatial) zero in the unprimed frame, as it should (because 4-vector transform the same way as the coordinates), and also

$p^{\nu \prime }=p_{\mathrm {boosted} }^{\nu },$

since in both systems one sees a momentum in the positive x-direction, but primed and unprimed systems deal with different momentum vectors though they happen to be numerically equal. (This motivates the "same Λ" above)

This is slightly subtle to me, and I reserve my right to be confused with all inverses and primes.

This expression,

${\psi ^{\prime }}^{a}(x^{\prime })=S[\Lambda ]_{b}^{a}\psi ^{b}(\Lambda ^{-1}x^{\prime })=S[\Lambda ]_{b}^{a}\psi ^{b}(x),$

(from bispinor) is a passive transformation. It could be explained as such; the primed and unprimed coordinate 4-tuples refer to the same event. Therefore, use the known original (unprimed) function to compute numbers (generally an m-tuple) appropriate to the event represented by x′. Thus feed the unprimed function with the unprimed coordinates it expects. The result at this point is m-tuples attached to all points in spacetime, but expressed in the unprimed frame. These m-tuples, just like 4-vectors are supposed to transform according to some representation of the Lorentz group (second postulate of special relativity). This is represented by the quantity S, a matrix. The corresponding active transformation would be

$\psi ^{a}(x)\to \psi _{\mathrm {boosted} }^{a}(x)=S[\Lambda ]_{b}^{a}\psi ^{b}(\Lambda ^{-1}x),$

again with the very real possibility that I am confused.

I hope I am right about this, because then switching between active and passive transformations is just a matter of inserting or removing primes and using an arrow instead of the equality sign for active transformations. (Rotations should be possible to handle, mutatis mutandis, the same way with "negative x-direction" passing over into "negative of the active rotation angle") Perhaps I am making this harder than it really is. Sorry about inconsistent notation, sometimes superscripts and subscripts and sometimes not. YohanN7 (talk) 13:53, 18 November 2015 (UTC)

Wow. This may be useful for the RQM articles, but overdoing it for this article (I'm not saying your saying to include it in this article). In the next round of my trimming I'll emphasize more the passive version. M∧Ŝc²ħεИ_τlk 22:12, 19 November 2015 (UTC)

The last item applies to both classical and quantum fields (both QFT and RQM), in particular, it applies to the EM field tensor, though it is usually not presented that way. Instead (like in the article), one uses a reducible 16-dimensional representation (tensor product of 4-vector representation with itself), but the anti-symmetric tensors transform (irreducibly) among themselves, so it is really a 6-dimensional representation at work (that can be put into the above form). The symmetric tensors also transform among themselves, but this 10-dimensional representation is again reducible into a 9-dimensional representation for traceless symmetric tensors and a "scalar", where scalar refers to the matrix representation (of the tensor) being a scalar matrix. This would, in semi-standard notation, be expressed as

$({\frac {1}{2}},{\frac {1}{2}})\otimes ({\frac {1}{2}},{\frac {1}{2}})=[4]\otimes [4]=[9]\oplus [6]\oplus [1]=(1,1)\oplus [(1,0)\oplus (0,1)]\oplus (0,0),$

where the first and last expressions are in the language of irreducible (m, n)-representations for non-negative half integral m and n. (The middle one in square brackets is irreducible when space inversion is included.) At least I think it works out this way

About the same story applies to field operators in QFT, where again things are usually expressed formally differently. Most of this is irrelevant to this article, except perhaps the part about the EM field. If you think this (EM part) is overkill to include, well, the article has a lot of non-trivial material, much more advanced than this. YohanN7 (talk) 12:36, 24 November 2015 (UTC)

Barut summarizes quickly the Lorentz transformations of both tensors and spinors, including a mention of representations. The transformation of the EM field is worth mentioning as a prototypical example of a tensor in spacetime, and to prevent biasing classical relativistic mechanics, mentioning the transformation of wavefunctions (as spinors) may be useful.

I intend to add this soon, along with this in my sandbox, but that needs a lot of trimming before it enters the article (feel free to criticize/advise, but the purpose of the section is to start from what is already given in the article, and run through the heuristics illustrating features of Lie groups and algebras).

To make more room, the double boost section in the current article could be compressed into a table summarizing the four possibilities of boost-then-rotation, rotation-then-boost, original configuration, and inverse configuration (the matrices are small, and could be written on multiple lines).

Overall, here is the ordering of the mathematics which I hope is the most ideal for everyone: elementary algebra, elementary vector algebra, matrix algebra, group formulation, tensor formulation, transformations of tensors and spinors, group representations for spinors.

M∧Ŝc²ħεИ_τlk 15:46, 25 November 2015 (UTC)

Delete irrelevant animation

The momentarily co-moving inertial frames along the world line of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The small dots are specific events in spacetime. If one imagines these events to be the flashing of a light, then the events that pass the two diagonal lines in the bottom half of the image (the past light cone of the observer in the origin) are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the momentarily co-moving inertial frame changes when the observer accelerates.

The File:Lorentz transform of world line.gif (to the right) should be deleted because accelerated observers are not relevant to this article. It does not even say anything specific, or have any emphasis on, Lorentz boosts (constant velocity) in spacetime, and purely spatial rotations. The caption is also enormous and drags on forever on spacetime, world lines, light cones, etc. without saying anything about Lorentz boosts. The animation doesn't actually distract me personally from the text, but it may distract others.

In all these years I thought to keep it since it may help to show how spacetime looks with different rapidities, but not anymore. The rapidity is a constant in a Lorentz boost, so a continuously varying rapidity may be misleading. It could be added to other relativity articles which involve spacetime diagrams (which can be drawn for flat or curved space), but not here.

(Actually, I'm not 100% sure about accelerated observers in special relativity, it's certainly possible to define coordinate- and four-acceleration in SR, how this fits in with accelerating frames I don't know. In any case this article should concentrate on Lorentz boosts, at most mentioning accelerated frames in passing). M∧Ŝc²ħεИ_τlk 00:50, 26 November 2015 (UTC)

Since there are no objections, I'll move it to world line, spacetime, and spacetime diagram, and delete from this article. M∧Ŝc²ħεИ_τlk 19:04, 9 January 2016 (UTC)

Actually not spacetime but Minkowski spacetime, since spacetime is too general. M∧Ŝc²ħεИ_τlk 19:12, 9 January 2016 (UTC)

1904 paper by Lorentz

Why no mention of the 1904 paper by Lorentz is which he clearly presents his transformations? Martin Hogbin (talk) 12:54, 11 December 2015 (UTC)

Many thanks, especially for the link, I have been carried away, but in the time it took to post this section you could have added it yourself. I'll insert it now. M∧Ŝc²ħεИ_τlk 14:07, 11 December 2015 (UTC)

Hyperbolic geometry and addition of two rapidities in different directions

I'm not 100% sure how to add rapidities if the boosts are in different directions, but surely it has something to do with

1. the line element in velocity space (here Velocity-addition formula#Hyperbolic geometry)
2. Translation:On the Non-Euclidean Interpretation of the Theory of Relativity by Vladimir Varićak
3. the paper Relativistic velocity space, Wigner rotation, and Thomas precession (2004) John A. Rhodes and Mark D. Semon (should be available free on google)
4. the PDF The Hyperbolic Theory of Special Relativity (2006) by J.F. Barrett (should be available free on google)
5. Sexl Urbantke mention on p.39 Lobachevsky geometry needs to be introduced into the usual Minkowski spacetime diagrams for non-collinear velocities.

It would be nice to briefly mention the relation between hyperbolic geometry and non-collinear rapidities, with a diagram of non-commuting boost represented by two hyperbolic triangles (fig.3 in second link, and p.39 in the fourth link). Maybe then we could break the article into "physical formulation", "geometric formulation", "mathematical formulation" sections.

The purpose of this proposal is not to describe all of SR with hyperbolic geometry, just to illustrate how spacetime diagrams extend to non-commuting boosts, i.e. how to show non-commuting boosts in spacetime. M∧Ŝc²ħεИ_τlk 12:45, 13 December 2015 (UTC)