Talk:Lorentz transformation/Archive 2

RE: The Lorentz transformation for frames in standard configuration
My objection is that the wording of the article, in particular the section on Einstein's derivation of the LT, is not sufficiently general. Einstein’s original derivation1 contains the following equations [γ=(1-v2/c2)-0.5]:


 * x’ = γφ(x – vt)
 * y’ = φy
 * z’ = φz
 * t’ = γφ(t – vxc-2).

The function φ appears on the right-hand sides of all four equations. He states without explanation that φ only depends on the relative speed v of S and S’. It should be clearly understood that this is an additional assumption in his derivation of the Lorentz transformation (LT) besides the two postulates of relativity. He then goes on to prove that the only value of φ under this assumption is unity. Note that the relativistic velocity transformation (VT) obtained later in his paper can be obtained without making this assumption since φ cancels upon division of the equations for x’, y’ and z’ by that for t’. This cancellation is also the reason for including φ in the above general equations. The VT has the following form (ux’=x’/t’ etc.):

ux’ = (1 – vux/c2)-1(ux - v) = η (ux - v)             uy’ = γ-1 (1 – vux/c2)-1 uy = η γ-1 uy                                                                                                            uz’ = γ-1 (1 – vux/c2)-1 uz = η γ-1 uz.

Instead of assuming that φ=1, one can obtain a GPS-compatible version of the general LT by combining the VT equations with the proportionality relation: t’= Q-1t = Q’t:

x’ = η Q-1(x - vt) y’ = η (γQ)-1y z’ = η (γQ)-1z t’= Q-1t This result is equivalent to setting φ equal to η (γQ)-1 in the general LT. Note that the GPS-compatible value of φ is not only a function of v but also of ux (the parallel component of the speed of the object of the measurement), contrary to Einstein’s original assumption.

The alternative LT (GPSLT) is clearly consistent with Einstein’s two postulates of relativity and also with the VT. It can be inverted in the usual way by interchanging the primed and unprimed symbols and changing the sign of v (note that ηη’= γ2). Unlike Einstein’s LT, the GPSLT is consistent with the ancient principle of the objectivity/rationality of measurement (PRM) because it does not assume that two clocks in motion must each be running slower than one another at the same time. It predicts instead that the respective proportionality factors for the two observers in S and S’ are the reciprocal of one another (QQ’=1). It also is consistent with the principle of remote simultaneity (PRS) because if Δt = 0, Δt’ must vanish as well. No violations of either the PRM or the PRS have ever been observed.

It needs to be emphasized that the GPSLT derivation is specific to the space/time variables. It does not affect the four-vector relation for energy/momentum. An analogous “normalization” function which does have a value of unity occurs in the derivation (based on Hamilton’s dE=vdp relation) of the latter transformation. This is necessary to ensure that the relativistic kinetic energy approaches the non-relativistic value at low relative speeds. More details may be found elsewhere.2

1A. Einstein, Ann. Physik 17, 891 (1905).

2 R. J. Buenker, Apeiron 15, 382 (2008).

Rjbuenker (talk) 18:27, 26 April 2011 (UTC)

Proposal for derivation from physical principles
Hello, I have recently rewritten the section on "Derivation from physical principles", a change that was promptly undone. As suggested, I would like to propose the derivation in this Talk section to see what people think about it. I would not like discuss original research now: the argument is taken from the book quoted by Stauffer and Eugene_Stanley, and I can provide more references. The section is here: Proposed derivation for the Lorentz transformations --Daniel (talk) 19:40, 26 September 2011 (UTC)


 * Hi, I took the liberty to correct a few mistakes and fix some formatting. See my edit. Could you also specify the ISBN and the exact page(s) where the derivation is to found? If nobody objects, afaic we can take this onboard. DVdm (talk) 07:27, 27 September 2011 (UTC)


 * Tweaking looks fine. Thanks for your time. Added isbn and pages, plus rewritten the final part to follow exactly the steps in the reference. Let's wait a day or two, then we can perhaps change the section. Best, --Daniel (talk) 11:31, 27 September 2011 (UTC)


 * I noticed your tweak. Thanks. I don't think we need the elaboration on the final calculation of A and B. It's better to just give the results. The reader can verify with the source. Also, using the 'we'-form in derivations is not really in accord with the wp:MOS. I'll make some more minor changes. DVdm (talk) 12:58, 27 September 2011 (UTC)

Oops, it looks like I forgot a few 'we'-s. I think we can also get rid of one of the two original references and just keep your Stauffer/Stanley and Einstein. I guess it looks good now, and, since there were no objections from others, feel free to insert the current version into the article. Cheers - DVdm (talk) 09:52, 28 September 2011 (UTC)

As there were no objections, I have taken over the draft now. DVdm (talk) 07:59, 4 October 2011 (UTC)


 * Thanks very much, DVdm. I have been a bit busy, but it was my intention to take over the draft into the main article. Best, --Daniel (talk) 08:19, 4 October 2011 (UTC)

Error in Lorentz transforms.svg
I mentioned this in the discussion for the image, but I'm not sure anyone is watching there... so I'll mention it here.

In the first transformation, the right hand side erroneously repeats the terms y' and z' instead of just y and z. --76.27.129.102 (talk) 07:05, 12 October 2011 (UTC)


 * I have made the correction. You might have to clear your browser cache to verify.
 * Note: I am not a big fan of the image's concept (nor its placing in the article), but I guess it's better than nothing. If nobody else has any objections, I can live with it.
 * ✅ - DVdm (talk) 07:56, 12 October 2011 (UTC)

removal of content
If no-one minds, I removed the matrix for the transformations of the EM field: it just restates the formulae directly above - whats the point? (it's a slight abuse of notation to have vectors and sclars placed in entries like that, but maybe that was not a big deal since the cross-product and curl determinants do this...).

Also I replced the very first image with Maschen's, and adjusted the text of the subsection Hyperbolic rotation of coordinates to compensate. They show exactly the same thing, but Maschen's is far cleaner and has more information in the diagram than a long caption, and including both just adds to the byte count. It also immediatley gives the reader the idea of what quantities are coming: light speed c, c/v as the gradient of the worldline and rapidity φ. F = q(E+v×B) ⇄ ∑ici 17:16, 21 May 2012 (UTC)


 * In the end I just removed all the transformation formulae and linked the section Lorentz transformation of the electromagnetic field to Classical electromagnetism and special relativity, since everything is there. The explainations have been kept here. F = q(E+v×B) ⇄ ∑ici 10:46, 6 June 2012 (UTC)

re-organize
Main changes:


 * Remove section on "index permutation", not particularly helpful.
 * Replace by the transformations in Composition of two boosts which is completely out of place (using r parallel and perp to relative velocity v),
 * also insert a few lines on how to get from the component equations to the full 3-vector equations, they may be obvious to some editors here but certainly not to the lay reader. As far as I can tell this is not in many textbooks but it’s a very useful trick that works for the other transformations if written in a simalar "parallel-perp" form:


 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Derivation
 * The transformation for t can be written:
 * $$t' = \gamma \left(t - \frac{\mathbf{v} \cdot \mathbf{r}_\bot}{c^2}\right)$$
 * $$t' = \gamma \left(t - \frac{\mathbf{v} \cdot \mathbf{r}_\bot}{c^2}\right)$$

since
 * $$\mathbf{r} \cdot \mathbf{v} = r_\parallel v $$

Starting from the transformation for r:


 * $$\mathbf{r}' = \mathbf{r}_\parallel' + \mathbf{r}_\bot' = \gamma \left(\mathbf{r}_\parallel - \mathbf{v}t \right) + \mathbf{r}_\bot $$

adding $$\scriptstyle 0 = \gamma\mathbf{r}_\bot-\gamma\mathbf{r}_\bot $$ to eliminate $$\scriptstyle \gamma\mathbf{r}_\parallel$$:


 * $$\begin{align}

\mathbf{r}' & = (\gamma \mathbf{r}_\parallel + \gamma\mathbf{r}_\bot) - \gamma t \mathbf{v} + \mathbf{r}_\bot -\gamma\mathbf{r}_\bot \\ & = \gamma \mathbf{r} - \gamma \mathbf{v} t + (1-\gamma)\mathbf{r}_\bot \\ \end{align}$$

now adding $$\scriptstyle 0 = (1-\gamma)\mathbf{r}_\parallel -(1-\gamma)\mathbf{r}_\parallel$$ to eliminate $$\scriptstyle (1-\gamma)\mathbf{r}_\bot$$:


 * $$\mathbf{r}' = \gamma \mathbf{r} - \gamma \mathbf{v} t + [(1-\gamma)\mathbf{r}_\bot + (1-\gamma)\mathbf{r}_\parallel] -(1-\gamma)\mathbf{r}_\parallel $$

since $$\scriptstyle \mathbf{r}_\parallel $$ and v are parallel and hence
 * $$\mathbf{r}_\parallel = A_\parallel \dfrac{\mathbf{v}}{v} = \dfrac{\mathbf{r}\cdot\mathbf{v}}{v^2} \mathbf{v}$$

leads to
 * $$\begin{align}

\mathbf{r}' & = \mathbf{r} - \gamma \mathbf{v} t + \frac{(\gamma-1)\mathbf{r}\cdot\mathbf{v}}{v^2} \mathbf{v} \\ & = \mathbf{r} + \left(\frac{\gamma-1}{v^2}\mathbf{r}\cdot\mathbf{v} - \gamma t \right)\mathbf{v}\\ \end{align}$$
 * }

And no - just in case someone says so - I really don't care for this instance if
 * "WP never includes routine algebra/calculus because the weather will turn bad tomorrow or becuase it blinds everyone."

It IS allowed by WP:MOSMATH to include derivations for clarification of concepts and there are plenty of derivations in places on WP anyway, including extensive detials occupying nearly half this article, so this inobvious one should not be excluded. F = q(E+v×B) ⇄ ∑ici 23:40, 5 June 2012 (UTC)

Re-write/organize derivation
Let’s be honest: could a reader, looking for a transparent and short (as possible) derivation of the transformations and where they come from, actually be bothered to read throughout Derivation?... It is far too long and blabs on and on, and generally has messy formatting - especially the group postulates section.

The section From physical principles is more readable, but the sections somehow seem abrupt.

I would propose replace all subsections under From physical principles EXCEPT Einstein's popular derivation by the one in my sandbox, since this one has a continuous flow to it and solves for the hyperbolic and the algebraic transforms together. If there are no objections I will make the replacement, and while at it clean up (but not delete/replace anything) from the group postulates section. F = q(E+v×B) ⇄ ∑ici 18:58, 6 June 2012 (UTC)


 * Forgot to add - From physical principles should come before group postulates since this is more direct and easier to comprehend. Group theory is not understood by everyone (including me, for now), and is too deep and abstract for a reader to begin reading up the derivations. F = q(E+v×B) ⇄ ∑ici 19:09, 6 June 2012 (UTC)


 * It seems like people are not active on this page (so it seems for the last few days)... The derivation has been rewritten and the sections switched around, as said above (I'll clean up group postulates later later). F = q(E+v×B) ⇄ ∑ici 20:19, 7 June 2012 (UTC)

Removing statement antecedent of spatial homogeneity impying linearity of the transform
As per my observation at WP:Reference_desk/Mathematics, I feel the statement "If space is homogeneous, then the Lorentz transformation must be a linear transformation." in the lead (intriduced here) is inaccurate and does not warrant being in the lead. Given general relativity, mathematical homogeneity is not to be assumed, and hence the statement is uninteresting. I hope someone with the necessary expertise in the subject can insert a section indicating that the local Lorentz transformation is inherently linear, but not in the lead. I am simply removing the reference to spatial homogeneity in the sentence. — Quondum☏ 06:29, 10 June 2012 (UTC)