Talk:Lorentz transformation/Archive 8

Derivation again

With regard to the above thread, see esp. my post dated 13:46, 13 January 2017 (UTC), should we have a section rigorously proving the connection between matrix elements and physically meaningful parameters? This would actually constitute, as a spin off effect, a "derivation" in the sense of manufacturing the formulas

$x=\gamma (x'+vt'),\,\,t=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right),\quad x'=\gamma (x-vt),\,\,t'=\gamma \left(t-{\frac {vx}{c^{2}}}\right),$

which we have agreed to keep elsewhere. But I'd argue though that such a derivation would benefit this article. For one thing, it wouldn't be limited to only boosts of the above type (or to boosts at all), and it would (when dealing with boosts of the above type) show where how and where the $v$ and the $\gamma$ pop up in the matrix.YohanN7 (talk) 14:01, 13 January 2017 (UTC)

I think any single derivation, that starts from somewhere, and ends in the formulae, would be better than none at all. Encouragingly, Kebl0155 (talk) 14:13, 13 January 2017 (UTC)

DVdm, Maschen? YohanN7 (talk) 12:37, 14 January 2017 (UTC)

As stated above, I personally object to any derivation in this article, from start to finish. It is what the derivations article is for. Simply heuristically constructing is different.

If people want to try the suggestion by YohanN7 in this thread, by all means edit.

I'm not sure what you mean about connecting matrix elements to physical parameters, do you mean these formulae

${\begin{aligned}\Lambda _{00}&=\gamma \,,\\\Lambda _{0i}&=\Lambda _{i0}=-\gamma \beta _{i}\,,\\\Lambda _{ij}&=\Lambda _{ji}=(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}+\delta _{ij}\,,\\\end{aligned}}$

for the boost matrix and

$\Lambda _{00}=1$

$\Lambda _{0i}=\Lambda _{i0}=0$

$\Lambda _{ij}=(\delta _{ij}-n_{i}n_{j})\cos \theta -\varepsilon _{ijk}n_{k}\sin \theta +n_{i}n_{j}$

for the rotation matrix? M∧Ŝc²ħεИ_τlk 13:13, 14 January 2017 (UTC)

Yes, and some additional conceptually and computationally useful info. YohanN7 (talk) 07:25, 16 January 2017 (UTC)

(edit conflict) I don't think we need it, unless there's something in the literature—i.e. a book, not a little unnoticed article—that closely supports the derivation. If there is no such thing, it would be original research, and on top of that wp:UNDUE. That's one of the reasons that Wikipedia needs reliable sources: it is an encyclopedia, not a textbook. So, afaiac, no. - DVdm (talk) 13:23, 14 January 2017 (UTC)

Is Steven Weinberg good enough? (Thank you for telling me about OR. I was totally unaware of that.) Physics and engineering student sometimes live in a mathematical vacuum, and are supposed to pick up the things I now refer to by osmosis. All evidence is that most don't. YohanN7 (talk) 07:25, 16 January 2017 (UTC)

Heh... if Weinberg wouldn't be good enough... who would? . Try to supply the full-blown book details with a proper {{cite book}} template. A Google-book readable citation would be fantastic. And it would prevent future above- and elsewhere-like long discussions. Go ahead. - DVdm (talk) 09:23, 16 January 2017 (UTC)

Let us wait a bit for more opinions. I too have mixed feelings about it, and have arguments against it as well. My arguments against it are the same as yours and M's I guess. There is a dedicated article. My arguments for it is that it would be a derivation from a mathematical perspective without reference to peculiar and possibly unfamiliar physical phenomena such as time dilation and length contraction. Such derivations are better described as ingenious rather than intuitive. With physics mostly out of the way, some may find it a lot easier to follow.

A compromise would be to put it late in the article, since though short and crisp (Weinberg's formulation of it, not mine) for the seasoned reader, it might intimidate the casual reader.

The book is Weinberg's Gravitation and cosmology. Unfortunately, it isn't available as a Google book. There is as well an actual proof that the transformations must be linear in it (most definitely intimidating, but short). Again using math as opposed to physical experiments of thought and hand-waving. This (the linearity) is conceptually important because a common misconception is that preservation of the velocity of light alone (postulate 2 of SR) leads to the inhomogeneous Lorentz group. It does not. It leads to the nonlinear conformal group of which the Poincare group is a subgroup. The source-free Maxwell equations are actually invariant under the conformal group. Now if one brings in postulate 1 of SR, then formula D2 must apply. It is then logically desirable to show that this rules out the conformal group. YohanN7 (talk) 10:05, 16 January 2017 (UTC)

Ok, fair enough. Btw, I have a copy of the book. - DVdm (talk) 10:34, 16 January 2017 (UTC)

If you want to try anytime, feel free and we can see how it looks. M∧Ŝc²ħεИ_τlk 10:42, 16 January 2017 (UTC)

Covariant and contravarient metric tensor

The article says ``As it happens, η^μν = η_μν" but actually this is specific to the instant form, in other forms of dynamics (light front or point form) this can vary (and the metric has off diagonals). See for example the Brodsky review https://arxiv.org/abs/hep-ph/9705477 page 19. I know almost everything is done in instant form, but the others still exist, how should we deal with this? 129.215.144.93 (talk) 13:02, 2 March 2017 (UTC)

Not at all. YohanN7 (talk) 10:42, 6 March 2017 (UTC)

I did look at the article. Good stuff, but it is a rather trivial point that curvilinear coordinates or other non-standard coordinates (which seems to be what it is all about) yield different entries in the metric. YohanN7 (talk) 11:58, 6 March 2017 (UTC)

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Different looking symbols for relative velocity in equations and text

The velocity letter v is used for the relative velocity in the x direction in the equations e.g. at \frac{v x} in:

${\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}$

and seemingly the same v in the following text e.g. at {math|v} in

where v is the relative velocity between frames in the ..

But on my computer screen the v in the equation is a script v and in the text it looks like a greek letter v (nu)

?????? RudiPo (talk) 11:24, 6 March 2018 (UTC)