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Since Einstein first derived the Lorentz transformations, their generalisation and execution has changed over the century. So starting with those first derived by Einstein:

What are the main, increasingly sophisticated ways, of carrying out a Lorentz transformation today? What are their advantages/disadvantages over the preceding methods?

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come on Ron, I'm dying to see an answer from you given your strong views on the way people perform the Lorentz transformations –  Larry Harson Jun 2 '12 at 19:19
Oh, you mean that I don't like 3-vector language? This is due to the horror of 1920s-1930s literature on relativity, and trying to read Thomas precession paper and get the result out. I gave up and had to redo it from scratch. The problem there was wrong type of sophistication--- 3-vector sophistication regarding a 4-dimensional transformation. –  Ron Maimon Jun 2 '12 at 21:31
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2 Answers

"Since Einstein first derived the Lorentz transformations...", that's priceless.

The Lorentz transformations, motivated as specific linear transformations of in a vector space (i.e. matrices) have not and will never change. What changes is notation. If by "...execution has changed over the century" you mean how to implement these efficiently, then I guess that's a comp. science question.

Regarding "...and generalizations...", as I said, the fundamental abstract transformation group is the same. Its finite representations (how to transform this and that object, and which objects are there, which can be transformed) are fairly easy to grasp.

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Indeed, the history of the transformations ends with Einstein: en.wikipedia.org/wiki/Lorentz_transformation#History –  genneth Jun 2 '12 at 14:22
-1 the people before Einstein merely wrote down the transformations, whereas Einstein did indeed derive them from his two postulates, and not in a matrix form. Secondly, Minkowski reinterpreted the lorentz transformations as rotations in hyperbolic space, emphasising the invariance of the space time interval. –  Physiks lover Jun 2 '12 at 15:27
@Physikslover: Mhm, they are the same transformation though. May they come as the invariance group of the Maxwell equations, the light beam of the metric tensor. I don't see the priority of one set of postulates over the other. And as linear coordinate transformations in flat space, they can be written down as matrices. –  NiftyKitty95 Jun 2 '12 at 16:19
sure, they're the same transformation, but carried out using increasingly sophisticated methods that offer greater insight, and that seems to be what the question is asking for. –  Physiks lover Jun 2 '12 at 19:25
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(Well, since @RonMaimon didn't take the bait first on this one, what the heck...!)

The more sophisticated way to do a Lorentz transformation is first to draw out the body in question onto the slanted coordinate systems of hyperbolic Minkowski space. This will give you something that doesn't even remotely look like a Lorentz transform, because you are mapping the hyperbolic coordinates into the Euclidean space of you graph paper (or computer model, whatever). What you get will instead looks just like the distorted spherical mirror clocks in Figure 4.23 on p.33 of @HansdeVries marvelous chapter on non-simultaneity in special relativity. (Hans: I have that same mirror-clock, not as pretty, in some of my personal pages of notes and musings on SR, so I loved seeing it in your book!)

What's weird about this more precise version of Lorentz contraction is that it's not contracted at all -- it's stretched, both in time and in space! As you move closer to $c$, the length of the propagation axis $x'$ of the object stretches out towards infinity in hyperbolic coordinates, and does the same thing in time! And keep in mind, this is in the very space used to describe SR accurately, that being Minkowski space.

So that clearly cannot the bottom line to what happens. And it's not: There's another step needed to get the Lorentz contraction we know and love: Four-dimensional projection across both space an time.

More to the point, in Hans' Figure 4.23 the observer frame is represented by the horizontal ring at the bottom of the right-hand figure. That ring in turn represents what you get if you project the shape of the highly stretched mirror sphere clock backwards in time and parallel to the time axis $t'$ of the moving frame. That projection is already distorted by the revised clock shape, and it gets distorted a second time by the fact that observer time is angled relative to the moving frame.

Now, put those two operations together -- 4D projection parallel to $t'$, followed by angular distortion by projecting onto a 3D space that is not fully orthogonal to $t'$ -- and then you get "simple" Lorentz contraction.

Easy, right? :) Actually, one of my favorite professorial-style understatements of all times is Minkowski's explanation of what I just described: "... an easy calculation gives ...[the Lorentz transformation]" (p.82 line 18 of The Principle of Relativity, paperback reprint version). Easy. Hmm. Well, Sommerfeld on p.92-93 of the same book later added over a page of obscure trigonometric equations trying to explain what Minkowski meant by "easy." I've reproved several items from that same book, including the famous $E=mc^2$ (it's delightful, odd, and unexpected in approach), and I've given up on doing that proof geometrically. It's ugly at best, distressingly unconvincing at worst. You can eyeball the silly figure and see pretty quickly that it has to be darned close to the Lorentz transformation, but prove it? Not easy!

So there you are: The nitty-gritty of how Minkowski arrived at the Lorentz contraction from Einstein's paper, all the while belittling Lorentz for having pulled such ideas out of the sky without proper mathematical support. Ah... hmm! I rather deeply suspect Minkowski found that relationship mostly because he already knew it had to be there, both from Lorentz' work and from Einstein's equations. (Einstein's original paper also contains the projection step, albeit in classic Einstein style; his equations are algebraically simple, but conceptually a bit like some sort of cross-mountain leaping contest).

And now that I've given you the nitty-gritty, allow me to unsettle you a bit:

(1) In Lorentz contraction, the leading edge of the object is in the future when compared to the trailing edge of the object. Hans de Vries nicely noticed this point in an answer to an earlier question I asked, but I think it gets skipped in most physics classes. John Bell was most definitely aware of it, as he mentions it as an unresolved issue in his article on the two-ships-and-a-string SR paradox.

(2) The stretched shape of the object in Minkowski time is not entirely fictional, since it has some real causality implications. I'm pretty sure that the Feynman assertion that a photon "exists at both its origin and destination at the same time" (a paraphrase, not an exact quote!) has some deep connections to this same point, since at $c$ the photon in Minkowski space stretches all the way from its origin to its destination, even as its projected Lorentz length falls to zero. The mental exercise required to hold perspectives as paradoxical as these two in your head at the same time is one of the reasons I love physics!

(3) There are some really weird "time synchronization" issues in all of this. For example, does the moving object project entirely from its future into our now to obtain the Lorentz contraction we observe experimentally? Or is it partially in our past and partially in our future? Or for that matter, is the whole thing already said and done and entirely in our past? If you look closely at the Minkowski diagrams, you will realize that they don't really address such questions because by default they use block time, that is, they assume the past, present, and future all exist equally. The trouble is that experimentally we never, ever see things in block time. We only see "now," and all of our experiments have to deal with "now" even when our models tell us that it is really a mixture of past, present, and future.

Issues like the above are why I think it should be OK even now to spend a bit of time (not all of it!) poking and prodding at such "known" results. Such poking and prodding can help make them clearer in our heads, if nothing else. Thus since the full scope of Lorentz contraction as portrayed in all of its detail in Minkowski space is quite distant indeed from the genuinely trivial transformation that Lorentz initially proposed, it's probably a good example of a topic where a bit of well-defined and highly selective poking and prodding is still justified.

Speaking of poking and prodding well-covered turf, John Bell is one of my true heroes in physics precisely because he consistently took an approach of "let's check out a few more version of that before say we 'know' exactly what happens in every case."

And if you look what Bell eventually found lying unnoticed among the detritus of decades of argument between two greatest minds of the $20^{th}$ century, Einstein and Bohr, you can see why there is merit in just such a perspective. The Bell Inequality, arguably one of the single biggest drivers in modern physic research, emerged precisely from Bell insisting on taking "one more look" at heavily trodden turf.

And who knows? Maybe someone out there in this group will do the same thing some day, and find some goodie that everyone else has overlooked. It's never likely that will happen -- it wasn't likely for Bell. But the only way to ensure you will never find something overlooked is to decide never to go looking for it.

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Thanks for your answer and I'll have a look at HansdeVries's chapter –  Physiks lover Jun 3 '12 at 12:04
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