Is there a time + two spatial dimension representation of a Minkowski-space surface which could be constructed within our own (assumed Euclidean) 3D space such that geometric movement within the surface would intuitively demonstrate the “strange" effects of the Lorentz transformation (length contraction, time dilation)? Perhaps by making manifest the idea of a hyperbolic rotation (the rapidity)?

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    $\begingroup$ Huh? The metric on manifold embedded in some other manifold is just a pullback of the bigger manifold's metric. Obviously this can't change signature of that metric. So either I don't understand what you are asking or you don't :) $\endgroup$ – Marek Jan 31 '11 at 19:28
  • $\begingroup$ Yes, the obvious fact that the Minkowski metric is not the same as the Euclidean metric is certainly a problem. I just wondered whether someone, somewhere had a neat model which just did the best it could. Something better than those hyperboloids of revolution. $\endgroup$ – Nigel Seel Jan 31 '11 at 20:19

I think what you are looking for is given in the first chapter of the epic text "Spinors and Spacetime" by Wolfgang Rindler and Roger Penrose. Or at least it is if what you're asking for is a simple and clear geometric construction that illustrates the effects of Lorentz transformations on the bulk (3+1) geometry.

The celestial sphere - from Penrose and Rindler

[Fingers crossed .... IMHO the use of this image and the ones below is covered under "fair use". If it is deleted blame the copyright regime.]

To cut a long story short, you identify points on the celestial sphere $\mathcal{S}^-$ with a light ray as seen by an observer at the center of the sphere. A stereographic projection allows to map points on $\mathcal{S}^-$ to the complex plane $\mathbb{C}$. The action of Lorentz boosts on the observer translates into the action of an $SL(2,\mathbb{C})$ element on the points of $\mathcal{S}^-$. The result is shown in the figure below:

Effect of Lorentz transformations on $\mathcal{S}^{\pm}$

I do not know of any other constructions which so vividly illustrates the geometrical effects of Lorentz transformations. I have left out many details for which I once again recommend the amazing text by Penrose and Rindler.

All hail the copyright gods.

Edit: In response to the comments, I answered the question the best I could understand it. I've emphasized the relevant sentence in the first para.

  • $\begingroup$ Wow! If we can only download the PDF! $\endgroup$ – Nigel Seel Jan 31 '11 at 21:44
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    $\begingroup$ Glad you like it @Nigel. Do you mean the pdf of the book? Maybe it is djvu you're looking for. Cheers! $\endgroup$ – user346 Jan 31 '11 at 21:52
  • $\begingroup$ Hm, sure, there is also the obvious $Spin(1,3) \cong SL(2, {\mathbb C})$ isomorphism and isomorphism of four-vectors and $2 \times 2$ hermitian matrices with norm given by determinant but I fail to see how these facts even remotely relate to the question (which asked for the 2+1 case, by the way) :) $\endgroup$ – Marek Jan 31 '11 at 22:10
  • $\begingroup$ @Marek from the OP's question I gather that he's asking for something that would intuitively demonstrate the “strange" effects of the Lorentz transformation ... Perhaps by making manifest the idea of a hyperbolic rotation (the rapidity)? This construction does exactly that. Also note @Nigel's clear approval in his comment ;) $\endgroup$ – user346 Feb 1 '11 at 5:59
  • $\begingroup$ Yes, a quick approval of your ingenuity before going to bed! $\endgroup$ – Nigel Seel Feb 1 '11 at 9:02

Since this question is about learning Special Relativity with an alternative to the Minkowski Diagram to aid understanding (and dropping one - even two - dimensions shouldnt cause much harm for that purpose), might I recommend consideration of the Bondi K-Calculus?

Here the "K" is introduced into the geometry, which represents the relativistic Doppler term: it is additive and cancels out much of the hyperbolic weirdness in basic Minkowski accounts.

A quick G-search found this simple Tutorial: http://www.math.ku.edu/~lerner/m291/SR_Lecture2.pdf with drawings.

Wikipedia links to another Tutorial (without diagrams as far as I can tell).


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