Is there an intuitive geometric view of the effects of Lorentz transformations?

Question

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)?

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Nigel Seel

Comments

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 :)

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Marek

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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.

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Nigel Seel

Answer 1

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.

[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:

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.

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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.

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user user346

Comments

Wow! If we can only download the PDF!

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Nigel Seel

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Glad you like it @Nigel. Do you mean the pdf of the book? Maybe it is djvu you're looking for. Cheers!

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user user346

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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) :)

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Marek

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@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 ;)

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user user346

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Yes, a quick approval of your ingenuity before going to bed!

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Nigel Seel

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@space_cadet: all right. But you'll note that the title of the question has nothing to do with your answer whatsoever. I am going to change it to something more appropriate.

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Marek

Answer 2

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).

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user Roy Simpson