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N. David Mermin has an interesting geometrical approach to SR that I came across today. He seems to have described it in the following places:
1. Mermin, N. David, "Space-time intervals as light rectangles," Amer. J. Phys. 66 (1998), no. 12, 1077
2. a popular-level book called "It's About Time"
3. various pdf's of talks, linked to from http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html (minkowski.pdf being the most detailed)
One thing I hadn't realized before is that there is a connection between the spacetime interval and the area of rectangles with lightlike edges. There are straightforward arguments that area in the x-t plane has to be conserved under boosts in the x direction, regardless of whether you're talking about SR or Galilean relativity or any other theory that doesn't violate the homogeneity and isotropy of spacetime (http://www.lightandmatter.com/area1book6.html , appendix 1) . Since area is conserved, and the spacetime interval can be interpreted as an area in the case of SR, you get a very simple and straightforward geometrical argument that the interval is invariant.
He also has a cute way of getting at the relativistic combination of velocities via the Doppler shift, which I think is far more transparent than any other approach I've seen. Essentially you can get the velocity w that results from combining velocities u and v by setting the cumulative Doppler shift equal to the product of the two partial Doppler shifts:
[tex]
\sqrt{\frac{1-w}{1+w}} = \sqrt{\frac{1-u}{1+u}} \sqrt{\frac{1-v}{1+v}}
[/tex]
This can be solved for w to get the usual formula, or if you take logs you get the additive relation between rapidities. But he has a cute way of looking at this purely geometrically (p. 25 of minkowski.pdf), which can then be used to derive the Lorentz gamma factor from Einstein's 1905 axiomatization of SR.
Personally I don't think the 1905 axiomatization is the right one to use in the year 2010, but a lot of Mermin's stuff nevertheless fits very nicely with my own preferred way of teaching SR, which is also heavily geometrical ( http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html ).
1. Mermin, N. David, "Space-time intervals as light rectangles," Amer. J. Phys. 66 (1998), no. 12, 1077
2. a popular-level book called "It's About Time"
3. various pdf's of talks, linked to from http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html (minkowski.pdf being the most detailed)
One thing I hadn't realized before is that there is a connection between the spacetime interval and the area of rectangles with lightlike edges. There are straightforward arguments that area in the x-t plane has to be conserved under boosts in the x direction, regardless of whether you're talking about SR or Galilean relativity or any other theory that doesn't violate the homogeneity and isotropy of spacetime (http://www.lightandmatter.com/area1book6.html , appendix 1) . Since area is conserved, and the spacetime interval can be interpreted as an area in the case of SR, you get a very simple and straightforward geometrical argument that the interval is invariant.
He also has a cute way of getting at the relativistic combination of velocities via the Doppler shift, which I think is far more transparent than any other approach I've seen. Essentially you can get the velocity w that results from combining velocities u and v by setting the cumulative Doppler shift equal to the product of the two partial Doppler shifts:
[tex]
\sqrt{\frac{1-w}{1+w}} = \sqrt{\frac{1-u}{1+u}} \sqrt{\frac{1-v}{1+v}}
[/tex]
This can be solved for w to get the usual formula, or if you take logs you get the additive relation between rapidities. But he has a cute way of looking at this purely geometrically (p. 25 of minkowski.pdf), which can then be used to derive the Lorentz gamma factor from Einstein's 1905 axiomatization of SR.
Personally I don't think the 1905 axiomatization is the right one to use in the year 2010, but a lot of Mermin's stuff nevertheless fits very nicely with my own preferred way of teaching SR, which is also heavily geometrical ( http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html ).
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