SR pedagogy: energy-momentum and area in the x-t plane

[bcrowell]

I've been comparing various books, including these:

Mermin, It's About Time
Takeuchi, An Illustrated Guide to Relativity

for possible use in a gen ed course on relativity. It's cool to see that there are so many books out there now that aren't just replaying Einstein's 1905 postulates with the same algebra; the textbook field tends to be very conservative, so this was a bit of a pleasant surprise to me. I was also glad to see that there were good options that weren't exploitatively overpriced, although I can't help expecting that the publishers will gradually jack up the price until these books are the same cost per ounce as heroin.

To some extent, I guess I've been reinventing the wheel in the pedagogy I've used with my students who are science and engineering majors. I started out doing the traditional 1905 approach and gradually refined it into something that turns out to have a lot in common with Mermin's and Takeuchi's, including visualizing Lorentz transformations based on the way they distort rhombi in the x-t plane.

I was pretty dissatisfied, however, with their treatments of a couple of topics.

Conservation of area by the Lorentz transformation

In any treatment using the rhombus-squishing approach, an issue that comes up is the fact that area is conserved. Takeuchi gets it utterly wrong:

This conservation of spacetime area maintains the symmetry between [two frames], since each is moving at the exact same speed when observed from the other frame, and ensures that the correspondence between the points on the two diagrams is one-to-one. [p. 92]

The first part, about maintaining symmetry, is wrong because it can be used to prove that absolutely *anything* stays the same under a Lorentz transformation. Some things do and some things don't, so there is no valid argument here unless specific facts about the x-t area are appealed to. The second part, about one-to-one correspondence, is also wrong, because you can have a one-to-one transformation that doesn't preserve area. I wouldn't have a big problem with this if he just presented it as a plausibility argument, but a naive reader would be almost certain to imagine that Takeuchi was somehow proving something logically.

Mermin says:

[...] unit rhombi used by different observers all have the same area. This rule follows simply and directly from the requirement that when Alice and Bob move away from each other at constant velocity, they must each see [optically] the other's clock running at the same rate, as measured by their own clock.[p. 115]

He then gives a valid geometrical proof. This is all OK, I guess, but I'm unenthusiastic for two reasons: (1) He presents the conservation of area as being dependent on facts about light, but actually the area-conserving property is valid outside the context of SR (e.g., it holds for Galilean relativity). (2) In any case I dislike presentations of SR that treat light as fundamental.

For anyone who's interested in how I prefer to present this, see http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html at "Proof that the Lorentz transformation preserves area."

Energy-momentum

Mermin takes the following approach. We want (1) the correspondence principle, (2) frame-independence of conservation laws. Combination of velocities is nonlinear because v=d/t, and in SR, both d and t change, not just d. So let's define a new quantity w=d/\tau, where \tau is proper time, to get rid of this problem (i.e., w is the spacelike part of the velocity four-vector). He does messy algebra to find a transformation law for w. Now define p=mw. The transformation law for w requires introducing some new constant, p0 which is eventually interpreted as energy. He proves that if p and p0 are conserved in one frame, they're conserved in any other frame as well. As far as I can tell, this is nothing more than a plausibility argument. That is, just because he comes up with one particular conservation law that satisfies 1 and 2, that doesn't mean it's unique, or that it's even valid empirically. His approach is also a heck of a lot of grotty algebra when the pay-off is only a plausibility argument.

Although I wasn't thrilled with Mermin's treatment, it could be worse. Halliday and Resnick do something similar, but stripped of even the slightest pretense of logic or motivation.

Takeuchi develops the "mass-momentum vector" in Newtonian mechanics. This is much more abstract than the pictures of world-lines he's done before, and seems inappropriate for his target audience, most of whom have probably never been exposed to 3-vectors in any context at all. He then generalizes this to an energy-momentum vector in SR. I haven't read through his treatment carefully enough to see whether it seems logically valid. In general, I dislike his tendency, which he shares with Mermin, to completely avoid any contact with experiment.

I'm still not really sure what is the best way to do this topic at the freshman level, especially with gen ed students. For physics majors, the fastest route is that if you've already introduced four-vectors, you say that it makes sense to search for some kind of momentum four-vector, and you can prove lickety-spit that there's only one way to construct such a thing. But I wouldn't expect this to work with gen ed students, and it's not trivially obvious that it must be a four-vector. I've seen books that use the work-kinetic energy theorem as a starting point, but I've never seen one of these treatments that gave so much as a plausibility argument for the validity of the work-KE theorem in SR, written in the same form as in nonrelativistic mechanics. I've done it in a variety of ways, including four-vectors and plausibility arguments. The approach that I think is most complete and logically rigorous is the one I did in http://www.lightandmatter.com/mechanics/" in ch. 14.

-Ben

 

[pervect]

I rather like what's usually called the K-calculus approach for simplicity. I think I mentioned the very old book by Bondi in another thread that uses this approach, "Relativity and Common Sense".

It only requires high school algebra to follow the derivations, which are well tied to what one observes, the doppler shift.

I'm not sure if there are more modern books that use this approach or not. Bondi's pacing is rather slow, in an attempt to reach an audience where high-school algebra might be seen as "hard math".

 

[DrGreg]

I rather like what's usually called the K-calculus approach for simplicity. I think I mentioned the very old book by Bondi in another thread that uses this approach, "Relativity and Common Sense".

It only requires high school algebra to follow the derivations, which are well tied to what one observes, the doppler shift.

I'm not sure if there are more modern books that use this approach or not. Bondi's pacing is rather slow, in an attempt to reach an audience where high-school algebra might be seen as "hard math".

For what it's worth, that's exactly the method I used in an old post Lorentz interval, post #3, equation (3).

 

[bcrowell]

For what it's worth, that's exactly the method I used in an old post Lorentz interval, post #3, equation (3).

This seems similar to what Mermin does -- obviously great minds work alike :-)