Comment by Steuard | Hacker News Reader

Comment by Steuard | Hacker News Reader

This explanation is correct, but I find it pretty empty of intuition: "Here's a magic formula! Look, it adds up funny." I've taught relativity quite a bit recently, and I much prefer to build intuition using "space-time diagrams" (or "Minkowski diagrams") showing coordinate systems for two different observers.

Tom Moore's Six Ideas That Shaped Physics textbook (Unit R) does a very nice job of this. But here's a really bargain-basement handout that I made many years ago to explain the idea before I started using Moore's text for this. (I'd probably do it a little differently now.) http://www.slimy.com/~steuard/teaching/classes/spacetime.pdf

My discussion of velocity addition there (at the bottom of page three out of four in that PDF) is much too terse for this context, I'm afraid. But if you draw my example there carefully and measure in the tilted coordinates as described earlier, you'll start to see why the leftmost observer still measures the rightmost observer moving less than the speed of light.

If you've got way too much time to spare, I've got a set of topic review videos that I made for various classes on this stuff. The quality's not great, but seeing some worked examples could be interesting. https://www.youtube.com/playlist?list=PLQcPPNE9JfpiF4GpUSOGm...

I like the problem of "you are travelling at 0.75c and you turn on your headlights, how fast do you see the photons travelling and how fast does a stationary observer see the photons travelling?" better.

Then the answer is just the speed of light in both cases, and that's really the crux of the problem of special relativity and that the speed of light is constant in all references frames. Then you work backwards to see that the observer that is moving must have time dilation effects.

I also liked the geometrical approach to Minkowski space-time a lot better than just chewing through formulas -- like Taylor and Wheeler's Space-Time Physics book. Drawing light cones and various grids and curves of constant space or time measurements relative to different observers made it all a lot more concrete to me.

IMO, a much more natural approach is to look at things from the perspective of light emitted. Specifically black body radiation. Red/Blue shifting correlates with changing temperatures. And temperature works as a natural proxy for time.

The other thing that confuses people is presenting things from an 'independent' perspective in the diagram.

If I'm not mistaken, the derivation of the black-body radiation energy distribution does not use special relativity at all, only statistical mechanics and quantum mechanics. How can it be used to explain special relativity?

A thought experiment I've always enjoyed is the idea of two infinitely long rods, each moving at near as damnit c, perpendicular to one another, with a point of contact.

At what speed does the point of contact move, relative to the reference frame of either bar, or to a static external observer?

So, do I understand you correctly: relative to the "static observer", the rods are both moving "sideways", as if the x-axis of a graph were moving toward +y and the y-axis were moving toward +x?

In that case, the "static observer" would see the point of contact moving at very nearly sqrt(2) times the speed of light (along the x-y diagonal). But that's okay, because no physical object (or bit of information) is being carried along at that intersection point. (If you put some little spindle at the intersection to be carried along, you'd suddenly have Problems.)

On the other hand, from the reference frame of one moving bar, the intersection point appears to be moving almost infinitely fast! If the "static observer" measured speeds u and v for the two rods (as fractions of light speed, so u and v are both close to 1), then an observer moving with one rod will see the intersection point move at speed v/sqrt(1-u^2). If we write u = 1 - ε for some very small ε, then the intersection point's speed in this frame is roughly v/sqrt(2ε). (The intuitive reason for this result is interesting: an observer on one moving rod will consider the other rod to be almost parallel, at a very small angle (rather than perpendicular), and still rushing toward her at nearly the speed of light.)

Told you it was a nice one to muse on! I'm a physics grad, so know the answer as you've stated it. Gotta love Lorentz contraction. :)

I should've guessed that you had a physics background, since that turned out to be an unexpectedly interesting problem. (The view from one of the moving rods in particular was more subtle than I anticipated. As usual, once I stopped trying to intuit the answer and finally set it up carefully with 4-vectors the answer came out pretty cleanly.) I wonder if there's a good place to use this in my classes. (It was a bit more subtle than I'd usually throw at my sophomores...)

What's your field?

Another good text for this approach (and it's also relatively cheap) is Mermin's It's About Time.

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