The Doppler Shift in Relativity

It's been quite a while since I've posted any new physics, but I had a visit on the weekend from an old physics buddy from univeristy days. Richard is teaching at Waterloo these days, and relativity is his thing. I've mentioned once or twice that it's not really my territory, but last year I thought I did a pretty nice series on how to do those first-year compound velocity problems with pictures. I didn't assume any formulas except for the fact that x^2 - t^2 is invariant (the relativistic Law of Pythagoras.) You can see what I wrote here. Near the bottom of the page you can see how hyperbolas on the x-t diagram are the same "distance" from the origin. So someone moving along the orange line counts of his seconds "one, two, three..." accorcing to when he intersects those hyperbolas:

Well, on the weekend Richard and I were talking physics, and he told me about a problem he gave his students to calculate the doppler shift in relativity. He wanted them to do it without using the formula, just deriving it from basic principles. Naturally I wanted to try it for myself - and without assuming anything about x^2-t^2. I think I got it right, and as a consequence, basically derived the fact that x^2-t^2 is invariant. Here is what I did.

Now, what you normally do is have an observer on the ground (Alice) and an observer in the train (Bob). Alice stands on the tracks after the train has passed and broadcasts a radio signal to Bob, say 900 kHz. Bob measures the frequency and finds it is slower than what Alice sent out. This is normal: it's just the same way sound works. If Alice blows a whistle at 900 hz, Bob hears it at a lower pitch. If Bob blows a whistle, Alice hears the pitch change from high to low as the train passes by her. That's how sound works, and that's how light works. Either way, the picture looks like this:

The orange line is the moving train (which I'm going to take as going 4/5 the speed of sound (or light) in this picture, and the blue lines represent the "pulses" passing between the two observers. You can think of them as pulses or you can think of them as wavefronts, where I've drawn the waves in purple. I've done something a little bit odd: instead of Alice blowing one whistle and Bob blowing another one, I have Bob holding up a mirror so that Alices waves get reflected right back to her. The sound is reflected back at a lower pitch, and so is the light.

It's not hard to do the calculation for Alice. It's not so hard to see (from the basic geometry) that if she blows a whistle at 900 Hz, the echo that relfects back reaches her ear at 100 Hz. And it's the same for light waves: if she radios Bob at 900 kHz,  it comes back at 100 kHz.


(It's actually a little more natural to work with periods instead of frequency. If alice sends out pulses (of sound or light) one microsecond apart, they come back to her nine microseconds apart.)

But where it gets interesting is when we consider what Bob hears. For Alice, it's all the same if she's sending out light pulses or sound pulses. But for Bob it makes a difference, and there are big imlications that flow from that difference. That's what we'll talk about when I return.