How Light and Sound are Different

The other day I started doing the Doppler shift as a relativity problem. It starts off looking a lot like the Doppler shift for ordinary sound in air. I had Alice sending a series of sound/light pulses to a train travelling at 4/5 the speed of sound/light; and I had Bob on the back of the train with a mirror reflecting the sound/light back to Alice. It's not too hard to calculate that if Alices pulses are 1 microsecond apart, then when they come back to Alice they are 9 microseconds apart, for a Doppler shift of 900%. You can see it easily from the picture below. The calculation is correct for light, and it's correct for sound:

What gets funny if we asked how much Doppler shift Bob sees. It's not hard to do the calucation for sound in air. We just drop vertical lines from Bob's pulse detection points down to the time axis. It's not hard to see that Bob detects the pulses 5 microseconds apart, for a Doppler shift of 500%:

Here's the thing: Bob is sending out pulses 5 microseconds apart, and Alice is measuring them 9 microseconds apart. So Alice's Doppler Shift is 180%. That's not the same as Bob's. So by analyzing who has a greater doppler shift, they can figure out that Alice is stationary and Bob must be moving.

And that's not how relativity works. In relativity we're not allowed to distinguish the stationary from the moving observer, so both Alice and Bob have to measure the same doppler shift. It's almost impossible to see how they can do that...unless we realize that time is moving slower for Bob than for Alice.

The total doppler for the reflected pulses is 900%. And the only way to make it the same for both observers is for them both to see 300%. Bob sees the pulses 3 microseconds apart, and Alice sees them 9 microseconds apart.

We did the caluclation for the special case of the train going at 4/5 the speed of light. But if we let the speed vary from 0 to 100%, and trace the contour defined by the equal-interval ticks, we will find that they are the hyperbolas defined by the equation x^2 - t^2 = constant: