Special Relativity Clock Challenge

[RelativeQuanta]

Here's the question from my text:
"Alice and Bob are movin in opposite directions around a circular ring of radius R, which is at rest in an inertial frame. Mobh move with constant speeds V as measured in that frame. Each carries a clock, which they synchronize to zero time at a moment when they are at the same position on the ring. Bob predicts that when next they meet, Alice's clock will read less than his because of the time dilation arising because she has been moving with respect to him. Alice predicts that Bob's clock will read less witht he same reasoning. They can't both be right. What's wrong with their arguments? What will the clocks really read?"

Here's how I answered it:
The problem with their arguments is that their referances frames are not intertial! They are accelerating since they are moving on a circular path. So, the Lorentz transformations introduced in the chapter won't hold. Since I haven't yet learned how to account for acceleration mathmaticly, I drew their respective world lines from the referance frame of the ring. Knowing that they are accelerating but moving with constant angular speed and traveling the same distance, (and because I suck at drawing curvy lines) I drew it as a \theta vs ct graph with Bob starting at 0 and Alice at 2pi with opposite angular velocities. My result was that when they meet, both their clocks will have the same reading since their world lines are the same length.

Is my logic, and answer, correct? I ask since my book had it marked as a challenge problem and I answered it in about 5 minutes.

 

[JesseM]

Yup, you got it. You don't even have to calculate their paths to figure out the time dilation--since time dilation is based only on speed in whichever inertial frame you're looking at, in the rest frame of the track both clocks will always tick t \sqrt{1 - V^2/c^2} in any time interval t. In general, if you know an object's speed as a function of time v(t) in an inertial frame, then to figure out how much a clock carried by that object will advance between two times t_0 and t_1 in that frame's coordinates, just integrate \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt