Special Relativity, Spring '10

Homework 6, covers Mermin Chapter 7 (and everything previous)

1. A question arose after class about my way of formulating the "leading clocks lag" rule, and what Mermin presents in Chapter 5. We should make sure that is straightened out with perfect clarity. Recall the setup where this rule was derived: Alice synchronizes two clocks (at the front and back of her train, respectively) by arranging for a light in the middle to flash at some point, and then the two clocks are arranged to both start reading "zero" (say) and to start ticking forward normally from there, when the light flash from the center arrives at them. Then view this whole process from the Bob/track frame: now the flast headed toward the rear of the train gets there before the other flash reaches the front of the train, and so by the time the front clock sets itself to read "zero", the back clock has already been set to "zero" and has been ticking away the seconds for some reasonable period of time. So, it's clear that the "leading clock lags": it reads "zero" when the trailing clock reads something greater than zero. But what, exactly does the trailing clock read at the moment (in Bob's frame) when the light pulse finally reaches the front clock? Find some way of answering this -- either "from scratch", or by deriving it from what Mermin does derive (which is that T = Dv/c^2... but of course you have to remember exactly what T and D mean!).

2. Alice and Bob get bored with messing around near the traintracks and decide to have a new kind of adventure. Alice decides to fly on a spaceship to alpha centauri, which is (let's say) 6 light years (i.e., 6 c-yrs) away. Bob stays on earth. Alice's ship flies at a speed of 3/5 c with respect to the earth and alpha centauri (which are at rest with respect to each other). Every year, starting precisely one year after her departure (according to her on-board clocks), Alice sends a little flash of light back toward Bob. (Maybe it's a radio transmission telling him how she's doing... all that matters for our purposes is that it travels at the speed of light.) That's the setup; now the questions:

(a) Draw a space-time diagram showing world lines for Alice, Bob, alpha centauri, and all of the light flashes that Alice sends home.

(b) In Bob's frame, how long after Alice's departure does she actually send the first flash back toward home?

(c) When does Bob actually receive that flash?

(d) How much time elapses between receipt of subsequent flashes?

(e) Assuming Alice stops sending the flashes after she arrives at alpha centauri, how many total flashes does she send?

(f) When (according to his own clocks) does Bob receive the last flash?

(g) What is the story that Alice will tell about why she only sent the number of flashes you said in part (e)?

3. This one doesn't have too much to do with Chapter 7, but seems like a good "review exercise" to work through. (It's basically the same as #2 from last week's homework, but with one small modification.) So back to Alice and Bob on the train and tracks respectively. The length of Alice's train is L, and there are previously-synchronized clocks at the front and back. (Both of those statements are "according to Alice".) She's at the back of the train, and when the clock there reads "t=0" she throws a ball at speed u (slower than c) toward the front of the train. The clock at the front of the train is designed to record its reading when the ball hits it. Obviously, this happens after a time L/u, and so that's what the clock reads when the ball gets there. (Blah blah blah with the steel bolts and whatnot!) Now the assignment is: tell the whole story from Bob's point of view. In particular, explain in gory mathematical detail how Bob will explain that final clock reading.