Special Relativity, Spring '10

Homework 7, covering Mermin chapter 8 (and previous stuff)

1. Alice purchases a fancy new kitchen timer and is excited to get on board her train to try it out. It works like this: she sets it to "one hour", then hits a button on the top. Then, after one hour, a bell rings. Whee! Call her hitting of the button "event A". Call the bell ringing "event B". It is pretty clear that, in Alice's own frame, the interval ("I" from page 84) between A and B is just one light-hour. Your job is to show that the interval between A and B -- as computed from scratch in Bob's frame -- is also one light-hour. For definiteness, assume that Alice's train moves at speed v = 3/5 c relative to Bob's frame, and that Bob uses coordinates such that event A happens at t=0 and x=0. So you'll just need to (a) draw a space-time diagram and (b) work out the space and time coordinates of event B and then (c) plug into the formula on page 84 to compute "I".

2. Draw a space-time diagram to capture/summarize the procedure (for measuring the interval between two events using a single clock present at one of the events) that is outlined on the bottom of page 87. Your diagram should evidently reflect one of the four possible sub-cases Mermin outlines in the paragraph that sprawls over onto page 88. (That is, just pick one of these sub-cases to illustrate with a space-time diagram -- you don't have to do all four sub-cases.) Now that you've got your space-time diagram, see if you can run the argument he outlines in this paragraph, i.e., show that indeed equation (8.12) gives the interval between the events.

3. Two weeks ago I assigned you the problem of working out exactly how Bob would account for the fact that the clock in the front of the train reads "L/c" when a photon (shot by Alice from the back of the train when the synchronized clock there read "zero") gets to it. We then worked through this one in full detail in class. Last week I assigned you the problem of doing the same thing, but where the photon (moving at speed c) is replaced by a ball (moving at speed u < c). Everybody made some kind of reasonable start on this, but nobody stuck it through to the end. So... stick it through to the end. That is, do this one "again" for this week. Here's a hint: the approach (that is, the sequence of things you should think/say/do) is identical to the one from two weeks ago; the only difference is that you'll have to use the velocity addition formula -- equation 4.2 -- for the speed of the ball in Bob's frame. And so the subsequent algebra is marginally messier than in the problem from two weeks ago. But have the courage/tenacity/faith/naivete/whatever to stick it out and see if you can get it to work.

Extra credit: In the first full paragraph of page 87, Mermin writes: "It is as if the clock is always moving through ... spacetime ... at the speed of light." What could this possibly mean? Maybe draw a spacetime diagram showing the worldline of a clock. How fast does the diagram indicate the clock to be "moving through spacetime"?

More extra credit: There's a barn of length L with doors at the front and back. Alice is stationed at the front door; Bob is stationed at the back door. They have previously synchronized their watches, and agreed that at the stroke of noon they will both slam their doors shut. Just prior to noon, though, here comes a jouster guy named Charlie, riding a horse moving at speed 3/5 c relative to the barn. Charlie carries a pole which has rest length L, and which is therefore length-contracted down to a length of 4/5 L in the barn frame. He (and his pole and the horse they rode in on) enter the still open front door of the barn just before noon, and both ends of the pole are still comfortably inside the barn when the clocks strike noon and Alice and Bob slam their doors. Shortly thereafter, of course, the front of Charlie's pole smashes into the now-closed back door of the barn. Charlie himself, tragically, suffers a similar fate mere nanoseconds later. The question is: how would Charlie (or perhaps more realistically, his friend Detlef, who uses Charlie's same reference frame but doesn't smash into a barn door at 3/5 c) account for the sequence of events? In particular, from Charlie's (Detlef's) frame, the barn is a factor of 4/5 shorter than the pole -- which of course makes it hard to understand how both ends of the pole could ever both be inside the barn. Anyway, hopefully you get the setup. Stew on it if you want; we'll talk about it in class whenever we get to it.