Modern Physics 2007
Lorentz transformation problems

rules

1) no unlabeled numbers... numbers are always preceded by an equal sign which in turn is preceded by a mnemonic label

2) the goal on this problem is to use as many of the lorentz transformation equations as possible, although each and every time interval could be found by a time dilation problem (perhaps you should use the latter for checks)

3) Label (numerically) the events and do not re-use a number for multiple events; I believe that there are 4 events in this problem, e.g.,

(1) rocket passes earth

(2) rocket passes space station

etc

4) each D quantity must have double subscripts that are related to the beginning and ending events involved)

5) I cannot envision a correct solution to the problem without the use of time-sequence pictures (as we drew in the case of the Tonya and Reid problems).

the rocket & earth problem

In this problem, the rocket will be the BAR frame and the earth will be the NON-BAR frame.

Time intervals should be denoted, as usual, by Dt; coordinate differences (which are NOT necessarily lengths) should be denoted by Dx.� Lengths (which may or may not be coordinate differences) should be denoted by L.

At noon, a rocket passes earth with v/c = 0.8.� Observers on both earth and the rocket agree that this event is at noon in each frame.

a)� At 12:30 pm (rocket time), the rocket passes a space station that is fixed relative to earth (and therefore whose clocks read earth time).� What time is it at the station when the rocket passes?

b)� How far from the earth is the station in light minutes?� (In this and all other parts, give the answer both �according to the rocket and to the earth!)

c)� At 12:30 pm (rocket time), the rocket sends a radio message back to earth.� When does the earth receive the message?� (both earth time and rocket time)

d)� The earth replies immediately upon receipt of the message.� When does the rocket receive the message?� (both earth time and rocket time)

(Hint: if any of your times, rocket or earth, don�t end precisely in a �0� to the nearest minute, you aren�t on the right track.)

the barn and the pole

Suppose a barn (of length 50 c-sec) and a pole (of length 100 c-sec) are oriented in the same direction.� The bar and pole are put in relative motion at a speed such that the relativistic factor g is exactly 4.� Suppose that the observer at the center of the barn crosses paths with the observer at the center of the pole at time� = 0.� Since the two observers are in the same place they can agree on the time and synchronize their watches to that time.� The barn observer, in anticipation of this path-crossing, has sent a signal to the equidistant razor-like barn doors (located at opposite ends of the barn) to close at precisely time = 0; the doors then immediately (well, almost) open again.� Woe unto the pole that happens to be in the way of a closing barn door.

Let�s all agree that the according to the barn, the pole is moving in the positive direction.

Use (x-bar, t-bar) for the pole; (x,t) for the barn.�

In this problem, instead of using Dx or Dt, use x and t (for actual positions and times, assuming both frames� clocks start at� t = t-bar = 0)

Resolve the apparent barn/pole paradox according to both observers.� Does the pole get cut in three (or two)?� Both observers of course must agree on whether the pole gets cut into pieces or not?� (Why?)

I would suggest that you determine positions and times for all important locations (i.e., doors and pole ends) and events (closings of the doors) in each frame using the Lorentz transformations.

(Hint: how many knowns -- times and locations -- did I give you in this problem?� six, or eight, right?)

Your solution should have at least 3 large. well-labeled diagrams (with positions or distances determined in your calculations).� Try to draw these to scale as best as you can.� It would be best if the diagrams were all arranged sequentially on one piece of paper with your calculations elsewhere.