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First Year Relativity Problem from Exam

  1. May 1, 2009 #1
    1. The problem statement, all variables and given/known data
    The Earth and the Planet of the Apes are in a reference frame where they are stationary relative to one another, and are measured to be 2 light years apart. The observers on Earth send a TV signal to the Planet of the Apes with a picture of a banana. 0.6 years later, the observers on Earth receive a TV signal from the Planet of the Apes saying "Give us the banana!".

    Question 1: What is the minimum speed that the TV signals must travel to fulfill the above conditions? (This was not the original wording but feel free to ignore this question, I don't quite remember it anyway)
    Question 2: What is the spacetime interval between the two events, as measured by an observer on Earth?
    Question 3: What is the spacetime interval between the two events as measured by an observer on a rocket traveling from Earth to PoA who finds that the two events (Earth and PoA sending their respective TV signals) are simultaneous?
    Question 4: According to the observer, what is the distance between the two planets?
    Question 5: At what speed is the observer moving relative to Earth and the PoA?

    3. The attempt at a solution

    Event 1: Earth sends TV signal to PoA
    Event 2: PoA sends TV signal to Earth

    Most of the people in my class (including me) assumed that one event caused the other, but is that possible if an observer in a reference frame measures them to be simultaneous?

    If I take Event 2 to happen at t = -1.4 years and Event 1 to happen at t = 0 years (for the signal from PoA to reach the Earth at t = 0.6 years), the spacetime interval turns out to be a negative number. Answer to Question 4 turns out to be 1.4 x 10^16 meters, and the answer to Question 5 turns out to be 0.63c

    Was I on the right track?
     
  2. jcsd
  3. May 1, 2009 #2

    Doc Al

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    Staff: Mentor

    How can these events be causally related? The signal from 1 won't even reach PoA until long after the the signal from 2 has arrived at earth. (But you're correct that if are simultaneous in some frame, then they can't be causally related.)

    Show how you arrived at these results.

    (Planet of the Apes is the earth! :eek:)
     
  4. May 1, 2009 #3
    Most people assumed a causal relation because Question 1 implied that a signal went to the PoA and back within a time interval of 0.6 years... as I said earlier I don't remember the exact wording of this question so I'll just let it go.

    Operating under the Earth's reference frame here
    -Distance between Earth and PoA is 2 light years, so for the signal from PoA asking for the banana to reach the Earth at t = 0.6 years, PoA must have emitted their TV signal at t = -1.4 years. Earth emitted their signal at t = 0 years

    -Spacetime interval squared = c^2 (deltaT)^2 - (deltaX)^2
    -Converted light years to metres and years to seconds
    -Setting deltaT (1.4 years or 44150400 seconds) and deltaX (1.89 x 10^16 m) yields a result of s^2 = -1.83 x 10^32 metres squared. (Answer for question 2 and 3)

    For the observer to measure them as simultaneous, I used the invariant spacetime interveral by setting deltaT = 0 and solving for deltaX, which came out to be 1.35 x 10^16 metres (Answer for question 4).

    To find the speed of the observer flying on the rocket, I used the formula for length contraction

    1.35 x 10^16 = sqrt(1-v^2/c^2)1.89 x 10^16

    With a little bit of algebra, v turns out to be 0.70c (answer for question 5)

    Um... I guess the minor difference in numbers came about when I was fiddling with the number of digits I was handling with - but is my method correct?

    NB: I didn't write this for my exam on Wednesday morning, but it doesn't even matter to me anymore - I just want to know how to do it properly.
     
  5. May 2, 2009 #4

    Doc Al

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    Staff: Mentor

    Perfectly correct.
     
  6. May 2, 2009 #5
    Alright, thanks a lot!
     
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