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Accelerating away from earth, time such that a light beam won't catch up with you

  1. Nov 18, 2012 #1
    1. The problem statement, all variables and given/known data
    You start at t=0 at rest on earth and accelerate with uniform acceleration a away form earth.
    Find a point in time [itex]t_0[/itex] such that when a beam emitted from earth at [itex]t>t_0 [/itex]won't catch up.


    2. Relevant equations
    [itex]x(t)=c^2/a(\sqrt{1+\frac{a^2}{c^2}t^2}-1)[/itex]


    3. The attempt at a solution
    I think that light travel with velocity c. So if the beam is emitted at [itex]t=t_1[/itex] then at time t, the beam have traveled [itex] c(t-t_1) [/itex]. So I try to find the solution for [itex]x(t)=c(t-t_1)[/itex], and I end up with the following expression for [itex]t[/itex]:
    [itex]t=\frac{a}{2c}\frac{t_1(2-a/c t_1)}{(a/c - a^2/c^2 t_1)}[/itex]

    According to this the time would be negatic in the intervall [itex]t_1=c/a[/itex] and [itex]t_1=2c/a[/itex] So I think in this intevall the beam won't be able to catch up, but after [itex]t_1=2c/a[/itex] the time becomes positive again, which I don't know how to interpret.
    Am I approaching this problem the wrong way?
     
  2. jcsd
  3. Nov 18, 2012 #2
    Think about it this way: will you ever accelerate to a speed greater than that of light? If not, how can you possibly outrun light?
     
  4. Nov 18, 2012 #3
    Well it is problem 3.9 in D'inverno Introducing Einsteins relativity. I agree that it seem impossible but the problem statement is that if you get a large enough headstart the light won't catch up.
     
  5. Nov 18, 2012 #4
    What assumptions are you supposed to make?
     
  6. Nov 18, 2012 #5

    SteamKing

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    negatic?
     
  7. Nov 19, 2012 #6

    vela

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    You should be able to show that your world line is a hyperbola. Find its asymptotes.
     
  8. Nov 19, 2012 #7

    TSny

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    Note t approaches ∞ as the denominator on the right approaches 0.
     
  9. Nov 26, 2012 #8
    Let's say you keep uniform acceleration a, relative to the stationary observer. After c/a time you will be moving at the speed of light. To keep uniform acceleration you need infinite amount of energy. I think the answer is c/a, just the problem is that you can't keep uniform acceleration.
     
  10. Nov 26, 2012 #9
  11. Nov 28, 2012 #10

    vela

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    You've misinterpreted the problem. The acceleration is uniform relative to the moving observer. As you noted, you can't have a uniform acceleration relative to the stationary observer indefinitely.
     
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