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Decelerating at 1g into a black hole

  1. Nov 19, 2008 #1
    In Kip Thorne's book "Timewarps" he describes a hypothetical spaceship decelerating almost to the event horizon of a supermassive blackhole, and says that the manoevre takes 13 years. I assume that the motion is along a geodesic.

    In terms of the Schwarzschild metric, how would this motion be described and how would the elapsed time be calculated?
     
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  3. Nov 19, 2008 #2

    Fredrik

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    It's definitely not along a geodesic. That means free fall.
     
  4. Nov 19, 2008 #3
    as for the proper elapsed time, just calculate the integral of the norm for the tangent vector along the path taken.
     
  5. Nov 19, 2008 #4
    How exactly?

    The metric is [tex]d \tau ^2 = \left(1- \frac{2M}{r}\right)dt^2- \frac{dr^2}{\left(1- \frac{2m}{r}\right)} -r^2 d \theta^2 [/tex]

    Assuming [tex]d \theta^2 = 0[/tex], the remainder can be divided through by [tex]d \tau ^2[/tex] to give

    [tex]1 = \left(1- \frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^2 - \frac{1}{\left(1- \frac{2m}{r}\right)}\left(\frac{dr}{d \tau}\right)^2[/tex]

    The fraction [tex]\left(\frac{dr}{d \tau}\right)[/tex] looks something like a velocity and not an acceleration. Is a second derivative is added to the equation somewhere?

    What are the constants of the motion? Is total energy conserved?
     
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