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The speed of light in a Schwarzschild space-time

  1. Feb 23, 2004 #1

    hellfire

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    I have seen a derivation of the dependence of the speed of light inside a Schwarschild space-time: c depends on the radial position (r), but a light ray which moves radially has a different dependence on r as a light ray which moves tangentially. My question is whether such an effect may be measurable somehow in a local reference frame and why did not the Michelson-Morley experiement record such an effect. Sorry if this question was already answered here, but after a short search I didn’t find any clear answer.

    Thanks.
     
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  3. Feb 24, 2004 #2

    DW

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    Yes the remote observer variance in the vacuum speed of light has been observed from signals relayed from other planets, but this is a remote observer effect. The local vacuum speed of light is the invariant c and so a local experiment like their interferometer wouldn't observe it.
     
  4. Feb 24, 2004 #3

    hellfire

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    Thanks for your answer, it seams to be a trivial point, but I am afraid I still do not get this. May be you can help me. Let’s take the derivation of for a radial light ray for example.

    For light ds^2 = 0.

    In a Schwarschild space-time:
    0 = (1-2m/r)(dt)^2 – (1-2m/r)^-1 (dr)^2
    (the angular components vanish, since it moves only radially)

    therefore:
    (dr/dt)^2 = (1-2m/r)^2

    and with:
    dr/dt = c_r

    one obtains:
    c_r = 1 - 2m/r

    Where is here the step with the assumption that this is for the remote observer only and not inside a local frame?
     
  5. Feb 24, 2004 #4

    DW

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    You made it prior to your second equation when you chose to express ds^2 in terms of Schwarzschild coordinates. Those coordinates are appropriate for a remote observer's reconing.
     
  6. Feb 24, 2004 #5

    hellfire

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    I see. Are there other coordinates which are not appropiate for remote observers? Could you give me a hint or a link which explains which are the criteria to recognize that Schwarzschild coordinates are appropiate for remote observers? Regards.
     
  7. Feb 24, 2004 #6

    DW

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    Yes infinitely many, take your pick. A well know class of coordinates that are not the remote observers coordinates are Kruskal-Szekeres coordinates for example.

    Look at the limit as r goes to infinity and see that the metric approaches that of special relativity except transformed to spherical coordinates. That is what tells you that the coordinates are representative of a remote observer's appropriate choice.
     
  8. Feb 25, 2004 #7

    hellfire

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    Excellent, this was of great help. Thanks.
     
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