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Critical Radius of Special and General Relativity Effects on a Clock Orbiting Earth

  1. Dec 11, 2012 #1
    Special Relativity states that a clock orbiting Earth will tick slower than one on the surface of Earth. General Relativity states the opposite - a clock will tick faster the further it is from a mass. Is there then a critical radius where these two effects cancel out? Ignore the rotation of Earth for this problem. State the radius in terms of rc=krE, where k is the number to be determined and where rE is the radius of the Earth.

    -This was the extra credit problem on my Relativity final which I really wish I could have solved. Anybody want to take a shot at it?
     
  2. jcsd
  3. Dec 11, 2012 #2

    PeterDonis

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    Re: Critical Radius of Special and General Relativity Effects on a Clock Orbiting Ear

    Yes. If you want to take a second stab at it, consider the following:

    An object on the surface of the Earth, ignoring the Earth's rotation, will have a "time dilation factor" of

    [tex]\sqrt{1 - \frac{2GM}{c^2 r_E}}[/tex]

    An object in orbit about the Earth at some radius r will have a "time dilation factor" of

    [tex]\sqrt{1 - \frac{2GM}{c^2 r} - \frac{v^2}{c^2}}[/tex]

    where v is the orbital velocity at radius r. (Can you see how to get this result?) These two facts are enough to get the answer you seek.
     
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