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Homework Help: General relativity- weak field limit and proper time

  1. Nov 14, 2011 #1
    1. The problem statement, all variables and given/known data

    A satellite is in circular polar orbit radius r around Earth (radius R, mass M). Clocks C on satellite and C0 on south pole of earth. Show the ratio of the rate of C to C0 is approximately

    [tex] 1 +\dfrac{GM}{Rc^2} - \dfrac{3GM}{2rc^2} [/tex]

    2. Relevant equations

    [tex] d\tau = (1+\dfrac{2\phi}{c^2})^{0.5} dt [/tex] where tau is proper time, and t is coordinate time of a stationary observer near a massive object, and phi is the scalar gravitational potential at that point.

    3. The attempt at a solution

    want to compare rates of measurement of proper time?

    can easily work out both gravitational potentials, and hence get

    [tex] d\tau_{C_0} = (1- \dfrac{2GM}{Rc^2})^{0.5} dt [/tex]
    [tex] d\tau_{C} = (1-\dfrac{2GM}{rc^2})^{0.5} [/tex]

    then I worked out [itex] \dfrac{d\tau_{C}}{d\tau_{C_0}} [/itex], using binomial expansion on both and got:

    [tex] 1 +\dfrac{GM}{Rc^2} - \dfrac{GM}{rc^2} [/tex]... not quite right.

    Last edited: Nov 14, 2011
  2. jcsd
  3. Nov 14, 2011 #2


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    replace [itex] with [itex] for inline text and [tex] for formulas on separate lines.
  4. Nov 15, 2011 #3

    George Jones

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    This is correct for a clock that is hovering, not for a clock that is orbiting. What is Newronian orbital speed? How can you use this?
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