General relativity- weak field limit and proper time

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Homework Statement



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]

Homework 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.

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]
and
[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.



HOW DO YOU USE LATEX ON HERE? haha...
 
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Answers and Replies

  • #2
dextercioby
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replace [itex] with [itex] for inline text and [tex] for formulas on separate lines.
 
  • #3
George Jones
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[tex] d\tau_{C} = (1-\dfrac{2GM}{rc^2})^{0.5} dt [/tex]
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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