General relativity- weak field limit and proper time

[hai2410]

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

$$1 +\dfrac{GM}{Rc^2} - \dfrac{3GM}{2rc^2}$$

Homework Equations

$$d\tau = (1+\dfrac{2\phi}{c^2})^{0.5} dt$$ 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

$$d\tau_{C_0} = (1- \dfrac{2GM}{Rc^2})^{0.5} dt$$
and
$$d\tau_{C} = (1-\dfrac{2GM}{rc^2})^{0.5}$$

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


$$1 +\dfrac{GM}{Rc^2} - \dfrac{GM}{rc^2}$$... not quite right.



HOW DO YOU USE LATEX ON HERE? haha...

 

[dextercioby]

replace [itex] with [itex] for inline text and [tex] for formulas on separate lines.

 

[George Jones]

$$d\tau_{C} = (1-\dfrac{2GM}{rc^2})^{0.5} dt$$

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?