General relativity- weak field limit and proper time

In summary, the conversation discusses the ratio of the rate of proper time measurements between a clock on a satellite in circular polar orbit around Earth and a clock on the south pole of Earth. The conversation also mentions the use of gravitational potentials and the formula for proper time. The conversation concludes with a question about using Newtonian orbital speed to find the correct formula for the ratio of time measurements.
  • #1
hai2410
2
0

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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  • #2
replace [itex] with [itex] for inline text and [tex] for formulas on separate lines.
 
  • #3
hai2410 said:
[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?
 

1. What is the weak field limit in general relativity?

The weak field limit is a simplified approximation used in general relativity to describe the behavior of objects in regions of weak gravitational fields. It assumes that the gravitational field is weak enough that it does not significantly affect the geometry of spacetime.

2. How is the weak field limit different from the strong field limit?

In the weak field limit, the gravitational field is weak and the curvature of spacetime is small, allowing for easier mathematical calculations. In the strong field limit, the gravitational field is strong and the curvature of spacetime is significant, requiring more complex mathematical models.

3. What is proper time in general relativity?

Proper time is the time measured by an observer who is at rest in a specific frame of reference. It takes into account the effects of time dilation caused by the curvature of spacetime, and is the time measured by a clock that is moving with the observer.

4. How does proper time change in the weak field limit?

In the weak field limit, the effects of time dilation are minimal, so proper time will not differ significantly from the time measured by a clock located far from any gravitational field. However, as the gravitational field becomes stronger, the effects of time dilation become more pronounced and proper time will differ more from the time measured by a distant clock.

5. Can the weak field limit be applied to all objects in the universe?

No, the weak field limit is only applicable to objects in regions of weak gravitational fields. In regions of strong gravitational fields, such as near black holes, the strong field limit must be used instead to accurately describe the behavior of objects in those regions.

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