Geosynchronous Orbits

[Charlie G]

I was having trouble making sense of the geosynchronous orbits from a relativistic point of view. Correct me if I am wrong, but in a geosynchronous orbit the object appears to hover over a certain place on the Earth, from the point of view of the Earth, which can be said to be at rest according to the principle of relativity, the object is at rest above the Earth.
Therefore, should the general pinciple of relativity hold true, the object should begin to fall towards the Earth in a straight line relative to an observer on the Earth's surface.

This obviously doesn't happen, but I have good faith that there exist a solution somewhere in the theory, which is why I have addressed my troubles here.

Thank-you.

[Integral]

Just what is it in the Special Relativity that makes you think this is true?

[A.T.]

Correct me if I am wrong, but in a geosynchronous orbit the object appears to hover over a certain place on the Earth, from the point of view of the Earth, which can be said to be at rest according to the principle of relativity

The principle of relativity applies to inertial frames. A frame attached to "a certain place on the Earth", where the geosynchronous satellite is at rest, is an rotating frame of reference. In rotating frames of reference you have centrifugal forces:
http://en.wikipedia.org/wiki/Centrifugal_force_(rotating_reference_frame)

[HallsofIvy]

I was having trouble making sense of the geosynchronous orbits from a relativistic point of view. Correct me if I am wrong, but in a geosynchronous orbit the object appears to hover over a certain place on the Earth, from the point of view of the Earth, which can be said to be at rest according to the principle of relativity, the object is at rest above the Earth.
Yes, that is correct.

Therefore, should the general pinciple of relativity hold true, the object should begin to fall towards the Earth in a straight line relative to an observer on the Earth's surface.
How does that follow?

This obviously doesn't happen, but I have good faith that there exist a solution somewhere in the theory, which is why I have addressed my troubles here.

Thank-you.

[Charlie G]

Thanks for the replies. I suppose my troubles only lied in my confusion about relativity, I had assumed tht the general principle applied to all frames of reference, even non-inertial frames, so that even an observer in one could call himself at rest and attribute inertial forces to a gravitating body. I must read my book for a second time for I seem to have misunderstood.

[MikeyW]

I had assumed tht the general principle applied to all frames of reference, even non-inertial frames, so that even an observer in one could call himself at rest and attribute inertial forces to a gravitating body.

This sounds fine to me... I don't get why you think the satellite will fall though? It is in an orbit with no resistance, the gravitational field provides the acceleration required for circular motion, don't you think it would stay where it is?

[A.T.]

I don't get why you think the satellite will fall though? It is in an orbit with no resistance,
Not in a co-rotating frame of reference where the satellite is at rest. There you need to introduce the centrifugal force to keep it from falling. That's what the OP forgot.

[Bob S]

See articles on the relativistic corrections in the GPS positioning system. Here are two
http://en.wikipedia.org/wiki/Effects_of_relativity_on_GPS
http://www.phys.lsu.edu/mog/mog9/node9.html
I read a very comprehensive article on GPS corrections a while back, but I cannot find it now.
Bob S