Solving the Critical Radius for Relativity: Find krE!

[n1caboose]

Special Relativity states that a clock orbiting Earth will tick slower than one on the surface of Earth. General Relativity states the opposite - a clock will tick faster the further it is from a mass. Is there then a critical radius where these two effects cancel out? Ignore the rotation of Earth for this problem. State the radius in terms of r_c=kr_E, where k is the number to be determined and where r_E is the radius of the Earth.

-This was the extra credit problem on my Relativity final which I really wish I could have solved. Anybody want to take a shot at it?

 

[PeterDonis]

Special Relativity states that a clock orbiting Earth will tick slower than one on the surface of Earth. General Relativity states the opposite - a clock will tick faster the further it is from a mass. Is there then a critical radius where these two effects cancel out?

Yes. If you want to take a second stab at it, consider the following:

An object on the surface of the Earth, ignoring the Earth's rotation, will have a "time dilation factor" of

$$\sqrt{1 - \frac{2GM}{c^2 r_E}}$$

An object in orbit about the Earth at some radius r will have a "time dilation factor" of

$$\sqrt{1 - \frac{2GM}{c^2 r} - \frac{v^2}{c^2}}$$

where v is the orbital velocity at radius r. (Can you see how to get this result?) These two facts are enough to get the answer you seek.