Posted: Mar 02, 2011 9:05 pm
iamthereforeithink wrote:Interesting stuff here. To make sure I understand the explanations offered, I had a simple follow up question:

Let's say there is a hole running from one end of the earth to the other through the center of the earth. As we know, an object dropped into such a hole (ignoring the effects of rotation), will execute simple harmonic motion with a period of 84 minutes and 23 seconds. However (per my understanding), this period for the SHM is from the reference frame of an observer on the earth's surface. Now, if the object dropped into the hole is me, and I'm wearing a watch, how will my watch record the period of the oscillation?

Presumably you're interested in the answer relative to someone's watch who stays at the surface. The max grav. dilation with respect to him will be when you're at the centre. The overall difference will be a mean over the range r=0 to r= R. You would also theoretically have to include the dilation due to motion, so you'd have the dr metric component which I ignored above. So the answer would involve integrating over the range. It would still be damn small though.