Posted: Mar 03, 2011 8:28 am
CdesignProponentsist wrote:Okay, getting closer

How is gravitational acceleration come into play then?

And what is the difference in gravitational acceleration between an object falling and an object standing on the surface?

Need to be precise about terminology here. In GR people don't usually talk about "gravitational acceleration", but rather have two concepts:

Coordinate Acceleration
This would be the rate of change of a space coordinate with respect to the time coordinate for an object in free fall. So for example in the Schwarzschild case, where the coordinates are (t, r, θ, φ), the radial coordinate acceleration component of a radially free-falling object would be dr/dt. You would compute this by working out the geodesic equation.

Proper Acceleration

To get the proper acceleration experienced by an object, you first define an inertial observer who's going to be the reference for the proper acceleration. This reference observer should be initially stationary relative to the object. So for example, if I have an object sitting at a point at distance r from the centre, I take an observer, sit him right by the object, then let the observer drop in free fall. The proper acceleration is the relative acceleration (derivative of four-velocity) between the two. In Newtonian mechanics the proper acceleration would be GM/r2, but in the Schwarzschild solution in GR, there will be a small correction to this involving some more G's, M's and an few c's