Let's start by remembering that the effects are pretty small, so that expression you gave can be accurately approximated as 1-GM/rc2
. (Here G is the universal gravitational constant, M is the mass of the earth, c is the speed of light, and r r is the distance from the center of the earth. In other words, the fractional slow-down due to gravity near the earth, compared to far space, is -GM/rc2
. This expression only applies above the surface.
To calculate the effect in a mine, you need some model for the density of the matter in the earth. Let's pick a really easy one, and say that the earth has uniform density. (Geologists could straighten us out on that.) Then integrating the field as one goes down gives a fractional slowdown of:
where R is the radius of the earth.
Checks: this agrees with the other expression at the surface, where r=R. It also has the same slope there, which agrees with the gravitational field being a continuous function. So I'm confident it's ok.
(published on 11/18/2011)