Gravity probe B is designed to detect the distortion of space (both the linear compression and lateral shear of the vacuum) that the mass of the earth produces by it's presence in space and it's rotation on it's axis. The hope is that this might be seen in the anticipated deflection of very sensitive gyroscopic devices on board the probe. If my description above is correct, then what the probes designers are saying is that the vacuum is able to both impart motion to matter, and also hinder it - i.e. inertial forces. Another simpler way of saying this is that the vacuum is viscous. This also says that any mass moving through the vacuum must have an influence on the vacuum (as does the earth's rotation, if it does) - so therefore common sense tells me that if this is true I should be able to calculate the resistance to motion of the earth say, as it orbits the sun. I can calculate the force that imparts the motion, but how do I calculate the force that restricts it? To be precise - an expression of an opposite force (F = mg)as Newton said, is inadequate, I want to calculate the force in terms of the vacuum and it's retarding effect (the inertia)if it exists. Any help?
The biggest problem I have with your model of the vacuum as a physical substance (one with "viscosity') is that it is frame dependent. As far as your question goes: The earth would eventually spiral into the sun, due to the emission of gravitational radiation, except that the sun will probably turn into a red giant first and it's questionable that the earth will survive long enough. Unfortunately I don't know the detailed formula for how long this would take - the best I could find out quickly was that the rate of change of the orbital period, dP/dt, was proportional to (v/c)^5 for small orbital velocities v. http://xxx.lanl.gov/abs/gr-qc/?0402007. on page 3 of the full paper. But if the earth were in free space, with no other bodies around, it would not "slow down" at all.