Theoretical geosync means zero velocity in any direction relative to ground. Zero up, down, forward, back, left, or right. You're stuck above a single point of dirt/water, over the equator, at a fixed altitude.
The only way to "fall" (lower your altitude) from here would be via some sort of acceleration force towards ground (like a burn). So yes, your speed would have to increase.
Well fine. I meant that orbital speed (the "along track part") would increase if you lowered the altitude of a geosync object with rockets, string, or any other force, I think.
The only way to "fall" (lower your altitude) from here would be via some sort of acceleration force towards ground (like a burn). So yes, your speed would have to increase.