No, the elevator has counterweights at both ends which each pull on the tether more than the object being moved due to gravity and centripetal force. The objec

No, the elevator has counterweights at both ends which each pull on the tether more than the object being moved due to gravity and centripetal force. The object on earth pulls more though which keeps it tethered. The elevator that is moving than exerts extra force to propel itself up but not more than what would drag the top counterweight down, thus, the rope still stays fully extended. Think of it like holding a rope tight between your arms and putting a robot that moves on it but ends up not exerting enough force to cause it to slack. Your arms (earth/space teather and the in space counterweight) and the object being moved would keep the same overall motion between all 3 objects (your body would just turn). The space elevator is similar except earth is way more massive than the object being moved so nothing really changes. Also, due to gravity, you have to expend extra energy to move from the bigger object (earth) to a smaller object (space elevator counterweight), aka, the minimum energy cost of moving things out of gravity wells.