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Loop cable for transportation between Moon and Earth

Imagine having an elevator which just hangs in the air in between of Moon and Earth (not attached to either of them). One side pulled by gravitation of the Moon, one side pulled by gravitation of the Earth (gravitational forces of same size, but opposite direction: F1 = -F2). Just a loop cable and motor hanging on the Moon's side (also just hanging in the air with cable). Sure, there is a lot of details that would need to be worked out. But just imagine every night seeing an elevator hanging in the air transporting goods between Moon and Earth. Would that not be trippy?