How does a woman's egg move? does gravity have anything to do with the movement of the egg?

Not just gravity: When you ovulate the follicle in your ovary bursts, releasing the egg surrounded by follicle fluid. The egg moves to the open end of the tube, usually on the same side you ovulate, inside fluid in your pelvis. The tube has both muscle contractions and tiny hair-like cilia which beat in waves to move the egg to the uterus, meeting sperm on the way. Gravity has little effect - don't need to lay down.