In high school, we said that an object on the ground has zero gravitational energy, while an object above the ground has some. But if a hole opened up in the floor, the object on the ground would fall – so it must have SOME potential energy, right? At the center of the earth, would you have no gravitational potential energy? If not, why – doesn’t the sun still pull on you?
You’ve brought up an interesting subject. Many quantities in physics are only well defined relative to some reference point. For example, your velocity is only defined relative to some reference frame; typically the earth’s rest frame. Viewed from a different reference frame, your velocity will be different. The same holds for gravitational potential energy. When you choose to define the object’s gravitational potential energy on the floor as zero, you are setting the scale with which to work. For altitudes above the floor, the object’s gravitational potential energy is positive, but for altitudes below the floor, that energy is negative. As the ball falls into the hole, its gravitational energy becomes more and more negative and its kinetic energy increases. To avoid working with these annoying negative potential energies, you should choose to set the gravitational potential energy to zero at the lowest point you’ll ever have to deal with; for example, the center of the earth. But the center of the earth isn’t really the limit of gravitational potential energy. The object could release even more gravitational potential energy by falling into the center of the sun. It could release still more by falling into the center of a giant star. Fortunately, there is a genuine limit. If you were to lower the object slowly into a black hole, the object would release absolutely all of its gravitational potential energy. In fact, it would release energy equal to its mass times the speed of light squared (the famous E=mc2 equation of Einstein). The object would actually cease to exist, having been converted entirely into energy (the work done on you as you lower the object, presumably at the end of a very sturdy rope). This effect sets a real value of zero for the gravitational potential energy of an object: the point at which the object ceases to exist altogether. Final note: if you drop something into a black hole, it doesn’t vanish the same way, because its gravitational potential energy becomes kinetic energy as it enters the black hole. The black hole retains that energy and grows slightly larger as a result. When you lower the object on a rope, you retain its energy and it doesn’t remain with the black hole. The black hole doesn’t change as it “consumes” the object.