Relativity Lite

Gravity Lite | 51 Potential energy has more utility than just helping us understand how marbles roll in physical bowls. It is an alternative way to describe a Newtonian force that causes motion. The negative of the manner in which the potential energy changes with distance is the defi- nition of such a force. For instance, the potential energy curve for a pair of magnets whose north poles face each other looks like the left half of a U , with the horizontal position within the left half representing the closeness of the two magnets. The change in potential energy becomes more marked as one moves leftward on the left half of the U , so, by defini- tion, the magnetic force becomes more and more repulsive as you try to bring the two north poles together. It is repulsive because the change in PE is toward more positive values, and by definition, we take the negative of this positive change to get the direction of the force. Does the force encoded in that abstract description of the PE match your visceral experience of magnets? The dotted red curve in figure 7 is a diagram of the Newtonian potential energy curve for the radius of Mercury’s orbit. One can imagine Mercury as a marble rolling back and forth in the depression. We have moved, here, from the case of a real marble rolling in a real bowl, under the influence of a real downward “Newtonian gravitational force” infigure 6, to an abstract marble rolling in an idea , called a potential energy curve in figure 7. In this abstract realm, it is the abstract principle that “systems seek to move to their lowest en- ergy state” that “pulls” the marble downward; not a real downward “gravitational force” (to the extent that we can say Newtonian approximations to Einsteinian spacetime constitute a “real” gravitational force). But what is the nature of the rim on either side of that depression? Let me ask you a couple of questions. Mercury is moving so fast that it completes one orbit in 88 days. What is it that keeps Mercury from flying off into space? In Newtonian terms, it is the tug of gravity pulling it always inward: its tendency to fly off is continually checked by the force of gravity. The energy associated with this force depends on distance as 1/R . In Einsteinian terms, the only spacetime available for Mercury to travel through is a funnel so that it is constrained to move around and around the funnel. It would need more kinetic energy to rise higher up the funnel and, hence, farther from the Sun. What is it that keeps Mercury from falling into the Sun? Its inertia; its tendency to keep moving in a straight line unless acted upon by a force. This is the “pseudo-­ force” that “slams” you into the door of a car turning a corner too sharply. We call this a pseudo-force because it is really the car’s door slamming into you as it turns while you try to continue moving in a straight line. In Newtonian terms, the outward pseudo-­ force of inertia must be perfectly balanced by the inward gravitational force for a planet to travel in a circular orbit. If you tie a rope to a brick and swing it around your head, you must exert an inward tug on a rope to counter the outward pseudo-force of the in- ertia of the brick—its desire to keep moving in a straight line toward your neighbor’s window.