There are two points that need understanding, one is that following a geodesic is going in a straight line, even when space is curved. The other point is the need to understand why particles curve space.
The first point can be understood by an example of travelers circling the globe. If four people get into an airplane and travel due north, south, east and west, they will cross paths. On a flat paper map of a city this seems ridiculous, that’s like stating that if one travels from the center of a city and go north, south, east, and west; that they all meet each other when they hit city limits. But it seems perfectly reasonable for world traveling distances that four people would cross paths. That is because they know they are traveling around a circle, even though they don't see any curvature. There was no mysterious "transporter" or force pulling them together. At a city size scale though people would think something magical is needed for everybody to travel away from each other and cross paths in a few miles. This is the basic idea behind Einstein's Principle of General Relativity, gravity is curves space time. In this case we don't perceive the curvature because it is very gradual curvature, like traveling around the earth. The first example used in Misner, Thorne, Wheeler's book "Gravitation" is of an ant travelling about an apple. In this case we can perceive the curvature of the apple and see that the ant travels in a straight line like travelers about the earth. A similar example will be made about particle forces.
Particles move together not because there is something pulling them together, they all are traveling in the straightest line possible, its just that space is curved. But this time we don't see the curvature because the curvature is so tiny that we can't see it. What would it look like if it were magnified? The dimension is a tiny curled dimension, with a very long axis, the length of the universe, and a short axis. If it were magnified it would first look like a fine line, then as it was magnified more, it would seem like a long spaghetti noodle, larger yet it would seem like a very long balloon like circus clowns use to make animal shapes. This long balloon stretches off in opposite directions to our infinity.
What is a particle's path along this? Take the ant off the apple and place it on the balloon. It can travel the long way or the short direction, or a combination of both. If it travels the long way it travels straight. If it travels the short way it goes around in a circle, going nowhere. If it travels in a direction off axis then it moves along a spiral, going somewhere but not as fast as it could if it just traveled along the long axis alone.
These three travel paths can be associated with three different particle states. The ant travelling purely along the long axis is a photon, a particle of light. Its gets from point A to point B in the shortest time. The ant going the short way is a particle with mass. It is moving but we can't see it because the path it travels is so small. The only feature we see is that it doesn't seem to move away from point A at all. The ant traveling slightly off axis slowly spirals away from point A. It is getting closer to B but would get there more quickly if it could turn more. But no matter how much you turn it, if you have any travel along the short axis at all you can't get to point B as fast as the ant going along the long axis alone. The spiral traveling ant is a particle with mass that has kinetic energy. It still retains it property of mass but can never get to point B as fast as the photon.
One of the results of having a particle traveling about the short axis is that it warps the balloon at its location. Just as a clown will twist a balloon to put a kink into it, the particle kinks space. The diameter of the balloon a distance away is unchanged, but as one gets closer to the kink, the diameter gets smaller. At the location of the kink it looks like its shrunk completely. The ant traveling along the long axis goes through this point without deviation, but it might make the distance from point A to point B seem somewhat longer. The ant traveling along the short axis a distance away also seems unaffected. If it were a little closer though the side of the ant closer to the kink would have be a slightly smaller diameter than the side away from the kink. This difference causes the ant to point in a slightly different direction after circling once. The side closer to the kink would be ahead of the side away from the kink. The ant would be pointing slightly away from the kink. If it circled around again it would have actual traveled slightly away from the kink and been and turned away slightly more. The closer it was to the kink when it started out the more it was turned away the first circuit. The further away the smaller the change in direction. The final result of the ants circling is that the ant is point away from the kink and traveling in a spiral. This is similar to what would happen if an electron were placed in the vicinity of another electron. It would start out at rest and would move away from the charge. The closer it starts to the charge the faster it moves away at first. Once the electron has moved sufficiently away from the charged particle it seems to have a fixed velocity.
The geodesic travelling ants help show how particles affect one another. The reason for a particle to warp space needs to be shown now. This is accomplished with a principle from quantum theory, p*x=h. To simplify the algebra we'll assign h=1. P is momentum and x is a distance. The ant has momentum, p. The ant can only travel along a direction that is x long. One ant, with p=1, travels around the short direction on the balloon with dimension that is exactly x long. A second ant is placed nearby travelling in the same direction with p=1. It is not clear if this matters. For now we'll make a rule that p=1/d where d is the distance between the ant locations. Why? Why not? We'll see what happens. If the ants are far apart the ant has its own p=1 plus a small amount of p=1/d from the other ant. The other ant has the same amount of extra p also. This extra p no longer fits into the short direction exactly. The x would need to shorten up by a small amount. If momentum, p=1+1/d; then dimension, x=1/(1+1/d) = d/(d+1). As the ants get closer their effect is to shrink x down. Why? The rule p*x=nh/2pi or p=(nh/2pi)/x is the basis for quantum theory, the reason it is this value is not clear or why these two parameters should be interrelated, but is does imply p=k/x
Does p go to infinity when d becomes 0? No, the limit for two particles is when p becomes 2. But when particles have kinetic energy their individual total p is greater than 1. So the total p is greater than 2. For the particles with no kinetic energy the maximum is 2. This corresponds to a distance of d=1, which means x gets shrunk in half. If additional particles, or ants moving along the same direction, and are placed as near as possible to the first two then x keeps shrinking. This shrinking dimension seems strange here. But, it occurs in another situation in physics that has been around a lot longer. It is referenced as Lorentz contraction in the Theory of Special Relativity.
One conclusion that can be made from this is that many physics properties are a result of quantum theory. The 1/r potential which is derived here is a result of p*x=h. Therefore, the inverse square law for force is a result of quantum effects.
Last Updated on November 2, 1997 by Bob Rutkiewicz