M5. Space Homogeneity And Momentum Conservation

Imagine a single isolated hydrogen atom alone and at rest in empty space. If you could draw up a chair and observe it without influencing it, what should you expect to see? (For this discussion, I ignore quantum mechanics and the wave nature of particles, pretending that the electron and the proton may be localized at points in space and be uninfluenced by the observer. These assumptions, although false, are permissible for the present discussion.) You should see an electron in rapid motion circling about a proton, and the proton itself moving more slowly in a smaller circle. Were you to back off until the whole atom could be discerned only as a single spot, that spot, if initially motionless, would remain motionless forever. Is that circumstance significant or insignificant, important or dull? It certainly does not seem surprising. Why should the atom move? you may ask. It is isolated from the rest of the universe; no forces act upon it from outside; therefore there is nothing to set it into motion. If you leave a book on a table and come back later, you expect to find it there. Everyday experience conditions us to expect that an object on which no external forces act will not spontaneously set itself into motion. There is no more reason for the atom to begin to move than for the book to migrate across the table and fly into a corner. The trouble with this argument is that it makes use of the common sense of ordinary experience, without offering any explanation for the ordinary experience.

If we put aside “common sense” and ask what the atom might do, it is by no means obvious that it should remain at rest. In spite of the fact that no external forces are acting, strong internal forces are at work. The proton exerts a force on the electron, which constantly alters its motion; the electron, in turn, exerts a force on the proton. Both atomic constituents are experiencing force. Why should these forces not combine to set the atom as a whole into motion? Having put the question in this way, we may consider the book on the table again. It consists of countless trillions of atoms, each one exerting forces on its neighboring atoms. Through what miracle do these forces so precisely cancel that no net force acts upon the book as a whole and it remains quiescent on the table?

The classical approach to this problem is to look for a positive, or permissive, law, a law that tells what does happen. Newton first enunciated this law, which (except for some modification made necessary by the theory of relativity) has withstood the test of time to the present day. It is Newton’s third law, and it says that all forces in nature occur in equal and opposite balanced pairs. The proton’s force on the electron is exactly equal and opposite to the electron’s force on the proton. The sum of these two forces (the vector sum) is zero, so that there is no tendency for the structure as a whole to move in any direction. The balancing of forces, moreover, can be related to a balancing of momenta. By making use of Newton’s second law,¹ which relates the motion to the force, one can discover that, in a hydrogen atom initially at rest (that is, with its center of mass at rest), the balanced forces will cause the momenta of electron and proton to be equal and opposite. At a given instant, the two particles are moving in opposite directions. The heavier proton moves more slowly, but has the same magnitude of momentum as the electron. As the electron swings to a new direction and a new speed in its track, the proton swings, too, in just such a way that its momentum remains equal and opposite to that of the electron. In spite of the continuously changing momenta of the two particles, the total momentum of the atom remains zero; the atom does not move. In this way—by “discovering” and applying two laws, Newton’s second and third laws of motion—one derives the law of momentum conservation and finds an explanation for the fact that an isolated atom does not move.

Without difficulty, the same arguments may be applied to the book on the table. Since all forces come in equal and opposite pairs, the forces between every pair of atoms cancel, so that the total force is zero, no matter how many billions of billions of atoms and individual forces there might be.

It is worth reviewing the steps in the argument above. Two laws of permission were discovered, telling what does happen. One law relates the motion to the force; the other says that the forces between pairs of particles are always equal and opposite. From these laws, the conservation of momentum can be derived as an interesting consequence, and this conservation law in turn explains the fact that an isolated atom at rest remains at rest.

The modern approach to the problem starts in quite a different way, by seeking a law of prohibition, a principle explaining why the atom does not move. The chain of reasoning goes like this: Symmetry → invariance → conservation. In the example of the isolated hydrogen atom, the symmetry of interest is the homogeneity of space (space is the same everywhere). Founded upon this symmetry is the invariance principle that the laws of nature are the same everywhere. Finally, the conservation law resting on this invariance principle is the conservation of momentum.

Let me clarify, through the example of the isolated hydrogen atom, how the assumed homogeneity of space is linked to the conservation of momentum. First, an exact statement of the invariance principle for this example: No aspect of the motion of an isolated atom depends upon the location of the center of mass of the atom. (The center of mass of any object is the average position of all of the mass in the object—the so-called “weighted average.” In a hydrogen atom, the center of mass is a point in space between the electron and the proton, close to the more massive proton.)

Let us visualize our hydrogen atom isolated in empty space with its center of mass at rest. Suppose now that its center of mass starts to move. In which direction should it move? We confront at once the question of the homogeneity of space. Investing our atom with human qualities for a moment, we can say that it has no basis upon which to “decide” how to move. To the atom surveying the possibilities, every direction is precisely as good or bad as every other direction. It is therefore frustrated in its “desire” to move and simply remains at rest.

This anthropomorphic description of the situation can be replaced by sound mathematics. What the mathematics shows is that an acceleration of the center of mass—for example, changing from a state of rest to a state of motion—is not consistent with the assumption that the laws of motion of the atom are independent of the location of the center of mass. If the center of mass of the atom is initially at rest at point A and it then begins to move, it will later pass through another point B. At point A, the center of mass had no velocity. At point B it does have a velocity. Therefore, the state of motion of the atom depends on the location of the center of mass, contrary to the invariance principle. Only if the center of mass remains at rest can the atom satisfy the invariance principle. If the center of mass of the atom had been moving initially, the invariance principle requires that it continue moving with constant velocity. The immobility of the center of mass requires, in turn, that the two particles composing the atom have equal and opposite momenta. A continual balancing of the two momenta means that their sum, the total momentum, is a constant.

The argument thus proceeds directly from the symmetry principle to the conservation law without making use of Newton’s laws of motion. That this is a deeper approach to conservation laws, as well as a more esthetically pleasing one, has been verified by history. Although Newton’s laws of motion have been altered by relativity and by quantum mechanics, the direct connection between the symmetry of space and the conservation of momentum has been unaffected—or even strengthened—by these modern theories, and momentum conservation remains a pillar of modern physics. A violation of the law of momentum conservation would imply an inhomogeneity of space; this is not an impossibility, but it would have far-reaching consequences for our view of the universe.

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