Fear of Physics (10 page)

The
how
alone would offer remarkable new insights. As soon as Galileo argued that a body at rest was just a special case of a body moving at a constant velocity, cracks began to emerge in Aristotelian philosophy, which asserted the special status of the former. In fact, Galileo’s argument implied that the laws of physics might look the same from the vantage point of an observer moving with constant velocity as they world from the vantage point of one at rest. After all, a third object in constant relative motion with respect to one will also be in constant motion relative to the other. Similarly, an object that speeds up or slows down relative to one will do the same relative to the other. This equivalence between the two vantage points was Galileo’s statement of relativity, predating Einstein’s by almost three centuries. It is very fortunate for us that it holds, because while we are accustomed to measuring motion as compared to the fixed and stable
terra firma,
all the while the Earth is moving around the sun, and the sun is moving around the galaxy, and our galaxy is moving around in a cluster of galaxies, and so on. So we really are not standing still, but rather moving at some large velocity relative to faraway galaxies. If this background motion had to be taken into account before we could properly describe the physics of a ball flying in the air relative to us on Earth, Galileo and Newton would never have been able to derive these laws in the first place. Indeed, it is only because the constant (on human time scales) motion of our galaxy relative to its neighbors does not alter the behavior of objects moving on Earth that the laws of motion
were uncovered, which in turn allowed the developments in astronomy that led to the recognition that our galaxy is moving relative to distant galaxies in the first place.

I’ll come back to relativity later. First, I want to describe how Galileo proceeded to milk his first success with uniform motion. Since most motion we see in nature is not in fact uniform, if Galileo was truly to claim to discuss reality, he had to address this issue. Again, he followed his first maxims: Throw out the irrelevant, and don’t ask why:

Galileo
defined
accelerated motion to be the simplest kind of nonuniform motion, namely, that in which the velocity of an object
changes, but at a constant rate. Is such an idealization relevant? Galileo ingeniously showed that such a simplification in fact described the motion of all falling bodies, if one ignores the extraneous effects of things like air resistance. This discovery paved the way for Newton’s Law of Gravity. If there hadn’t been a knowledge of the regularity in the underlying motion of falling bodies, the simplicity of assigning a force proportional to the mass of such objects would have been impossible. In fact, to get this far, Galileo had to overcome two other obstacles which are somewhat irrelevant to the point I am making, but his arguments were so simple and clever that I can’t resist describing them.

Aristotle had claimed that falling objects instantly acquire their final velocity upon being released. This was a reasonable claim based upon intuitive notions of what we see. Galileo was the first to show convincingly that this was not the case, using a ridiculously simple example. It was based on a
gedanken
or “thought” experiment, in the words of Einstein, a slightly updated version of which I will relate here. Imagine dropping a shoe into a bathtub from 6 inches above the water. Then drop it from 3 feet (and stand back). If you make the simple assumption that the size of the splash is related to the speed of the shoe when it hits the water, you can quickly convince yourself that the shoe speeds up as it falls.

Next was Galileo’s demonstration that all objects fall at the same rate, independent of their mass, if you ignore the effects of air resistance. While most people think of this in terms of the famous experiment of dropping two different objects off the leaning Tower of Pisa, which may actually never have been performed, Galileo in fact suggested a much simpler thought experiment that pointed out the paradox in assuming that objects that are twice as massive fall twice as fast. Imagine dropping two cannonballs of
exactly the same mass off a tower. They should fall at the same rate even if their rate of falling did depend upon their mass. Now, as they are falling, imagine that a very skilled and fast craftsman reaching out a window connects them with a piece of strong tape. Now you have a single object whose mass is twice the mass of either cannonball. Common sense tells us that this new object will not suddenly begin falling twice as fast as the two cannonballs were falling before the tape was added. Thus, the rate at which objects fall is
not
proportional to their mass.

Having forced aside these red herrings, Galileo was now ready actually to measure the acceleration of a falling body and show that it is constant. Recall that this implies that the velocity changes at a constant rate. I remind you that in laying the foundation upon which the theory of gravity was developed, Galileo did no more than attempt to describe how things fell, not why. It is like trying to learn about Feynman’s game of chess by first carefully examining the configuration of the chessboard and then carefully describing the motion of the pieces. Over and over again since Galileo, we have found that the proper description of the “playing field” on which physical phenomena occur goes a long way toward leading to an explanation of the “rules” behind the phenomena. In the ultimate version of this, the playing field
determines
the rules, and I shall argue later that this is exactly where the thrust of modern physics research is heading.... but I digress.

Galileo did not stop here. He went on to solve one other major complication of motion by copying what he had already done. To this point he, and we, have discussed motion in only one dimension—either falling down or moving horizontally along. If I throw a baseball, however, it does both. The trajectory of a baseball, again ignoring air resistance, is a curve that mathematicians
call a parabola, an arclike shape. Galileo proved this by doing the simplest possible extension of his previous analyses. He suggested that two-dimensional motion could be reduced to two independent copies of one-dimensional motion, which of course he had already described. Namely, the downward component of the motion of a ball would be described by the constant acceleration he had outlined, while the horizontal component of the motion would be described by the constant uniform velocity he had asserted all objects would naturally maintain in the absence of any external force. Put the two together, and you get a parabola.

While this may sound trivial, it both clarified a whole slew of phenomena that are often otherwise misconstrued and set a precedent that has been followed by physicists ever since. First, consider an Olympic long jump, or perhaps a Michael Jordan dunk beginning from the foul-shot line. As we watch these remarkable feats, it is clear to us that the athletes are in the air for what seems an eternity. Considering their speed leading up to the jump, how much longer can they glide in the air? Galileo’s arguments give a surprising answer. He showed that horizontal and vertical motion are independent. Thus, if the long-jumper Carl Lewis or the basketball star Michael Jordan were to jump up while standing still, as long as he achieved the same vertical height he achieved at the midpoint of his running jump, he would stay in the air
exactly
as long. Similarly, to use an example preached in physics classes around the world, a bullet shot horizontally from a gun will hit the ground at the same time as a penny dropped while the trigger is being pulled, even if the bullet travels a mile before doing so. The bullet only
appears
to fall more slowly because it moves away so quickly that in the time it takes to leave our sight, neither it nor the penny has had much time to fall at all!

Galileo’s success in showing that two dimensions could be just thought of as two copies of one dimension, as far as motion is concerned, has been usurped by physicists ever since. Most of modern physics comes down to showing that new problems can be reduced, by some technique or other, to problems that have been solved before. This is because the list of the types of problems we can solve exactly can probably be counted on the fingers of two hands (and maybe a few toes). For example, while we happen to live in three spatial dimensions, it is essentially impossible to solve exactly most fully three-dimensional problems, even using the computational power of the fastest computers. Those we can solve invariably involve either effectively reducing them to solvable one- or two-dimensional problems by showing that some aspects of the problem are redundant, or at the very least reducing them to independent
sets
of solvable one- or two-dimensional problems by showing that different parts of the problem can be treated independently.

Examples of this procedure are everywhere. I have already discussed our picture of the sun, in which we assume that the internal structure is the same throughout the entire sun at any fixed distance from the center. This allows us to turn the interior of the sun from a three-dimensional problem to an effectively one-dimensional problem, described completely in terms of the distance,
r,
from the solar center. A modern example of a situation in which we don’t ignore but rather break up a three-dimensional problem into smaller pieces can be obtained closer to home. The laws of quantum mechanics, which govern the behavior of atoms and the particles that form them, have allowed us to elucidate the laws of chemistry by explaining the structure of atoms, which make up all materials. The simplest atom is the hydrogen atom, with only a single, positively charged particle at its center, the
proton, surrounded by a single negatively charged particle, an electron. The quantum-mechanical solution of the behavior of even such a simple system is quite rich. The electron can exist in a set of discrete states of differing total energy. Each of the main “energy levels” is itself subdivided into states in which the shape of the electron’s “orbits” are different. All of the intricate behavior of chemistry—responsible for the biology of life, among other things—reflects, at some basic level, the simple counting rules for the number of such available states. Elements with all of these states but one in a certain level filled by electrons like to bond chemically to elements that have only one lone electron occupying the highest energy levels. Salt, also called sodium chloride, for example, exists because sodium shares its lone electron with chlorine, which uses it to fill up the otherwise sole unoccupied state in its highest energy level.

The only reason we have been able to enumerate the level structure of even the simplest atoms such as hydrogen is because we have found that the three-dimensional nature of these systems “separates” into two separate parts. One part involves a one-dimensional problem, which relates to understanding simply the radial distance of the electron from the proton. The other part involves a two-dimensional problem, which governs the angular distribution of the electron “orbits” in the atom. Both of these problems are solved separately and then combined to allow us to classify the total number of states of the hydrogen atom.

Here’s a more recent and more exotic example, along similar lines. Stephen Hawking has become known for his demonstration in 1974 that black holes are not black—that is, they emit radiation with a temperature characteristic of the mass of the black hole. The reason this discovery was so surprising is that black holes were so named because the gravitational field at their surface is so
strong that nothing inside can escape, not even light. So how can they emit radiation? Hawking showed that, in the presence of the strong gravitational field of the black hole, the laws of quantum mechanics allow this result of classical thinking to be evaded. Such evasions of classical “no go” theorems are common in quantum mechanics. For example, in our classical picture of reality, a man resting in a valley between two mountains might never be able to get into a neighboring valley without climbing over one or the other of the mountains. However, quantum mechanics allows an electron in an atom, say, with an energy smaller than that required to escape from the atom, according to classical principles, sometimes to “tunnel” out from inside the electric field binding it, and find itself finally free of its former chains! A standard example of this phenomenon is radioactive decay. Here, the configuration of particles—protons and neutrons—buried deep in the nucleus of an atom can suddenly change. Depending upon the properties of an individual atom or nucleus, quantum mechanics tells us that it is possible for one or more of these particles to escape from the nucleus, even though classically they are all irretrievably bound there. In another example, if I throw a ball at a window, either the ball will have enough energy go through it, or it will bounce off the window and return. If the ball is small enough so that its behavior is governed by quantum-mechanical principles, however, things are different. Electrons, say, impinging on a thin barrier can do both! In a more familiar example, light impinging on the surface of a material like a mirror might normally be reflected. If the mirror is thin enough, however, we find that even though most is reflected, some of the light can “tunnel” through the mirror and appear on the other side! (I shall outline the new “rules” that govern this weird behavior later. For the moment, take it as a given.)