Einstein was twenty-six in 1905 when he published his paper “Zur Elektrodynamik bewegter Körper” (“On the Electrodynamics of Moving Bodies”). In it he made the simple assumption that the laws of physics and in particular the speed of light should appear to be the same to all uniformly moving observers. This idea, it turns out, demands a revolution in our concept of space and time. To see why, imagine two events that take place at the same spot but at different times, in a jet aircraft. To an observer on the jet there will be zero distance between those two events. But to a second observer on the ground the events will be separated by the distance the jet has traveled in the time between the events. This shows that two observers who are moving relative to each other will not agree on the distance between two events.
Now suppose the two observers observe a pulse of light traveling from the tail of the aircraft to its nose. Just as in the above example, they will not agree on the distance the light has traveled from its emission at the plane’s tail to its reception at the nose. Since speed is distance traveled divided by the time taken, this means that if they agree on the speed at which the pulse travels—the speed of light—they will not agree on the time interval between the emission and the reception.
What makes this strange is that, though the two observers measure different times, they are watching the same physical process. Einstein didn’t attempt to construct an artificial explanation for this. He drew the logical, if startling, conclusion that the measurement of the time taken, like the measurement of the distance covered, depends on the observer doing the measuring. That effect is one of the keys to the theory in Einstein’s 1905 paper, which has come to be called special relativity.
We can see how this analysis could apply to timekeeping devices if we consider two observers looking at a clock. Special relativity holds that the clock runs faster according to an observer who is at rest with respect to the clock. To observers who are not at rest with respect to the clock, the clock runs slower. If we liken a light pulse traveling from the tail to the nose of the plane to the tick of a clock, we see that to an observer on the ground the clock runs slower because the light beam has to travel a greater distance in that frame of reference. But the effect does not depend on the mechanism of the clock; it holds for all clocks, even our own biological ones.
Einstein’s work showed that, like the concept of rest, time cannot be absolute, as Newton thought. In other words, it is not possible to assign to every event a time with which every observer will agree. Instead, all observers have their own measures of time, and the times measured by two observers who are moving relative to each other will not agree. Einstein’s ideas go counter to our intuition because their implications aren’t noticeable at the speeds we normally encounter in everyday life. But they have been repeatedly confirmed by experiment. For example, imagine a reference clock at rest at the center of the earth, another clock on the earth’s surface, and a third clock aboard a plane, flying either with or against the direction of the earth’s rotation. With reference to the clock at the earth’s center, the clock aboard the plane moving eastward—in the direction of the earth’s rotation—is moving faster than the clock on the earth’s surface, and so it should run slower. Similarly, with reference to the clock at the earth’s center, the clock aboard the plane flying westward—against the earth’s rotation—is moving slower than the surface clock, which means that clock should run faster than the clock on the surface. And that is exactly what was observed when, in an experiment performed in October 1971, a very accurate atomic clock was flown around the world. So you could extend your life by constantly flying eastward around the world, though you might get tired of watching all those airline movies. However, the effect is very small, about 180 billionths of a second per circuit (and it is also somewhat lessened by the effects of the difference in gravity, but we need not get into that here).
Due to the work of Einstein, physicists realized that by demanding that the speed of light be the same in all frames of reference, Maxwell’s theory of electricity and magnetism dictates that time cannot be treated as separate from the three dimensions of space. Instead, time and space are intertwined. It is something like adding a fourth direction of future/past to the usual left/right, forward/backward, and up/down. Physicists call this marriage of space and time “space-time,” and because space-time includes a fourth direction, they call it the fourth dimension. In space-time, time is no longer separate from the three dimensions of space, and, loosely speaking, just as the definition of left/right, forward/backward, or up/down depends on the orientation of the observer, so too does the direction of time vary depending on the speed of the observer. Observers moving at different speeds would choose different directions for time in space-time. Einstein’s theory of special relativity was therefore a new model, which got rid of the concepts of absolute time and absolute rest (i.e., rest with respect to the fixed ether).
Einstein soon realized that to make gravity compatible with relativity another change was necessary. According to Newton’s theory of gravity, at any given time objects are attracted to each other by a force that depends on the distance between them at that time. But the theory of relativity had abolished the concept of absolute time, so there was no way to define when the distance between the masses should be measured. Thus Newton’s theory of gravity was not consistent with special relativity and had to be modified. The conflict might sound like a mere technical difficulty, perhaps even a detail that could somehow be worked around without much change in the theory. As it turned out, nothing could have been further from the truth.
Over the next eleven years Einstein developed a new theory of gravity, which he called general relativity. The concept of gravity in general relativity is nothing like Newton’s. Instead, it is based on the revolutionary proposal that space-time is not flat, as had been assumed previously, but is curved and distorted by the mass and energy in it.
A good way to picture curvature is to think of the surface of the earth. Although the earth’s surface is only two-dimensional (because there are only two directions along it, say north/south and east/west), we’re going to use it as our example because a curved two-dimensional space is easier to picture than a curved four-dimensional space. The geometry of curved spaces such as the earth’s surface is not the Euclidean geometry we are familiar with. For example, on the earth’s surface, the shortest distance between two points—which we know as a line in Euclidean geometry—is the path connecting the two points along what is called a great circle. (A great circle is a circle along the earth’s surface whose center coincides with the center of the earth. The equator is an example of a great circle, and so is any circle obtained by rotating the equator along different diameters.)
Imagine, say, that you wanted to travel from New York to Madrid, two cities that are at almost the same latitude. If the earth were flat, the shortest route would be to head straight east. If you did that, you would arrive in Madrid after traveling 3,707 miles. But due to the earth’s curvature, there is a path that on a flat map looks curved and hence longer, but which is actually shorter. You can get there in 3,605 miles if you follow the great-circle route, which is to first head northeast, then gradually turn east, and then southeast. The difference in distance between the two routes is due to the earth’s curvature, and a sign of its non-Euclidean geometry. Airlines know this, and arrange for their pilots to follow great-circle routes whenever practical.
According to Newton’s laws of motion, objects such as cannonballs, croissants, and planets move in straight lines unless acted upon by a force, such as gravity. But gravity, in Einstein’s theory, is not a force like other forces; rather, it is a consequence of the fact that mass distorts space-time, creating curvature. In Einstein’s theory, objects move on geodesics, which are the nearest things to straight lines in a curved space. Lines are geodesics on the flat plane, and great circles are geodesics on the surface of the earth. In the absence of matter, the geodesics in four-dimensional space-time corres
pond to lines in three-dimensional space. But when matter is present, distorting space-time, the paths of bodies in the corresponding three-dimensional space curve in a manner that in Newtonian theory was explained by the attraction of gravity. When space-time is not flat, objects’ paths appear to be bent, giving the impression that a force is acting on them.
Einstein’s general theory of relativity reproduces special relativity when gravity is absent, and it makes almost the same predictions as Newton’s theory of gravity in the weak-gravity environment of our solar system—but not quite. In fact, if general relativity were not taken into account in GPS satellite navigation systems, errors in global positions would accumulate at a rate of about ten kilometers each day! However, the real importance of general relativity is not its application in devices that guide you to new restaurants, but rather that it is a very different model of the universe, which predicts new effects such as gravitational waves and black holes. And so general relativity has transformed physics into geometry. Modern technology is sensitive enough to allow us to perform many sensitive tests of general relativity, and it has passed every one.
Though they both revolutionized physics, Maxwell’s theory of electromagnetism and Einstein’s theory of gravity—general relativity—are both, like Newton’s own physics, classical theories. That is, they are models in which the universe has a single history. As we saw in the last chapter, at the atomic and subatomic levels these models do not agree with observations. Instead, we have to use quantum theories in which the universe can have any possible history, each with its own intensity or probability amplitude. For practical calculations involving the everyday world, we can continue to use classical theories, but if we wish to understand the behavior of atoms and molecules, we need a quantum version of Maxwell’s theory of electromagnetism; and if we want to understand the early universe, when all the matter and energy in the universe were squeezed into a small volume, we must have a quantum version of the theory of general relativity. We also need such theories because if we are seeking a fundamental understanding of nature, it would not be consistent if some of the laws were quantum while others were classical. We therefore have to find quantum versions of all the laws of nature. Such theories are called quantum field theories.
The known forces of nature can be divided into four classes:
1. Gravity. This is the weakest of the four, but it is a long-range force and acts on everything in the universe as an attraction. This means that for large bodies the gravitational forces all add up and can dominate over all other forces.
2. Electromagnetism. This is also long-range and is much stronger than gravity, but it acts only on particles with an electric charge, being repulsive between charges of the same sign and attractive between charges of the opposite sign. This means the electric forces between large bodies cancel each other out, but on the scales of atoms and molecules they dominate. Electromagnetic forces are responsible for all of chemistry and biology.
3. Weak nuclear force. This causes radioactivity and plays a vital role in the formation of the elements in stars and the early universe. We don’t, however, come into contact with this force in our everyday lives.
4. Strong nuclear force. This force holds together the protons and neutrons inside the nucleus of an atom. It also holds together the protons and neutrons themselves, which is necessary because they are made of still tinier particles, the quarks we mentioned in Chapter 3. The strong force is the energy source for the sun and nuclear power, but, as with the weak force, we don’t have direct contact with it.
The first force for which a quantum version was created was electromagnetism. The quantum theory of the electromagnetic field, called quantum electrodynamics, or QED for short, was developed in the 1940s by Richard Feynman and others, and has become a model for all quantum field theories. As we’ve said, according to classical theories, forces are transmitted by fields. But in quantum field theories the force fields are pictured as being made of various elementary particles called bosons, which are force-carrying particles that fly back and forth between matter particles, transmitting the forces. The matter particles are called fermions. Electrons and quarks are examples of fermions. The photon, or particle of light, is an example of a boson. It is the boson that transmits the electromagnetic force. What happens is that a matter particle, such as an electron, emits a boson, or force particle, and recoils from it, much as a cannon recoils after firing a cannonball. The force particle then collides with another matter particle and is absorbed, changing the motion of that particle. According to QED, all the interactions between charged particles—particles that feel the electromagnetic force—are described in terms of the exchange of photons.
The predictions of QED have been tested and found to match experimental results with great precision. But performing the mathematical calculations required by QED can be difficult. The problem, as we’ll see below, is that when you add to the above framework of particle exchange the quantum requirement that one include all the histories by which an interaction can occur—for example, all the ways the force particles can be exchanged—the mathematics becomes complicated. Fortunately, along with inventing the notion of alternative histories—the way of thinking about quantum theories described in the last chapter—Feynman also developed a neat graphical method of accounting for the different histories, a method that is today applied not just to QED but to all quantum field theories.
Feynman’s graphical method provides a way of visualizing each term in the sum over histories. Those pictures, called Feynman diagrams, are one of the most important tools of modern physics. In QED the sum over all possible histories can be represented as a sum over Feynman diagrams like those below, which represent some of the ways it is possible for two electrons to scatter off each other through the electromagnetic force. In these diagrams the solid lines represent the electrons and the wavy lines represent photons. Time is understood as progressing from bottom to top, and places where lines join correspond to photons being emitted or absorbed by an electron. Diagram (A) represents the two electrons approaching each other, exchanging a photon, and then continuing on their way. That is the simplest way in which two electrons can interact electromagnetically, but we must consider all possible histories. Hence we must also include diagrams like (B). That diagram also pictures two lines coming in—the approaching electrons—and two lines going out—the scattered ones—but in this diagram the electrons exchange two photons before flying off. The diagrams pictured are only a few of the possibilities; in fact, there are an infinite number of diagrams, which must be mathematically accounted for.
Feynman diagrams aren’t just a neat way of picturing and categorizing how interactions can occur. Feynman diagrams come with rules that allow you to read off, from the lines and vertices in each diagram, a mathematical expression. The probability, say, that the incoming electrons, with some given initial momentum, will end up flying off with some particular final momentum is then obtained by summing the contributions from each Feynman diagram. That can take some work, because, as we’ve said, there are an infinite number of them. Moreover, although the incoming and outgoing electrons are assigned a definite energy and momentum, the particles in the closed loops in the interior of the diagram can have any energy and momentum. That is important because in forming the Feynman sum one must sum not only over all diagrams but also over all those values of energy and momentum.
Feynman diagrams provided physicists with enormous help in visualizing and calculating the probabilities of the processes described by QED. But they did not cure one important ailment suffered by the theory: When you add the contributions from the infinite number of different histories, you get an infinite result. (If the successive terms in an infinite sum decrease fast enough, it is possible for the sum to be finite, but that, unfortunately, doesn’t happen here.) In particular, when the Feynman diagrams are added up, the answer seems to imply that the electron has an infinite mass and charge. This is absurd, because we can measure
the mass and charge and they are finite. To deal with these infinities, a procedure called renormalization was developed.
The process of renormalization involves subtracting quantities that are defined to be infinite and negative in such a way that, with careful mathematical accounting, the sum of the negative infinite values and the positive infinite values that arise in the theory almost cancel out, leaving a small remainder, the finite observed values of mass and charge. These manipulations might sound like the sort of things that get you a flunking grade on a school math exam, and renormalization is indeed, as it sounds, mathematically dubious. One consequence is that the values obtained by this method for the mass and charge of the electron can be any finite number. That has the advantage that physicists may choose the negative infinities in a way that gives the right answer, but the disadvantage that the mass and charge of the electron therefore cannot be predicted from the theory. But once we have fixed the mass and charge of the electron in this manner, we can employ QED to make many other very precise predictions, which all agree extremely closely with observation, so renormalization is one of the essential ingredients of QED. An early triumph of QED, for example, was the correct prediction of the so-called Lamb shift, a small change in the energy of one of the states of the hydrogen atom discovered in 1947.
The success of renormalization in QED encouraged attempts to look for quantum field theories describing the other three forces of nature. But the division of natural forces into four classes is probably artificial and a consequence of our lack of understanding. People have therefore sought a theory of everything that will unify the four classes into a single law that is compatible with quantum theory. This would be the holy grail of physics.