Borderlands of Science

by Charles Sheffield

  We now consider relativity and its implications.

  2.4 Relativity. The second great physical theory of the twentieth century, as important to our understanding of Nature as quantum theory, is relativity. Actually, there are in a sense two theories of relativity: the special theory, published by Einstein in 1905, and the general theory, published by him in 1915.

  2.5 Special relativity. The special theory of relativity concentrates on objects that move relative to each other at constant velocity. The general theory allows objects to be accelerated relative to each other in any way, and it includes a theory of gravity.

  Relativity is often thought to be a "hard" subject. It really isn't, although the general theory calls for a good deal of mathematics. What relativity is, more than anything, is unfamiliar. Before the effects of relativity are noticed, things need to be moving relative to each other very fast (a substantial fraction of the speed of light), or they must involve a very strong gravitational field. We are as unaware of relativity as a moving snail is unaware of wind resistance, and for the same reason; our everyday speeds of motion are too slow for the effects to be noticed.

  Everyone from Einstein himself to Bertrand Russell has written popular accounts of relativity. We name just half a dozen references, in increasing order of difficulty: Einstein's Universe (Calder, 1979); The Riddle of Gravitation (Bergmann, 1968); Relativity and Common Sense (Bondi, 1964); Einstein's Theory of Relativity (Born, 1924); The Meaning of Relativity (Einstein, 1953); and Theory of Relativity (Pauli, 1956). Rather than talk about the theory itself, we are going to confine ourselves here to its major consequences. In the case of special relativity, there are six main ones to notice and remember.

  1) Mass and energy are not independent quantities, but can be converted to each other. The formula relating the two is the famous E=mc2.

  2) Time is not an absolute measure, the same for all observers. Instead, time passes more slowly on a moving object than it does relative to an observer of that object. The rule is, for an object traveling at a fraction F of the speed of light, when an interval T passes onboard the object, an interval of 1/sqrt(1- F2) of T passes for the observer. For example, if a spaceship passes you traveling at 99 percent of the speed of light, your clock will register that seven hours pass while the spaceship's clocks show that only one hour has passed on board. This phenomenon is known as "time dilation," or "time dilatation," and it has been well-verified experimentally.

  3) Mass is not an absolute measure, the same for all observers. For an object traveling at a fraction F of the speed of light, its mass will appear to be increased by a factor of 1/sqrt(1-F2) so far as an outside observer is concerned. If a spaceship passes you traveling at 99 percent of the speed of light, its mass will appear to have increased by a factor of seven over its original value. This phenomenon has also been well-verified experimentally.

  4) Nothing can be accelerated to travel faster than light. In fact, to accelerate something to the speed of light would take an infinite amount of energy. This is actually a consequence of the previous point. Note: this does not say that an object cannot vanish from one place, and appear at another, in a time less than it would take light to travel between those locations. Hence this is not inconsistent with the quantum teleportation discussion of the previous section.

  5) Length also is not an absolute measure, the same for all observers. If a spaceship passes you traveling at 99 percent of the speed of light, it will appear to be foreshortened to one-seventh of its original length. This phenomenon is known as "Lorentz contraction," or "Fitzgerald-Lorentz contraction."

  6) The speed of light is the same for all observers, regardless of the speed of the light source or the speed of the observer. This is not so much a consequence of the special theory of relativity as one of the assumptions on which the theory is based.

  The consequences of special relativity theory are worked out more simply if instead of dealing with space and time separately, calculations are performed in a merged entity we term "spacetime." This is also not a consequence of the theory, but rather a convenient framework in which to view it.

  After it was proposed, the theory of relativity became the subject of much popular controversy. Detractors argued that the theory led to results that were preposterous and "obvious nonsense." That is not true, but certainly some of the consequences of relativity do not agree with "intuitive" common sense evolved by humans traveling at speeds very slow compared with the speed of light.

  Let us consider just one example. Suppose that we have two spaceships, A and B, each traveling toward Earth (O), but coming from diametrically opposite directions. Also, suppose that each of them is moving at 4/5 of the speed of light according to their own measurements. "Common sense" would then insist that they are moving toward each other at 4/5+4/5=8/5 of light speed. Yet one of our tenets for relativity theory is that you cannot accelerate an object to the speed of light. But we seem to have done just that. Surely, A will think that B is approaching at 1.6 times light speed.

  No. So far as A (or B) is concerned, we must use a relativistic formula for combining velocities in order to calculate B's speed relative to A. According to that formula, if O observes A and B approaching with speeds u and v, then A (and B) will observe that they are approaching each other at a speed U=(u+v)/(1+uv/c2), where c=the speed of light. In this case, we find U=40/41 of the speed of light.

  Can U ever exceed c, for any permitted values of u and v? That's the same as asking, if u and v are less than 1, can (u+v)/(1+uv) ever be greater than 1? It's easy to prove that it cannot.

  Now let us take the next step. Let us look at the passage of time. Suppose that A sends a signal ("Hi") and nine seconds later, according to his time frame, sends the same message again. According to rule 2), above, the time between one "Hi!" and the next, as measured by us, will be increased by a factor 5/3. For us, 15 seconds have passed. And so far as B is concerned, since B thinks that A is traveling at 40/41 of light-speed, an interval 9/ sqrt (1-40/412)=41 seconds have passed.

  If you happen to be one of those people who read a book from back to front, you may now be feeling confused. In discussing the expansion of the universe in Chapter 4, we point out that signals from objects approaching us have higher frequencies, while signals from objects receding from us have lower frequencies. But here we seem to be saying the exact opposite of that: the time between "Hi!" signals, which is equivalent to a frequency, seems to be less for O and B than it is for A.

  In fact, that is not the case. We have to allow for the movement of A between transmission of successive signals. When A sends the second "Hi," nine seconds later than the first according to his measurements, he has moved 15 seconds closer according to O. That is a distance 15x4/5=12 light-seconds (a light-second, in analogy to a light-year, is the distance light travels in one second). Thus the travel time of the second "Hi" is decreased by 12 seconds so far as O is concerned. Hence the time between "Hi's" as measured by O is three seconds. The signal frequency has increased.

  The same is true for B. The time between transmission of "Hi's" is 41 seconds as perceived by B, but in that time the distance between A and B as measured by B has decreased by 40 light-seconds. The time between successive "Hi's" is therefore just one second for B. The signal frequency so far as B is concerned has increased, more than it did for O.

  If the preceding few paragraphs seem difficult, don't worry about them. My whole point is that the results of relativity theory can be very counterintuitive when your intuition was acquired in situations where everything moves much less than the speed of light. The moral, from a storyteller's point of view, is be careful when you deal with objects or people moving close to light speed. An otherwise good book, The Sparrow (Russell, 1996) was ruined for me by a grotesque error in relativistic time dilation effects. It could have been corrected with a simple change of target star.

  Just for the fun of it, let us ask what happens to our signals between A, B, and O if we have a working quantum telepor
tation device, able to send signals instantaneously. What will the received signals have as their frequencies? No one can give a definite answer to this, but a likely answer is that quantum teleportation is totally unaffected by relative velocities. If that's the case, everyone sends and receives signals as though they were all in close proximity and at rest relative to each other. As a corollary, for quantum teleportation purposes the universe lacks any spatial dimension and can be treated as a single point.

  2.6 General relativity. For the general theory of relativity, the main consequences to remember are:

  1) The presence of matter (or of energy, which the special theory asserts are two forms of one and the same thing) causes space to curve. What we experience as gravity is a direct measure of the curvature of space.

  2) Objects move along the shortest possible path in curved space. Thus, a comet that falls in toward the Sun and then speeds out again following an elongated elliptical trajectory is traveling a minimum-distance path in curved space. In the same way, light that passes a massive gravitational object follows a path significantly bent from the straight line of normal geometry. Light that emanates from a massive object will be lengthened in wavelength as it travels "uphill" out of the gravity field. Note that this is not the "red shift" associated with the recession of distant galaxies, which will be discussed in Chapter 4.

  3) If the concentration of matter is high, it is possible for spacetime itself to curve so much that a knowledge of some regions becomes denied to the rest of the universe. The interior of a black hole is just such a region. We are unaware of this in everyday life, simply because the concentrations of matter known to us are too low for the effects to occur.

  4) Since matter curves space, the total amount of matter in the universe has an effect on its overall structure. This will become profoundly important in Chapter 4, when we consider the large-scale structure and eventual fate of the universe.

  To truly space-faring civilizations, the effects of special and general relativity will be as much a part of their lives as sun, wind, and rain are to us.

  2.7 Beyond the atom. Quantum theory and the special theory of relativity together provide the tool for analysis of subatomic processes. But we have not defined the subatomic world to which it applies.

  Before the work of Rutherford and J.J. Thomson, the atom was considered a simple, indivisible object. Even after Rutherford's work, the only known subatomic particles were electrons and protons (the nucleus of an atom was regarded as a mixture of electrons and protons).

  The situation changed in 1932, with the discovery of the positron (a positively charged electron) and the neutron (a particle similar in mass to the proton, but with no charge). At that point the atom came to be regarded as a cloud of electrons encircling a much smaller structure, the nucleus. The nucleus is made up of protons (equal in number to the electrons) and neutrons.

  However, this ought to puzzle and worry us. Electrons and protons attract each other, because they are oppositely charged; but protons repel each other. So how is it possible for the nucleus, an assembly of protons and neutrons, to remain intact?

  The answer is, through another force of nature, known as the strong force. The strong force is attractive, but it operates only over very short distances (much less than the size of an atom). It holds the nucleus together—most of the time. Sometimes another force, known as the weak force, causes a nucleus to emit an electron or a positron, and thereby become a different element. To round out the catalog of forces, we also have the familiar electromagnetic force, the one that governs attraction or repulsion of charged particles; and finally, we have the gravitational force, through which any two particles of matter, no matter how small or large, attract each other. The gravitational force ought really to be called the weak force, since it is many orders of magnitude less powerful than any of the others. It dominates the large-scale structure of the universe only because it applies over long distances, to every single bit of matter.

  We have listed four fundamental forces. Are there others?

  We know of no others, but that might only be an expression of our ignorance. From time to time, experiments hint at the existence of a "fifth force." Upon closer investigation, the evidence is explained some other way, and the four forces remain. However, it is quite legitimate in science fiction to hypothesize a fifth force, and to give it suitable properties. If you do this, however, be careful. The fifth force must be so subtle, or occur in such extreme circumstances, that we would not have stumbled over it already in our exploration of the universe. You can also, if you feel like it, suggest modifications to the existing four forces, again with suitable caution.

  The attempt to explain the four known forces in a single framework, a Theory of Everything, assumes that no more forces will be discovered. This strikes me as a little presumptuous.

  Returning to the discovery of fundamental particles, after the neutron came the neutrino, a particle with neither charge nor mass (usually; recently, some workers have discovered evidence suggesting a small mass for the neutrino). The existence of the neutrino had been postulated by Wolfgang Pauli in 1931, in order to retain the principle of the conservation of energy, but it was not actually discovered until 1953.

  Then—too quickly for the theorists to feel comfortable about it—came the muon (1938), pions (predicted 1935, discovered 1947), the antiproton (1955), and a host of others, etas and lambdas and sigmas and rhos and omegas.

  Quantum theory seemed to provide a theoretical framework suitable for all of these, but in 1960 the basic question—"Why are there so many animals in the `nuclear zoo'?"—remained unanswered. In the early 1960s, Murray Gell-Mann and George Zweig independently proposed the existence of a new fundamental particle, the quark, from which all the heavy subatomic particles were made.

  The quark is a peculiar object indeed. First, its charge is not a whole multiple of the electron or proton charge, but one-third or two-thirds of such a value. There are several varieties of quarks: the "up" and "down" quark, the "top" and "bottom" quark, and the "strange" and "charmed" quarks; each may have any of three "colors," red, green, or blue (the whimsical labels are no more than that; they lack physical significance). Taken together, the quarks provide the basis for a theory known as quantum chromodynamics, which is able to describe very accurately the forces that hold the atomic nucleus together.

  A theory to explain the behavior of lighter particles (electrons, positrons, and photons) was developed earlier, mainly by Richard Feynman, Julian Schwinger, and Sinitiro Tomonaga. Freeman Dyson then proved the consistency and equivalence of the seemingly very different theories. The complete synthesis is known as quantum electrodynamics. Between them, quantum electrodynamics and quantum chromodynamics provide a full description of the subatomic world down to the scale at which we are able to measure.

  However, the quark is a rather peculiar particle to employ as the basis for a theory. A proton consists of three quarks, two "up" and one "down"; a neutron is one "up" and two "down." Pions each contain only two quarks. An omega particle consists of three strange quarks. This is all based purely on theory, because curiously enough, no one has ever seen a quark. Theory suggests that we never will. The quark exists only in combination with other quarks. If you try to break a quark free, by hitting a proton or a neutron with high-energy electrons or a beam of radiation, at first nothing appears. However, if you keep increasing the energy of the interaction, something finally does happen. New particles appear—not the quarks, but more protons, pions, and neutrons. Energy and mass are interchangeable; apply enough energy, and particles appear. The quark, however, keeps its privacy.

  I have often thought that a good bumper sticker for a particle physicist would be "Free the Quarks!"

  The reluctance of the free quark to put in an appearance makes it very difficult for us to explore its own composition. But we ask the question: What, if anything, is smaller than the quark?

  Although recent experiments suggest that the quark does
have a structure, no one today knows what it is. We are offshore of the physics mainland, and are allowed to speculate in fictional terms as freely as we choose.

  Or almost. There are two other outposts that we need to be aware of in the world of the ultra-small. The proton and the neutron have a radius of about 0.8x10-15 meters. If we go to distances far smaller than that, we reach the realm of the superstring.

  A superstring is a loop of something not completely defined (energy? force?) that oscillates in a space of ten dimensions. The string vibrations give rise to all the known particles and forces. Each string is about 10-35 meters long. We live in a space of four dimensions (three space and one time), and the extra six dimensions of superstring theory are "rolled up" in such a way as to be unobservable. In practice, of course, a superstring has never been observed. The necessary mathematics to describe what goes on is also profoundly difficult.

  Why is the concept useful? Mainly, because superstring theory includes gravity in a natural way, which quantum electrodynamics and quantum chromodynamics do not. In fact, superstring theory not only allows gravity to be included, it requires it. We might be closing in on the "Theory of Everything" already mentioned, explaining the four known fundamental interactions of matter in a single set of equations.

  There is a large literature on superstrings. If the concept continues to prove useful, we will surely find ways to make the mathematics more accessible. Remember, the calculus needed to develop Isaac Newton's theories was considered impossibly difficult in the seventeenth century. Meanwhile, the science fiction writer can be more comfortable with superstrings than many practicing scientists.

  On the same small scale as the superstring we have something known as the Planck length. This is the place where vacuum energy fluctuations, inevitably predicted by quantum theory, radically affect the nature of space. Rather than a smooth, continuous emptiness, the vacuum must now be perceived as a boiling chaos of minute singularities. Tiny black holes constantly form and dissolve, and space has a foam-like character where even the concept of distance may not have meaning. (We have mentioned black holes but not really discussed them, though surely there is no reader who has not heard of them. They are so important a part of the science fiction writer's arsenal that they deserve a whole section to themselves. They can be found in Chapter 3.)