Borderlands of Science

by Charles Sheffield

  What is the final fate of a star that explodes and becomes a neutron star? That depends on the mass of the part that's left. One possibility is that it remains a neutron star to the end of its life. Another more exotic possibility is that it shrinks further and becomes a black hole. That intriguing option we will describe in the next section, after which we will expand the scale of our exploration.

  3.3 Black holes. The story of black holes begins with Albert Einstein and the theory of general relativity.

  In 1916, soon after the publication of the field equations in their final form, Karl Schwarzschild produced the first exact solution. Einstein was reportedly quite surprised, because of the complicated nature of the field equations—a set of ten coupled nonlinear partial differential equations. As Einstein wrote to Max Born, twenty years later, "If only it were not so damnably difficult to find rigorous solutions."

  The "Schwarzschild solution" gave the gravitational field for an isolated mass, which later became known as the Schwarzschild black hole. At the time, it was considered to be mathematically interesting, but of no physical significance. Soon after Schwarzschild's work, Reissner and Nordstrom solved the general relativity equations for a spherical mass that also carried a charge. It too was regarded with no special interest.

  In 1939, Oppenheimer and Snyder studied the collapse of a star under gravitational forces—a situation which certainly does have physical significance, since it is a common stellar occurrence.

  Two remarks from the summary of their paper are worth quoting: "Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of the sun, this contraction will continue indefinitely." In other words, not only can a star collapse, but if it is heavy enough there is no way that the collapse and contraction can be stopped. And "the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles." This is the first modern picture of a black hole, a body with a gravitational field so strong that light cannot escape from it. We say "modern picture" because John Michell in 1783, and Pierre Laplace in 1798, independently noted that a sufficiently massive body would have an escape velocity from its surface that exceeded the speed of light.

  The idea of a "gravitational radius" came straight from the Schwarzschild solution. It is the distance from the center where the reddening of light becomes infinite, and it defines a sphere. Any light coming from inside that sphere can never be seen by an outside observer. Today the surface of the sphere has a variety of names, all defining the same thing: the surface of infinite red shift, the trapping surface, the one-way membrane, and the event horizon. Since the gravitational radius for the Sun is only three kilometers, if it were squeezed down to this size (which will never happen, fortunately, as a result of gravity) conditions inside the collapsed body would be difficult to imagine. The density of matter would be about twenty billion tons per cubic centimeter.

  You might suppose that the Oppenheimer and Snyder paper, with its apparently bizarre conclusions, would have produced a sensation. In fact, it aroused little notice. It too was looked on as a mathematical oddity, a result that physicists did not need to take too seriously. The resurgence of interest in the solutions of the equations of general relativity did not take place until after Einstein's death in 1955, and it was one of the leaders of that renaissance, John Wheeler, who in 1958 provided the inspired name for the Schwarzschild solution at the gravitational radius: the black hole.

  The object described by the Schwarzschild and Reissner/Nordstrom solutions could have a mass, and a charge, and that was all. The next development came in 1963, and it was a big surprise to everyone in the field.

  Roy Kerr had been exploring a particular form of the Einstein field equations. The analysis was highly mathematical and seemed to be wholly abstract—until Kerr found that he could produce an exact form of solution. It included the Schwarzschild black hole as a special case, but there was more, another quantity that Kerr was able to associate with spin. For the first time, the possibility of a spinning black hole had appeared. It could also, as was shown a couple of years later by Ezra Newman and collaborators, have an associated charge.

  From this point on, I am for convenience going to call the charged, spinning Kerr-Newman black hole a kernel. It has a number of fascinating properties useful to science fiction writers.

  First, since it carries a charge, a kernel can be moved from place to place using electric and magnetic fields. Second, the kernel has associated with it not the single characteristic surface of the Schwarzschild solution (the sphere defined by the gravitational radius), but two. In this case, the surface of infinite red shift is distinct from the event horizon.

  To visualize the surfaces, take a hamburger bun and hollow out the inside, enough to let you put a round hamburger entirely within it. For a kernel, the outer surface of the bread (which is a sort of ellipsoid in shape) is the surface of infinite red shift, the "static limit" within which nothing can remain at rest, no matter how hard and efficiently its rocket engine works. Inside the bun, the surface of the meat patty forms a sphere, the "event horizon" from which no light or particle can ever escape to the outside. We can never find out anything about what goes on within the meat patty's surface, so its composition and nature, like that of many hamburgers, must remain a complete mystery. For a kernel, the bun and patty surfaces touch only at the north and south poles of the axis of rotation (the top and bottom centers of the bun). A really interesting region, however, lies between these two surfaces. It is called the ergosphere, and it has a most unusual property, pointed out in 1969 by Roger Penrose (yes, the same Penrose as in Chapter 2—he is a highly versatile and creative individual, who has made major contributions to relativity theory and other fields).

  Penrose showed that it is possible for a particle to dive in toward the kernel from outside, split in two when it is inside the ergosphere, and then have part of it ejected to the exterior in such a way that the piece has more total energy than the particle that went in. If we do this, we have extracted energy from the black hole.

  Note that this must be a kernel, a spinning black hole, not a Schwarzschild black hole. The energy that we have gained comes from the rotational energy of the hole itself.

  If the kernel starts out with only a little spin energy, we can use the energy-extraction process in reverse, to provide more rotational energy. We will refer to that as "spinning up" the kernel. "Spin down" is the opposite process, the one that extracts energy.

  One other property of a kernel will prove useful later. Every kernel (but not a Schwarzschild black hole) possesses a "ring singularity." It appears possible to remain far enough from the singularity to avoid destruction by tidal forces, but close enough to take advantage of peculiar aspects of space-time there. This is discussed further in Chapter 9.

  Since it can be proved that a black hole has as properties only mass, charge, spin, and magnetic moment, and the last one is fixed completely by the other three, that seems to say all that can be said about kernels. This result, that all black holes are completely defined by three constants, is a theorem that is often stated in the curious form, "A black hole has no hair."

  That was the situation until 1974, when Stephen Hawking produced a result that shocked everyone. In perhaps the biggest surprise in all black hole history, he proved that black holes are not black.

  This calls for some explanation. General relativity and quantum theory were both developed in this century, but they have never been combined in a satisfactory way. Physicists have known this and been uneasy about it for a long time. In attempting to move toward what John Wheeler referred to as "the fiery marriage of general relativity with quantum theory," Hawking studied quantum mechanical effects in the vicinity of a black hole. He found that particles and radiation can (and must) be emitted from the hole.

  The smaller (and therefore less
massive) the hole, the faster the rate of radiation. Hawking was able to relate the mass of the black hole to a temperature, and as one would expect, a "hotter" black hole pours out radiation and particles faster than a "cold" one. For a black hole the size of the Sun, the associated temperature is far lower than the background temperature of the Universe (the 2.7 Kelvin background radiation). Such a black hole receives more energy than it emits, so it will steadily increase in mass. However, there is no rule of nature that says a black hole has to be big and massive. For a black hole of a few billion tons (the mass of a small asteroid) the temperature is so high, at ten billion degrees, that the black hole will radiate itself away to nothing in a gigantic and rapid burst of radiation and particles. Furthermore, a spinning black hole will preferentially radiate particles that decrease its spin, while a charged black hole will prefer to radiate charged particles that reduce its overall charge.

  These results are so strange that in 1972 and 1973 Hawking spent a lot of time trying to find the mistake in his own calculations. Only when he had performed every check that he could think of was he finally forced to accept the conclusion: black holes are not black after all; and the smallest black holes are the least black.

  We have discussed the properties of kernels, without asking the crucial question: Do they exist?

  For a while, it was thought that very small black holes, weighing only a hundredth of a milligram, might have been created in the Big Bang. The Hawking radiative process showed that any of those, if they ever existed, would have gone long since. Big black holes, however, seem not only possible, but inevitable. If the core of a collapsing star is massive enough (more than about three times the mass of our Sun), then after the star explodes to a supernova, Oppenheimer and Snyder's results apply. The remnant star is forced to collapse without limit, and no force in the universe is powerful enough to stop it.

  Black holes, if they exist at all, ought therefore to be common throughout the universe, perhaps enough to make a sizable contribution to the missing mass needed to close it (see Chapter 4). However, some people object to the very idea of black hole existence. Associated with them is a singularity—an infinity—that no one has been able to explain away, and singularities are generally regarded as evidence that a theory has something wrong with it. Einstein himself was reported to consider black holes as a "blemish to be removed from his theory by a better mathematical formulation."

  Until that better mathematical formulation comes along, black holes are an acceptable part of theoretical physics; but what is the experimental evidence for them?

  We have a problem. A black hole, unless it is small (the mass of, say, a small asteroid) will not radiate measurable energy. Also, we know of no way that a black hole less than about three solar masses can form. Black holes are therefore, by definition, not directly visible. Their existence, like the existence of quarks, depends not on observing them, but on the role they play in simplifying and explaining other observations.

  A black hole's presence must be detected by indirect effects. For example, matter falling into a black hole will be ripped apart and give off a powerful radiation signal; but so will matter that falls into a neutron star. Distinguishing between the two is a subtle and difficult problem. One of the early and best candidates for a solar-sized black hole is the source known as Cygnus X-1.

  Very large black holes probably lie at the heart of many galaxies, and are the mechanism that powers quasars. It is also possible to regard the whole universe as a black hole, within which we happen to live; but these are conjectures, not established facts. In view of Einstein's comment, maybe any other possible explanation is to be preferred.

  Despite lack of final proof of their existence, black holes form a valuable weapon in the writer's arsenal. In fact, they are so accepted a feature of the science fiction field that they can be introduced without further explanation, like an alien or a faster-than-light drive. Black holes, of various sizes and properties, can be found in hundreds of stories.

  I will mention just a handful, so you will not be tempted to write a classic story that already exists: "The Hole Man," by Larry Niven (Niven, 1973); Imperial Earth, by Arthur C. Clarke (Clarke, 1975); Beyond the Blue Event Horizon, by Frederik Pohl (Pohl, 1980); and Earth, by David Brin (Brin, 1990). All of these employ the Schwarzschild black hole. I like better the spinning, rotating black hole. With my preferred name for them, a kernel (for Kerr-Newman black hole), they are used in all the stories in One Man's Universe, and in Proteus Unbound (Sheffield, 1983, 1989).

  CHAPTER 4

  Physics in the Very Large

  4.1 Galaxies. The ancient astronomers, observing without benefit of telescopes, knew and named many of the stars. They also noted the presence of a hazy glow that extends across a large fraction of the sky, and they called it the Milky Way. Finally, those with the most acute vision had noted that the constellation of Andromeda contained within it a much smaller patch of haze.

  The progress from observation of the stars to the explanation of hazy patches in the sky came in stages. Galileo started the ball rolling in 1610, when he examined the Milky Way with his telescope and found that he could see huge numbers of stars, far more than were visible with the unaided eye. He asserted that the Milky Way was nothing more than stars, in vast numbers. William Herschel carried this a stage farther, counting how many stars he could see in different parts of the Milky Way, and beginning to build towards the modern picture of a great disk of billions of separate stars, with the Sun well away (30,000 light-years) from the center.

  At the same time, the number of hazy patches in the sky visible with a telescope went up and up as telescope power increased. Lots of them looked like the patch in Andromeda. A dedicated comet hunter, Charles Messier, annoyed at constant confusion of hazy patches (uninteresting) with comets (highly desirable) plotted out their locations so as not to be bothered by them. This resulted in the Messier Catalog: the first and inadvertent catalog of galaxies.

  But what were those fuzzy glows identified by Messier?

  The suspicion that the Andromeda and other galaxies might be composed of stars, as the Milky Way is made up of stars, was there from Galileo's time. Individual stars cannot usually be seen, but only because of distance. The number of galaxies, though, probably exceeds anything that Galileo would have found credible. Today's estimate is that there are about a hundred billion galaxies in the visible universe—roughly the same as the number of individual stars in a typical galaxy. Galaxies, fainter and fainter as their distance increases, are seen as far as our telescopes can probe.

  In most respects, the distant ones look little different from the nearest ones. But there is one crucial difference. And it tells us something fundamental about the whole universe.

  4.2 The age of the universe. Galaxies increase in numbers as they decrease in apparent brightness, and it is natural to assume that these two go together: if we double the distance of a galaxy, it appears one-quarter as bright, but we expect to see four times as many like it if space is uniformly filled with galaxies.

  What we would not expect to find, until it was suggested by Carl Wirtz in 1924 and confirmed by Edwin Hubble in 1929, is that more distant galaxies appear redder than nearer ones.

  To be more specific, particular wavelengths of emitted light have been shifted towards longer wavelengths in the fainter (and therefore presumably more distant) galaxies. The question is, what could cause such a shift?

  The most plausible mechanism, to a physicist, is called the Doppler Effect. According to the Doppler Effect, light from a receding object will be shifted to longer (redder) wavelengths; light from an approaching object will be shifted to shorter (bluer) wavelengths. Exactly the same thing works for sound, which is why a speeding police car's siren seems to drop in pitch as it passes by.

  If we accept the Doppler effect as the cause of the reddened appearance of the galaxies, we are led (as was Hubble) to an immediate conclusion: the whole universe must be expanding, at a close to cons
tant rate, because the red shift of the galaxies corresponds to their faintness, and therefore to their distance.

  Note that this does not mean that the universe is expanding into some other space. There is no other space. It is the whole universe—everything there is—that has grown over time to its present dimension.

  From the recession of the galaxies we can draw another conclusion. If the expansion proceeded in the past as it does today, there must have been a time when everything in the whole universe was drawn together to a single point. It is logical to call the period since then, the age of the universe. The Hubble galactic red shift allows us to calculate that length of time. The universe seems to be between ten and twenty billion years old.

  We have here a truly remarkable result: observation of the faint agglomerations of stars known as galaxies has led us, very directly and cleanly, to the conclusion that we live in a universe of finite and determinable age. A century ago, no one would have believed such a thing possible.

  The recession of the galaxies also, in a specific sense, says that we live in a universe of finite and determinable size. For since, according to relativity theory, nothing can move faster than light, the "edge of the universe" is the distance at which the recession velocity of the galaxy is light speed. Nothing can come to us from farther away than that. There could be anything out there, anything at all, and we would never know it.

  Answering one question inevitably leads to another: Can we say anything more about the other "edge" of the universe, the time that defines its beginning?

  One approach is to use our telescopes to peer farther into space. When we do this, we are also looking farther back in time. If a galaxy five billion light-years away sent light in our direction, that radiation has been on the way for five billion years. Therefore, if we can look far enough out, at galaxies eight or even ten billion light-years, we will be observing the early history of the universe.