The Compleat McAndrew

by Charles Sheffield

  I knew all about the canals on Mars, the dust pools on the Moon, and the swamps on Venus, about the Dean drive and dianetics and the Hieronymus machine. I believed that men and pigs were more closely related than men and monkeys; that atoms were miniature solar systems; that you could shoot men to the moon with a cannon (a belief that didn’t survive my first course in dynamics); that the pineal gland was certainly a rudimentary third eye and probably the seat of parapsychological powers; that Rhine’s experiments at Duke University had made telepathy an unquestioned part of modern science; that with a little ingenuity and a few electronic bits and pieces you could build in your backyard a spacecraft to take you to the moon; and that, no matter what alien races might have developed on other worlds and be scattered around the Galaxy, humans would prove to be the smartest, most resourceful, and most wonderful species to be found anywhere.

  That last point may even be true. As Pogo remarked long ago, true or false, either way it’s a mighty sobering thought.

  What I needed was a crib sheet. We had them in school for the works of Shakespeare. They were amazingly authoritative, little summaries that outlined the plot, told us just who did what and why, and even informed us exactly what was in Shakespeare’s head when he was writing the play. If they didn’t say what he had for lunch that day, it was only because that subject never appeared on examination papers. Today’s CliffsNotes are less authoritative, but only I suspect because the changing climate of political correctness encourages commentators to be as bland as possible.

  I didn’t know it at the time, but the crib sheets were what I was missing in science fiction. Given the equivalent type of information about SF, I would not have assured my friends (as I did) that the brains of industrial robots made use of positrons, that the work of Dirac and Blackett would lead us to a faster-than-light drive, or that the notebooks of Leonardo da Vinci gave all the details needed to construct a moon rocket.

  As Mark Twain remarked, it’s not what we don’t know that causes the trouble, it’s the things we know that ain’t so. (This is an example of the problem. I was sure this was said by Mark Twain, but when I looked it up I found it was a Josh Billings line. Since then I have seen it as attributed to Artemus Ward.) What follows, then, is my crib sheet for this book. This Appendix sorts out the real science, based on and consistent with today’s theories (but probably not tomorrow’s), from the “science” that I made up for these stories. I have tried to provide a clear dividing line, at the threshold where fact stops and fiction takes over. But even the invented material is designed to be consistent with and derived from what is known today. It does not contradict current theories, although you will not find papers about it in the Physical Review or the Astrophysical Journal.

  The reader may ask, which issues of these publications? That’s a very fair question. After all, these stories were written over a twenty-year period. In that time, science has advanced, and it’s natural to ask how much of what I wrote still has scientific acceptance.

  I reread each story with that in mind, and so far as I know everything still fits with current knowledge. A few things have even gained in plausibility. For example, when I wrote “Rogueworld” we had no direct evidence of any extra-solar planets. Now reports come in every month or two of another world around some other star, based not on direct observation of the planet but on small observed perturbations in the apparent position of the star itself. The idea of vacuum energy extraction, first introduced to science fiction in “All the Colors of the Vacuum,” has proceeded from wild science fiction idea to funded research. Black holes, which at the time I wrote “Killing Vector” were purely theoretical entities, form a standard part of modern cosmology. A big black hole, about 2.5 million times the mass of the Sun, is believed to lie at the center of our own galaxy. Radiating black holes, which in 1977 were another way-out idea, are now firmly accepted. The Oort cloud, described in “The Manna Hunt,” is a standard part of today’s physical model of the extended Solar System.

  So has there been nothing new in science in the past twenty years? Not at all. Molecular biology has changed so fast and so much since the 1970s that the field seen from that earlier point of view is almost unrecognizable, and the biggest changes still lie in the future. Computers have become smaller, more powerful, and ubiquitous, beyond what anyone predicted twenty years ago. We also stand today on the verge of quantum computation, which takes advantage of the fact that at the quantum level a system can exist in several states simultaneously. The long-term potential of that development is staggering.

  Finally, in the very week that I write this, a report has appeared of the first successful experiment in “quantum teleportation.” Via a process known as “entanglement,” which couples the quantum state of two widely separated systems, a Caltech team “teleported” a pattern of information from one location to another, independent of the speed of light. If there isn’t a new hard SF story in that report, I don’t know where you’ll find one.

  Kernels, black holes, and singularities.

  Kernels feature most prominently in the first chronicle, but they are assumed and used in all the others, too. A kernel is actually a Ker-N-le, which is shorthand for Kerr-Newman black hole.

  To explain Kerr-Newman black holes, it is best to follow McAndrew’s technique, and go back a long way in time. We begin in 1915. In that year, Albert Einstein published the field equations of general relativity in their present form. He had been trying different possible formulations since about 1908, but he was not satisfied with any of them before the 1915 set. His final statement consisted of ten coupled, nonlinear, partial differential equations, relating the curvature of space-time to the presence of matter.

  The equations are very elegant and can be written down in tensor form as a single short line of algebra. But written out in full they are horrendously long and complex—so much so that Einstein himself did not expect to see any exact solutions, and thus perhaps didn’t look very hard. When Karl Schwarzschild, just the next year, produced an exact solution to the “one-body problem” (he found the gravitational field produced by an isolated mass particle), Einstein was reportedly quite surprised.

  This “Schwarzschild solution” was for many years considered mathematically interesting, but of no real physical importance. People were much more interested in looking at approximate solutions of Einstein’s field equations that could provide possible tests of the theory. Everyone wanted to compare Einstein’s ideas on gravity with those introduced two hundred and fifty years earlier by Isaac Newton, to see where there might be detectible differences. The “strong field” case covered by the Schwarzschild solution seemed less relevant to the real world.

  For the next twenty years, little was discovered to lead us toward kernels. Soon after Schwarzschild published his solution, Reissner and Nordstrom solved the general relativity equations for a spherical mass particle that also carried an electric charge. This included the Schwarzschild solution as a special case, but it was considered to have no physical significance and it too remained a mathematical curiosity.

  The situation finally changed in 1939. In that year, Oppenheimer and Snyder were studying the collapse of a star under gravitational forces—a situation which certainly did have physical significance, since it is a common stellar occurrence.

  Two remarks made in their summary are worth quoting directly: “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 have to say “modern pic
ture” because before 1800 it had been noted as a curiosity that a sufficiently massive body could have an escape velocity from its surface that exceeded the speed of light; in a sense, the black hole was predicted more than two hundred years ago.)

  Notice that the collapsing body does not have to contract indefinitely if it is the size of the Sun or smaller, so we do not have to worry that the Earth, say, or the Moon, will shrink indefinitely to become a black hole. Notice also that there is a reference to the “gravitational radius” of the black hole. This was something that came straight out of the Schwarzschild solution, the distance where the reddening of light became infinite, so that any light coming from inside that radius could never be seen by an outside observer. Since the gravitational radius for the Sun is only about three kilometers, if the Sun were squeezed down to this size conditions inside the collapsed body defy the imagination. The density of matter must be about twenty billion tons per cubic centimeter.

  You might think that Oppenheimer and Snyder’s paper, with its apparently bizarre conclusions, would have produced a sensation. In fact, it aroused little notice for a long time. It too was looked at as a mathematical oddity, a result that physicists needn’t take too seriously.

  What was going on here? The Schwarzschild solution had been left on the shelf for a generation, and now the Oppenheimer results were in their turn regarded with no more than mild interest.

  One could argue that in the 1920s the attention of leading physicists was elsewhere, as they tried to drink from the fire-hose flood of theory and experiment that established quantum theory. But what about the 1940s and 1950s? Why didn’t whole groups of physicists explore the consequences for general relativity and astrophysics of an indefinitely collapsing stellar mass?

  Various explanations could be offered, but I favor one that can be stated in a single word: Einstein. He was a gigantic figure, stretching out over everything in physics for the first half of this century. Even now, he casts an enormous shadow over the whole field. Until his death in 1955, researchers in general relativity and gravitation felt a constant awareness of his presence, of his genius peering over their shoulder. If Einstein had not been able to penetrate the mystery, went the unspoken argument, what chance do the rest of us have? Not until after his death was there a resurgence of interest and spectacular progress in general relativity. And it was one of the leaders of that resurgence, John Wheeler, who in 1958 provided the inspired name for the Schwarzschild solution needed to capture everyone’s fancy: the black hole.

  We still have not reached the kernel. The black hole that Wheeler named was still the Schwarzschild black hole, the object that McAndrew spoke of with such derision. It had a mass, and possibly an electric charge, but that was all. The next development came in 1963, and it was a big surprise to everyone working in the field.

  Roy Kerr, at that time associated with the University of Texas at Austin, had been exploring a particular set of Einstein’s field equations that assumed an unusually simple form for the metric (the metric is the thing that defines distances in a curved space-time). The analysis was highly mathematical and seemed wholly abstract, until Kerr found that he could produce a form of exact solution to the equations. The solution included the Schwarzschild solution as a special case, but there was more; it provided in addition another quantity that Kerr was able to associate with spin.

  In the Physical Review Letters of September, 1963, Kerr published a one-page paper with the not-too-catchy title, “Gravitational field of a spinning mass as an example of algebraically special metrics.” In this paper he described the Kerr solution for a spinning black hole. I think it is fair to say that everyone, probably including Kerr himself, was astonished.

  The Kerr black hole has a number of fascinating properties, but before we get to them let us take the one final step needed to reach the kernel. In 1965 Ezra Newman and colleagues at the University of Pittsburgh published a short note in the Journal of Mathematical Physics, pointing out that the Kerr solution could be generated from the Schwarzschild solution by a curious mathematical trick, in which a real coordinate was replaced by a complex one. They also realized that the same trick could be applied to the charged black hole, and thus they were able to provide a solution for a rotating, charged black hole: the Kerr-Newman black hole, that I call the kernel.

  The kernel has all the nice features admired by McAndrew. Because it is charged, you can move it about using electric and magnetic fields. Because you can add and withdraw rotational energy, you can use it as a power source and a power reservoir. A Schwarzschild black hole lacks these desirable qualities. As McAndrew says, it just sits there.

  One might think that this is just the beginning. There could be black holes that have mass, charge, spin, axial asymmetry, dipole moments, quadrupole moments, and many other properties. It turns out that this is not the case. The only properties that a black hole can possess are mass, charge, spin and magnetic moment—and the last one is uniquely fixed by the other three.

  This strange result, often stated as the theorem “A black hole has no hair” (i.e. no detailed structure) was established to most people’s satisfaction in a powerful series of papers in 1967-1972 by Werner Israel, Brandon Carter, and Stephen Hawking. A black hole is fixed uniquely by its mass, spin, and electric charge. Kernels are the end of the line, and they represent the most general kind of black hole that physics permits.

  After 1965, more people were working on general relativity and gravitation, and other properties of the Kerr-Newman black holes rapidly followed. Some of them were very strange. For example, the Schwarzschild black hole has a characteristic surface associated with it, a sphere where the reddening of light becomes infinite, and from within which no information can ever be sent to the outside world. This surface has various names: the surface of infinite red shift, the trapping surface, the one-way membrane, and the event horizon. But the Kerr-Newman black holes turn out to have two characteristic surfaces associated with them, and the surface of infinite red shift is in this case distinct from the event horizon.

  To visualize these surfaces, take a hamburger bun and hollow out the inside enough to let you put a round hamburger patty entirely within it. For a Kerr-Newman black hole, 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 no particle can remain at rest, no matter how hard its rocket engines work. Inside the bun, the surface of the meat patty is a sphere, the “event horizon,” from which no light or particle can ever escape. We can never find out anything about what goes on within the meat patty’s surface, so its composition must be a complete mystery (you may have eaten hamburgers that left the same impression). For a rotating black hole, these 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). The really interesting region, however, is that between these two surfaces—the remaining bread, usually called the ergosphere. It has a property which allows the kernel to become a power kernel.

  Roger Penrose (whom we will meet again in a later chronicle) pointed out in 1969 that it is possible for a particle to dive in towards a Kerr black hole, split in two when it is inside the ergosphere, and then have part of it ejected in such a way that it has more total energy than the whole particle that went in. If this is done, we have extracted energy from the black hole.

  Where has that energy come from? Black holes may be mysterious, but we still do not expect that energy can be created from nothing.

  Note that we said a Kerr black hole—not a Schwarzschild black hole. The energy we extract comes from the rotational energy of the spinning black hole, and if a hole is not spinning, no energy can possibly be extracted from it in this way. As McAndrew remarked, a Schwarzschild hole is dull, a dead object that cannot be used to provide power. A Kerr black hole, on the other hand, is one of the most efficient energy sources imaginable, better by far than most nuclear fission or fusion processes. (A Kerr-Newman black h
ole allows the same energy extraction process, but we have to be a little more careful, since only part of the ergosphere can be used.)

  If a Kerr-Newman black hole starts out with only a little spin energy, the energy-extraction process can be worked in reverse, to provide more rotational energy—the process that McAndrew referred to as “spin-up” of the kernel. “Spin-down” is the opposite process, the one that extracts energy. A brief paper by Christodoulou in the Physical Review Letters of 1970 discussed the limits on this process, and pointed out that you could only spin-up a kernel to a certain limit, termed an “extreme” Kerr solution. Past that limit (which can never be achieved using the Penrose process) a solution can be written to the Einstein field equations. This was done by Tomimatsu and Sato, and presented in 1972 in another one-page paper in Physical Review Letters. It is a very odd solution indeed. It has no event horizon, which means that activities there are not shielded from the rest of the Universe as they are for the usual kernels. And it has what is referred to as a “naked singularity” associated with it, where cause and effect relationships no longer apply. This bizarre object was discussed by Gibbons and Russell-Clark, in 1973, in yet another paper in Physical Review Letters.

  That seems to leave us in pretty good shape. Everything so far has been completely consistent with current physics. We have kernels that can be spun up and spun down by well-defined procedures—and if we allow that McAndrew could somehow take a kernel past the extreme form, we would indeed have something with a naked singularity. It seems improbable that such a physical situation could exist, but if it did, space-time there would be highly peculiar. The existence of certain space-time symmetry directions—called killing vectors—that we find for all usual Kerr-Newman black holes would not be guaranteed. Everything is fine.