The Compleat McAndrew

by Charles Sheffield

  Or is it?

  Oppenheimer and Snyder pointed out that black holes are created when big masses, larger than the Sun, contract under gravitational collapse. The kernels that we want are much smaller than that. We need to be able to move them around the solar system, and the gravitational field of an object the mass of the Sun would tear the system apart. Unfortunately, there was no prescription in Oppenheimer’s work, or elsewhere, to allow us to make small black holes.

  Stephen Hawking finally came to the rescue. Apart from being created by collapsing stars, he said, black holes could also be created in the extreme conditions of pressure that existed during the Big Bang that started our Universe. Small black holes, weighing no more than a hundredth of a milligram, could have been born then. Over billions of years, these could interact with each other to produce more massive black holes, of any size you care to mention. We seem to have the mechanism that will produce the kernels of the size we need.

  Unfortunately, what Hawking gave he soon took away. In perhaps the biggest surprise of all in black hole theory, he showed that black holes are not black.

  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 towards what John Wheeler terms 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 the hole, the faster the rate of radiation. He 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 much faster than a “cold” one. For a black hole the mass of the Sun, the associated temperature is lower than the background temperature of the Universe. Such a black hole receives more than it emits, so it will steadily increase in mass. However, for a small black hole, with the few billion tons of mass that we want in a kernel, the temperature is so high (ten billion degrees) that the black hole will radiate itself away in a gigantic and rapid burst of radiation and particles. Furthermore, a rapidly spinning kernel will preferentially radiate particles that decrease its spin, and a highly charged one 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 analysis. Only when he had performed every check that he could think of was he finally forced to accept the conclusion: black holes aren’t black after all; and the smallest black holes are the least black.

  That gives us a problem when we want to use power kernels in a story. First, the argument that they are readily available, as leftovers from the birth of the Universe, has been destroyed. Second, a Kerr-Newman black hole is a dangerous object to be near. It gives off high energy radiation and particles.

  This is the point where the science of Kerr-Newman black holes stops and the science fiction begins. I assume in these stories that there is some as-yet-unknown natural process which creates sizeable black holes on a continuing basis. They can’t be created too close to Earth, or we would see them. However, there is plenty of room outside the known Solar System—perhaps in the region occupied by the long-period comets, from beyond the orbit of Pluto out to perhaps a light-year from the Sun.

  Second, I assume that a kernel can be surrounded by a shield (not of matter, but of electromagnetic fields) which is able to reflect all the emitted particles and radiation back into the black hole. Humans can thus work close to the kernels without being fried in a storm of radiation and high-energy particles.

  Even surrounded by such a shield, a rotating black hole would still be noticed by a nearby observer. Its gravitational field would still be felt, and it would also produce a curious effect known as “inertial dragging.”

  We have pointed out that the inside of a black hole is completely shielded from the rest of the Universe, so that we can never know what is going on there. It is as though the inside of a black hole is a separate Universe, possibly with its own different physical laws. Inertial dragging adds to that idea. We are used to the notion that when we spin something around, we do it relative to a well-defined and fixed reference frame. Newton pointed out in his Principia Mathematica that a rotating bucket of water, from the shape of the water’s surface, provides evidence of an “absolute” rotation relative to the stars. This is true here on Earth, or over in the Andromeda Galaxy, or out in the Virgo Cluster. It is not true, however, near a rotating black hole. The closer that we get to one, the less that our usual absolute reference frame applies. The kernel defines its own absolute frame, one that rotates with it. Closer than a certain distance to the kernel (the “static limit” mentioned earlier) everything must revolve—dragged along and forced to adopt the rotating reference frame defined by the spinning black hole.

  The McAndrew balanced drive.

  This device makes a first appearance in the second chronicle, and is assumed in all the subsequent stories.

  Let us begin with well-established science. Again it starts at the beginning of the century, in the work of Einstein. In 1908, he wrote as follows:

  “We…assume the complete physical equivalence of a gravitational field and of a corresponding acceleration of the reference system…”

  And in 1913:

  “An observer enclosed in an elevator has no way to decide whether the elevator is at rest in a static gravitational field or whether the elevator is located in gravitation-free space in an accelerated motion that is maintained by forces acting on the elevator (equivalence hypothesis).”

  This equivalence hypothesis, or equivalence principle, is central to general relativity. If you could be accelerated in one direction at a thousand gees, and simultaneously pulled in the other direction by an intense gravitational force producing a thousand gees, you would feel no force whatsoever. It would be just the same as if you were in free fall.

  As McAndrew said, once you realize that fact, the rest is mere mechanics. You take a large circular disk of condensed matter (more on that in a moment), sufficient to produce a gravitational acceleration of, say, 50 gees on a test object (such as a human being) sitting on the middle of the plate. You also provide a drive that can accelerate the plate away from the human at 50 gees. The net force on the person at the middle of the plate is then zero. If you increase the acceleration of the plate gradually, from zero to 50 gees, then to remain comfortable the person must also be moved in gradually, starting well away from disk and finishing in contact with it. The life capsule must thus move in and out along the axis of the disk, depending on the ship’s acceleration: high acceleration, close to disk; low acceleration, far from disk.

  There is one other variable of importance, and that is the tidal forces on the human passenger. These are caused by the changes in gravitational force with distance—it would be no good having a person’s head feeling a force of one gee, if his feet felt a force of thirty gees. Let us therefore insist that the rate of change of acceleration be no more than one gee per meter when the acceleration caused by the disk is 50 gees.

  The gravitational acceleration produced along the axis of a thin circular disk of matter of total mass M and radius R is a textbook problem of classical potential theory. Taking the radius of the disk to be 50 meters, the gravitational acceleration acting on a test object at the center of the disk to be 50 gees, and the tidal force there to be one gee per meter, we can solve for the total mass M, together with the gravitational and tidal forces acting on a body at different distances Z along the axis of the disk.

  We find that if the distance of the passengers from the center of the plate is 246 meters, the plate produces gravitational acceleration on passengers of 1 gee, so if the drive is off there is a net force of 1 gee on them; at zero meters (on the plate itself) the plate produces a gravitational acceleration on passengers of 50 gees, so if the drive accelerates t
hem at 50 gees, they feel as though they are in free fall. The tidal force is a maximum, at one gee per meter, when the passengers are closest to the disk.

  This device will actually work as described, with no science fiction involved at all, if you can provide the plate of condensed matter and the necessary drive. Unfortunately, this turns out to be nontrivial. All the distances are reasonable, and so are the tidal forces. What is much less reasonable is the mass of the disk that we have used. It is a little more than 9 trillion tons; such a disk 100 meters across and one meter thick would have an average density of 1,170 tons per cubic centimeter.

  This density is modest compared with that found in a neutron star, and tiny compared with what we find in a black hole. Thus we know that such densities do exist in the Universe. However, no materials available to us on Earth today even come close to such high values—they have densities that fall short by a factor of more than a million. And the massplate would not work as described, without the dense matter. We have a real problem.

  It’s science fiction time again: let us assume that in a couple of hundred years we will be able to compress matter to very high densities, and hold it there using powerful electromagnetic fields. If that is the case, the massplate needed for McAndrew’s drive can be built. It’s certainly massive, but that shouldn’t be a limitation—the Solar System has plenty of spare matter available for construction materials. And although a 9 trillion ton mass may sound a lot, it’s tiny by celestial standards, less than the mass of a modest asteroid.

  With that one extrapolation of today’s science it sounds as though we can have the McAndrew balanced drive. We can even suggest how that extrapolation might be performed, with plausible use of present physics.

  Unfortunately, things are not as nice as they seem. There is a much bigger piece of science fiction that must be introduced before the McAndrew drive can exist as a useful device. We look at that next, and note that it is a central concern of the third chronicle.

  Suppose that the drive mechanism is the most efficient one consistent with today’s physics. This would be a photon drive, in which any fuel is completely converted to radiation and used to propel the ship. There is certainly nothing in present science that suggests such a drive is theoretically impossible, and some analysis of matter-antimatter reactions indicates that the photon drive could one day be built. Let us assume that we know how to construct it. Then, even with this “ultimate” drive, McAndrew’s ship would have problems. It’s not difficult to calculate that with a fifty gee drive, the conversion of matter to radiation needed to keep the drive going will quickly consume the ship’s own mass. More than half the mass will be gone in a few days, and McAndrew’s ship will disappear from under him.

  Solution of this problem calls for a lot more fictional science than the simple task of producing stable condensed matter. We have to go back to present physics and look for loopholes. We need to find inconsistencies in the overall picture of the Universe provided by present day physics, and exploit these as necessary.

  The best place to seek inconsistencies is where we already know we will find them—in the meeting of general relativity and quantum theory. If we calculate the energy associated with an absence of matter in quantum theory, the “vacuum state,” we do not, as common sense would suggest, get zero.

  Instead we get a large, positive value per unit volume. In classical thinking, one could argue that the zero point of energy is arbitrary, so that one can simply start measuring energies from the vacuum state value. But if we accept general relativity, this option is denied to us. Energy, of any form, produces space-time curvature. We are therefore not allowed to change the definition of the origin of the energy scale. Once this is accepted, the energy of the vacuum state cannot be talked out of existence. It is real, if elusive, and its presence provides the loophole that we need.

  Again, we are at the point where the science fiction enters. If the vacuum state has an energy associated with it, I assume that this energy is capable of being tapped. Doesn’t this then, according to relativity (E = mc2), suggest that there is also mass associated with the vacuum, contrary to what we think of as vacuum? Yes, it does, and I’m sorry about that, but the paradox is not of my creation. It is implicit in the contradictions that arise as soon as you try to put general relativity and quantum theory together.

  Richard Feynman, one of the founders of quantum electrodynamics, addressed the question of the vacuum energy, and computed an estimate for the equivalent mass per unit volume. The estimate came out to two billion tons per cubic centimeter. The energy in two billion tons of matter is more than enough to boil all Earth’s oceans (powerful stuff, vacuum). Feynman, commenting on his vacuum energy estimate, remarks:

  “Such a mass density would, at first sight at least, be expected to produce very large gravitational effects which are not observed. It is possible that we are calculating in a naive manner, and, if all of the consequences of the general theory of relativity (such as the gravitational effects produced by the large stresses implied here) were included, the effects might cancel out; but nobody has worked all this out yet. It is possible that some cutoff procedure that not only yields a finite energy for the vacuum state but also provides relativistic invariance may be found. The implications of such a result are at present completely unknown.”

  With that degree of uncertainty at the highest levels of present-day physics, I feel not at all uncomfortable in exploiting the troublesome vacuum energy to service McAndrew’s drive.

  The third chronicle introduces two other ideas that are definitely science fiction today, even if they become science fact a few years from now. If there are ways to isolate the human central nervous system and keep it alive independently of the body, we certainly don’t know much about them. On the other hand, I see nothing that suggests this idea is impossible in principle—heart transplants were not feasible forty years ago, and until this century blood transfusions were rare and highly dangerous. A century hence, today’s medical impossibilities should be routine.

  The Sturm Invocation for vacuum survival is also invented, but I believe that it, like the Izaak Walton introduced in the seventh chronicle, is a logical component of any space-oriented future. Neither calls for technology beyond what we have today. The hypnotic control implied in the Invocation, though advanced for most practitioners, could already be achieved. And any competent engineering shop could build a Walton for you in a few weeks—I am tempted to patent the idea, but fear that it would be rejected as too obvious or inevitable a development.

  Life in space and the Oort cloud.

  Most chronicles take place at least partly in the Halo, or the Outer Solar System, which I define to extend from the distance of Pluto from the Sun, out to a little over a light-year. Within this radius, the Sun is still the primary gravitational influence, and controls the orbits of objects moving out there.

  To give an idea of the size of the Halo, we note that Pluto lies at an average distance of about 6 billion kilometers from the Sun. This is about forty astronomical units, where the astronomical unit, usually abbreviated to AU, is defined as the mean distance of the Earth from the Sun. The AU provides a convenient yardstick for measurements within the Solar System. One light-year is about 63,000 AU (inches in a mile, is how I remember it). So the volume of space in the Halo is 4 billion times as large as the sphere enclosing the nine known planets.

  By Solar System standards, the Halo is thus a huge region. But beyond Neptune and Pluto, we know little about it. There are a number of “trans-Neptunian objects,” but no one knows how many. Some of them may be big enough to qualify as planets. The search for Pluto was inspired early this century by differences between theory and observation in the orbits of Uranus and Neptune. When Pluto was found, it soon became clear that it was not nearly heavy enough to produce the observed irregularities. The obvious explanation is yet another planet, farther out than the ones we know.

  Calculations of the orbit and size of a tenth planet n
eeded to reconcile observation and theory for Uranus and Neptune suggest a rather improbable object, out of the orbital plane that all the other planets move in and about seventy times the mass of the Earth. I don’t believe this particular object exists.

  On the other hand, observational equipment and techniques for faint objects are improving rapidly. The number of known trans-Neptunian objects increases almost every month.

  The other thing we know for sure about the Halo is that it is populated by comets. The Halo is often called the Oort cloud, since the Dutch astronomer Oort suggested thirty years ago that the entire Solar System is enveloped by a cloud of cometary material, to a radius of perhaps a light-year. He regarded this region as a “cometary reservoir,” containing perhaps a hundred billion comets. Close encounters between comets out in the Halo would occasionally disturb the orbit of one of them enough to divert it to the Inner System, where it would appear as a long-period comet when it came close enough to the Sun. Further interactions with Jupiter and the other planets would then sometimes convert the long-period comet to a short-period comet, such as Hailey’s or Encke’s comet, which we observe repeatedly each time they swing by close to the Sun.

  Most comets, however, continue their lonely orbits out in the cloud, never approaching the Inner System. They do not have to be small to be invisible to us. The amount of sunlight a body receives is inversely proportional to the square of its distance from the Sun; the apparent area it presents to our telescopes is also inversely proportional to the square of its distance from Earth. For bodies in the Halo, the reflected light that we receive from them thus varies as the inverse fourth power of their distance from the Sun. A planet with the size and composition of Uranus, but half a light-year away, would be seven trillion times as faint. And we should remember that Uranus itself is faint enough that it was not discovered until 1781, when high-quality telescopes were available. So far as present-day detection powers are concerned, there could be almost anything out there in the Halo.