The Compleat McAndrews

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 ca

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 spi

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.