Is there that much available?

If we estimate the mass and energy from visible material in stars and galaxies, we find a value nowhere near the “critical density” needed to make the Universe finally flat. If we say that the critical mass-energy density has to be equal to unity just to slow the expansion, we observe in visible matter only a value of about 0.01.

There is evidence, though, from the rotation of galaxies, that there is a lot more “dark matter” present there than we see as stars. It is not clear what this dark matter is — black holes, very dim stars, clouds of neutrinos — but when we are examining the future of the Universe, we don’t care. All we worry about is the amount. And that amount, from galactic dynamics, could be at least ten times as much as the visible matter. Enough to bring the density to 0.1, or possible even 0.2. But no more than that.

One might say, all right, that’s it. There is not enough matter in the Universe to stop the expansion, by a factor of about ten, so we have confirmed that we live in a forever-expanding Universe. Recent (1999) observations seem to confirm that result.

Unfortunately, that is not the answer that most cosmologists would really like to hear. The problem comes because the most acceptable cosmological models tell us that if the density is as much as 0.1 today, then in the past it must have been much closer to unity. For example, at one second A.C., the density would have had to be within one part in a million billion of unity, in order for it to be 0.1 today. It would be an amazing coincidence if, by accident, the actual density were so close to the critical density.

Most cosmologists therefore say that, today’s observations notwithstanding, the density of the Universe is really exactly equal to the critical value. In this case, the Universe will expand forever, but more and more slowly.

The problem, of course, is then to account for the matter that we don’t observe. Where could the “missing matter” be, that makes up the other nine-tenths of the universe?

There are several candidates. One suggestion is that the Universe is filled with energetic (“hot”) neutrinos, each with a small but non-zero mass. However, there are problems with the Hot Neutrino theory. If they are the source of the mass that stops the expansion of the Universe, the galaxies, according to today’s models, should not have developed as early as they did in the history of the Universe.

What about other candidates? Well, the class of theories already alluded to and known as supersymmetry theories require that as-yet undiscovered particles ought to exist.

There are axions, which are particles that help to preserve certain symmetries (charge, parity, and time-reversal) in elementary particle physics; and there are photinos, gravitinos, and others, based on theoretical supersymmetries between particles and radiation. These candidates are slow moving (and so considered “cold”) but some of them have substantial mass. They too would have been around soon after the Big Bang. These slow-moving particles clump more easily together, so the formation of galaxies could take place earlier than with the hot neutrinos. We seem to have a better candidate for the missing matter — except that no one has yet observed the necessary particles. At least neutrinos are known to exist!

Supersymmetry, in a particular form known as superstring theory, offers another possible source of hidden mass. This one is easily the most speculative. Back at a time, 10^-43 seconds A.C., when gravity decoupled from everything else, a second class of matter may have been created that is able to interact with normal matter and radiation, today, only through the gravitational force. We can never observe such matter, in the usual sense, because our observational methods, from ordinary telescopes to radio telescopes to gamma ray detectors, all rely on electromagnetic interaction with matter.

This “shadow matter” produced at the time of gravitational decoupling lacks any such interaction with the matter of the familiar Universe. We can determine its existence only by the gravitational effects it produces, which, of course, is exactly what we need to “close the Universe,” and also exactly what we needed for the fifth chronicle.

One can thus argue that the fifth chronicle is all straight science; or, if you are more skeptical, that it and the theories on which it is based are both science fiction. I think that I prefer not to give an opinion.

Invariance and science.

In mathematics and physics, an invariant is something that does not change when certain changes of condition are made. For example, the “connectedness” or “connectivity” of an object remains the same, no matter how we deform its surface shape, provided only that no cutting or merging of surface parts is permitted. A grapefruit and a banana have the same connectedness — one of them can, with a little effort, be squashed to look like the other (at least in principle, though it does sound messy). A coffee cup with one handle and a donut have the same connectedness; but both have a different connectedness from that of a two-handled mug, or from a mug with no handle. You and I have the same connectedness — unless you happen to have had one or both of your ears pierced, or wear a ring through your nose.

The “knottedness” of a piece of rope is similarly unchanging, provided that we keep hold of the ends and don’t break the string, There is an elaborate vocabulary of knots. A “knot of degree zero” is one that is equivalent to no knot at all, so that pulling the ends of the rope in such a case will give a straight piece of string — a knot trick known to every magician. But when Alexander the Great “solved” the problem of the Gordian Knot by cutting it in two with his sword, he was cheating.

Invariants may sound useless, or at best trivial. Why bother with them? Simply for this reason: they often allow us to make general statements, true in a wide variety of circumstances, where otherwise we would have to deal with lots of specific and different cases.

For example, the statement that a partial differential equation is of elliptic, parabolic, or hyperbolic type is based on a particular invariant, and it tells us a great deal about the possible solutions of such equations before we ever begin to try to solve them. And the statement that a real number is rational or irrational is invariant, independent of the number base that we are using, and it too says something profound about the nature of that number.

What about the invariants of physics, which interested McAndrew? Some invariants are so obvious, we may feel they hardly justify being mentioned. For example, we certainly expect the area or volume of a solid body to be the same, no matter what coordinate system we may use to define it.

Similarly, we expect physical laws to be “invariant under translation” (so they don’t depend on the actual position of the measuring instrument) and “invariant under rotation” (it should not matter which direction our experimental system is pointing) and “invariant under time translation” (we ought to get the same results tomorrow as we did yesterday). Most scientists took such invariants for granted for hundreds of years, although each of these is actually making a profound statement about the physical nature of the Universe.

So, too, is the notion that physical laws should be “invariant under constant motion.” But assuming this, and rigorously applying it, led Einstein straight to the theory of special relativity. The idea of invariance under accelerated motion took him in turn to the theory of general relativity.

Both these theories, and the invariants that go with them, are linked inevitably with the name of one man, Albert Einstein. Another great invariant, linear momentum, is coupled in my mind with the names of two men, Galileo Galilei and Isaac Newton. Although the first explicit statement of this invariant is given in Newton’s First Law of Motion (“Every body will continue in its state of rest or of uniform motion in a straight line except in so far as it is compelled to change that state by impressed force.”), Galileo, fifty years earlier, was certainly familiar with the general principle.