Chapter I 1 2 3 4 5 6

Borderlands of Science

Copyright © 1999

by Charles Sheffield


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 lightyears) 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 Döppler Effect. According to the Döppler 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 Döppler 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 constant 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 redshift 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 lightyears 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 lightyears, we will be observing the early history of the universe.

There is one big built-in assumption here: the observed red shift has to be associated with a velocity of recession, and therefore with distance. One mysterious class of objects with large red shifts has led some people to question that assumption. These are the quasars (a contraction of quasi-stellar radio source, or quasi-stellar object).

Quasars are characterized by their large red shifts, which suggests they are very distant; and by their brightness, which means they have a very high intrinsic luminosity at least comparable with a galaxy. And they are small. We know this not because they fail to show a distinct disc, which is not surprising at their presumed distances, but because their variations in light patterns take place over such short periods that we know they cannot be more than a few light-hours across. That is no more than the size of our own solar system.

The big question is, how can something so small be so bright?

The only mechanism that anyone has been able to suggest is of a massive black hole (a hundred million times the mass of our sun, or more) into which other matter is falling. This proves an extraordinarily efficient way of creating lots of energy. Almost half the mass of the in-falling matter can be converted to pure radiation. If one or two stars a year were to fall into a monster black hole, that would be enough to power the quasar.

There are, however, reputable scientists who do not believe this explanation at all. According to them, quasars are not at galactic distances. They are much closer, much smaller and less bright, and the red shift of their light is due to some other cause.

What other cause? We will mention one possibility in Chapter 13. Meanwhile, we assume the validity of the Big Bang model.

4.3 Early days. "Oh, call back yesterday," said Salisbury, in Shakespeare's Richard the Second. "Bid time return."

What was the universe like, ten or twenty billion years ago, when it was compressed into a very small volume? Surprisingly, we can deduce a good deal about those early days. The picture is a coherent one, consistent with today's ideas of the laws of physics. It also, quite specifically, says something about the formation of elements during those earliest times.

Like much of twentieth century physics, the story begins with Albert Einstein. After he had developed the general theory of relativity and gravitation, he and others used it in the second decade of this century to study theoretical models of the universe. Einstein could construct a simple enough universe, with matter spread through the whole of space. What he could not do was make it sit still. The equations insisted that the model universe either had to expand, or it had to contract.

To make his model universe stand still, Einstein introduced in 1917 a new, and logically unnecessary, "cosmological constant" into the general theory. With that, he could build a stable, static universe. He later described the introduction of the cosmological constant, and his refusal to accept the reality of an expanding or contracting universe, as the biggest blunder of his life.

When Hubble's work showed the universe to be expanding, Einstein at once recognized its implications. However, he himself did not undertake to move in the other direction, and ask about the time when the contracted universe was far more compact than it is today. That was done by a Belgian, Georges Lemaître. Early in the 1930's Lemaître went backwards in time, to the period when the whole universe was a "primeval atom." In this first and single atom, everything was squashed into a sphere only a few times as big as the Sun, with no space between atoms, or even between nuclei. As Lemaître saw it, this unit must then have exploded, fragmenting into the atoms and stars and galaxies and everything else in the universe that we know today. He might justifiably have called it the Big Bang, but he didn't. That name was coined by Fred Hoyle (the same man who did the fundamental work on nucleosynthesis) in 1950. It is entirely appropriate that Hoyle, whose career has been marked by colorful and imaginative thinking, should have named the central event of modern cosmology. And it is ironic that Hoyle himself, as we will see in Chapter 13, denies the reality of the Big Bang.

Lemaître did not ask about the composition of the primeval atom. It might be thought that the easiest assumption is that everything in the universe was already there, much as it is now. But that cannot be true, because as we go back in time, the universe had to be hotter as well as more dense. Before a certain point, atoms as we know them could not exist; they would be torn apart by the intense radiation that permeated the whole universe.

The person who did worry about the composition of the primeval atom was George Gamow. In the 1940's, he conjectured that the original stuff of the universe was nothing more than densely packed neutrons. Certainly, it seemed reasonable to suppose that the universe at its outset had no net charge, since it seems to have no net charge today. Also, a neutron left to itself will decay radioactively, to form an electron and a proton. One electron and one proton form an atom of hydrogen; and even today, the universe is predominantly hydrogen atoms. So neutrons could account for most, if not all, of today's universe.

If the early universe was very hot and very dense and all hydrogen, some of it ought to have fused and become helium, carbon, and other elements. The question, How much of each?, was one that Gamow and his student, Ralph Alpher, set out to answer. They calculated that about a quarter of the matter in the primeval universe should have turned to helium, a figure consistent with the present composition of the oldest stars.

What Gamow and Alpher could not do, and what no one else could do after them, was make the elements heavier than helium. In fact, Gamow and colleagues proved that heavier element synthesis did not take place. It could not happen very early, because in the earliest moments, elements would be torn apart by energetic radiation. At later times, the universe expanded and cooled too quickly to provide the needed temperatures.

Heavier element formation has to be done in stars, during the process known as stellar nucleosynthesis. The failure of the Big Bang to produce elements heavier than helium confirms something that we already know, namely, that the Sun is much younger than the universe. Sol, at maybe five billion years old, is a second, third, or even fourth generation star. Some of the materials that make up Sun and Earth derive from older stars that ran far enough through their evolution to produce the heavier elements by nuclear fusion and in supernovas.

4.4 All the way back. We are now going to run time backward toward the Big Bang. (Note: this section draws heavily from the book, THE FIRST THREE MINUTES (Weinberg, 1977). I strongly recommend the original).

Where should we start the clock? Well, when the universe was smaller in size, it was also hotter. In a hot enough environment, atoms as we know them cannot hold together because high-energy radiation rips them apart as fast as they form. A good time to begin our backward running of the clock is the time when atoms could form and persist as stable units. Although stars and galaxies would not yet exist, at least the universe would be made up of familiar components: hydrogen and helium atoms.

Atoms formed, and held together, somewhere between half a million and a million years after the Big Bang. Before that time, matter and radiation interacted continuously. Afterward, radiation could not tear matter apart as fast as it was formed. The two "de-coupled", or nearly so, became quasi-independent, and went their separate ways. Matter and radiation still interacted (and do so to this day), but more weakly. The temperature of the universe when this happened was about 3,000 degrees. Ever since then, the expansion of the universe has lengthened the wavelength of the background radiation, and thus lowered its temperature. The cosmic background radiation discovered by Penzias and Wilson, at 2.7 degrees above absolute zero, is nothing more than the radiation at the time when it decoupled from matter, now grown old.

Continuing backwards: before atoms could form, helium and hydrogen nuclei and free electrons could exist; but they could not combine to make atoms, because radiation broke them apart. The form of the universe was, in effect, controlled by radiation energetic enough to prevent atom formation. This situation held from about three minutes to one million years A.C. (After Creation).

If we go back before three minutes A.C., radiation was even more dominant. It prevented the build-up even of helium nuclei. As noted earlier, the fusion of hydrogen to helium requires hot temperatures, such as we find in the center of stars. But fusion cannot take place if it is too hot. For helium nuclei to form, three minutes after the Big Bang, the universe had to "cool" to about a billion degrees. All that existed before this time were electrons (and their positively charged forms, positrons), neutrons, protons, neutrinos, and radiation.

Until three minutes A.C., you might think that radiation controlled events. Not so. As we proceed backwards and the temperature of the primordial fireball continues to increase, we reach a point where the temperature is so high (above ten billion degrees) that large numbers of electron-positron pairs are created from pure radiation. That happened from one second to 14 seconds A.C. After that, the number of electron-positron pairs decreased rapidly, because less were being generated than were annihilating themselves and returning to pure radiation. When the universe "cooled" to ten billion degrees, neutrinos also decoupled from other forms of matter.

We have a long way to go, physically speaking, to the moment of creation. As we continue backwards, temperatures rise and rise. At a tenth of a second A.C., the temperature of the universe is 30 billion degrees. The universe is a soup of electrons, protons, neutrons, neutrinos, and radiation. However, as the kinetic energy of particle motion becomes greater and greater, effects caused by differences of particle mass are less important. At 30 billion degrees, an electron easily carries enough kinetic energy to convert a proton into the slightly heavier neutron. In this period free neutrons are constantly decaying to form protons and electrons, but energetic proton-electron collisions undo their work by re-making neutrons.

The clock keeps running backward. The important time intervals become shorter and shorter. At one ten-thousandth of a second A.C., the temperature is one thousand billion degrees. The universe is so small that the density of matter, everywhere, is as great as that in the nucleus of an atom (about 100 million tons per cubic centimeter; a fair-sized asteroid, at this density, fits in a thimble). The universe is a sea of quarks, electrons, neutrinos, and energetic radiation.

We can go farther, at least in theory, to the time, 10-35 seconds A.C., when the universe went through a super-rapid "inflationary" phase, growing from the size of a proton to the size of a basketball in about 5 x 10-32 seconds. We can even go back to a time 10-43 seconds A.C. (termed the Planck time), when according to a class of theories known as supersymmetry theories, the force of gravity decoupled from everything else, and remains decoupled to this day.

The times mentioned so far are summarized in TABLE 4.1. Note that all these times are measured from the moment of the Big Bang, so t = 0 is the instant that the universe came into being.


TABLE 4.1 displays one inconvenient feature. Everything seems to be crowded together near the beginning, and major events become farther and farther apart in time as we come closer to the present. This is even more apparent when we note that the origin of the solar system, while important to us, has no cosmic significance.

Let us seek a change of time scale that will make important events more evenly spaced on the time line. We make a change of the time coordinate, defining a new time, T, by T = log(t/tN), where tN is chosen as 15 billion years, the assumed current age of the universe.

That produces TABLE 4.2. All the entries in it are negative, since we have been dealing so far only with past times. However, the entries for important events, in cosmological terms, are much more evenly spaced in T-time.

We will return to TABLE 4.2 later. Note, however, that we cannot get all the way to the Big Bang in T-time, since that would correspond to a T value of minus infinity. However, a failure to reach infinite pressure and temperature is no bad thing. In T-time, the Big Bang happened infinitely long ago.

The time transformation that we made to T-time has no physical motivation. It gives us a convenient time scale for spacing past events, in terms of a familiar function, but there is no reason to think it will be equally convenient in describing the future.

A value of T = +60.7, which is as far ahead of the present on the T-time scale as the Planck time is behind us, corresponds to a time of 7.5 x 1070 years from now.

Does the future of the universe admit such a time? We shall see.

At this point, however, I want to pause and ask, does it make any sense to go back so far? If we try to press "all the way back" to zero time, we find ourselves faced with a singularity, a time when matter density and temperature tend to infinity. The appearance of infinity in a physical theory is one good way of knowing that there is something wrong -- not with the universe, but with the theory. The most likely problem is that physical laws derived under one set of conditions cannot be applied to grossly different conditions. However, it is also possible that the theory itself is too naive.

In either case, we are already far away from the scientific mainland, well into science fiction waters. We are certainly beyond the realm of the physical laws that we can test today. We are at this stage no more plausible than Archbishop Ussher, convinced that he had pinned down the time of creation.

More to the point, does the early history of the universe make any difference to anything today?

Oddly enough, it does. The early history was crucial in deciding the whole structure of today's universe. Let us see why.

4.5 The missing matter. The universe is expanding. Almost every cosmologist today agrees on that. Will it go on expanding forever, or will it one day slow to a halt, reverse direction, and fall back in on itself in a "Big Crunch"? Or is the universe perhaps poised on the infinitely narrow dividing line between expansion and ultimate contraction, so that it will increase more and more slowly in size and finally (but after infinite time) stop its growth?

We also ought to mention still another possibility, that the universe oscillates, going through endless phases of expansion followed by contraction. This idea, known as kinematic relativity, was developed by E. A. Milne (not, please, to be confused with A. A. Milne), but it has now fallen from favor.

The thing that chooses among the three main possibilities is the total amount of mass in the universe; or rather, since we do not care what form the mass takes, and mass and energy are totally equivalent, the future of the universe is decided by the total mass-energy content per unit volume.

If the mass-energy is too big, the universe will end in the Big Crunch. If it is too small, the universe will fly apart forever. Only in the Goldilocks situation, where the mass-energy is "just right," will the universe ultimately reach a "flat" condition.

The amount of matter needed to stop the expansion of the universe is not large, by terrestrial standards. It calls for only three hydrogen atoms per cubic meter.

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 arbitrarily say that the critical mass-energy density has to be unity to end the expansion after infinite time, we observe a value of only 0.01.

There is evidence, though, from the rotation of galaxies, of more "dark matter" than visible matter. 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.

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 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. And now, I should point out, we are very much into science fiction territory.

One suggestion is that the universe is filled with energetic ("hot") neutrinos, each with a small but non-zero mass (as mentioned earlier, the neutrino is usually assumed to be mass-less). Those neutrinos would be left over from the very early days of the universe, so we are forced back to studying the period soon after the Big Bang. However, there are other problems with the Hot Neutrino theory, because 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.

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 masses. 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. Neutrinos are at least known to exist!

Supersymmetry, in the particular form known as superstring theory, offers one other possible source of hidden mass. This one is easily the most speculative. Back at this time, 10-43 seconds A.C., when gravity decoupled from everything else, a second class of matter may have been created that interacts with normal matter and radiation 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. The "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." Unfortunately, the invocation of shadow matter takes us back to such an early time that if we are sure of anything, it is that the universe then was unrecognizably different from the way that it is today.

I used shadow matter in a story, (THE HIDDEN MATTER OF MCANDREW, Sheffield, 1992). However, I took care to be suitably vague about its properties.

4.6 The end of the universe. "When I dipped into the Future far as human eye could see," said Tennyson in the poem, Locksley Hall. Writing in 1842 he did pretty well, foreseeing air warfare and universal world government. We can go a long way beyond that.

Let's start with the "near-term" future. We can model mathematically the evolution of our own sun. In the near-term (meaning in this case the next few billion years) the results are unspectacular. The sun is a remarkably stable object. It will simply go on shining, becoming slowly brighter. Five billion years from now it will be twice its present diameter, and twice as bright. Eventually, however, it will begin to deplete its stock of hydrogen. At that point it will not shrink as one might expect, but begin to balloon larger and larger. Eight billion years in the future, the sun will be two thousand times as bright, and it will have grown so big (diameter, a hundred million miles) that its sphere will fill half our sky. The oceans of Earth will long since have evaporated, and the land surface will be hot enough to melt lead.

That far future Sun, vast, stationary and dim-glowing in the sky of an ancient Earth, was described by H.G. Wells in one of the most memorable scenes in science fiction (THE TIME MACHINE, 1895). The details are wrong - his future Earth is cold, not hot - but the overall effect is incredibly powerful. If you have not read it recently, it well repays re-reading.

In studying the long-term future of the sun, we have as an incidental dealt with the future of the Earth. It will be incinerated by the bloated sun, which by that time will be a red giant. The sun, as its energy resources steadily diminish even farther, will eventually blow off its outer layers of gas and shrink to end its life, ten billion years from now, as a dense white dwarf star not much bigger than today's Earth.

None of this should be a problem for humanity. Either we will be extinct, or long before five billion years have passed we will have moved beyond the solar system. We can, if we choose, go to sit around a smaller star. It will be less prodigal with its nuclear fuel, and we can enjoy its warmth for maybe a hundred billion years. By that time the needs of our descendants will be quite unknowable.

However, before that time something qualitatively different may have happened to the universe. Just possibly, it will have ended. We know that the universe is open, closed, or flat, but no one knows which. We must examine all three alternatives.

4.7 The Big Crunch. We begin with the case of the closed universe, which is in many ways the least appealing. It has to it a dreadful feeling of finality — though it is not clear why a human being, with a lifetime of a century or so, should be upset by events maybe fifty to a hundred billion years in the future.

The Big Crunch could happen as "soon" as 50 billion years from now, depending on how much the average mass-energy of the universe exceeds the critical amount. We know from observation that the mass-energy density is not more than twice the critical density. In that limiting case we will see about 17.5 billion more years of expansion, followed by 32.5 billion years of collapse. A smaller mass-energy density implies a longer future.

Not surprisingly, T-time is inappropriate to describe this future. The logarithm function has a singularity at t = 0, but nowhere else. An appropriate time for the closed universe contains not one singularity (T = -¥, the Big Bang), but two (T = -¥, the Big Bang, and T = +¥, the Big Crunch). As the universe approaches its end, the events that followed the Big Bang must appear in inverse order. There will come a time when atoms must disappear, when helium splits back to hydrogen, when electron/positron pairs appear, and so on.

A reasonable time transformation for the closed universe is Tc = log(t/(C-t)), where C is the time, measured from the Big Bang, of the Big Crunch.


TABLE 4.3 shows how this transformation handles significant times of the past and future. In this case, we have chosen Tc = 0 as the midpoint in the evolution of the universe, equally far from its beginning and its end. For past times, the values are very similar to those obtained with T-time. For future times close to the Big Crunch, T-time and Tc-time are radically different. As the universe is collapsing to its final singularity Tc-time is rushing on to infinity, but the hands of the T-time clock would hardly be moving.

Tc is a plausible time to describe the evolution of a closed universe. When t tends to zero, Tc tends to minus infinity, and when t tends to C, Tc tends to plus infinity. Thus both end points of the universe are inaccessible in Tc-time. The transformation is symmetric about the "mid-point" of the universe, t = C/2. This does not mean, as is sometimes said, that time will "run backwards" as the Universe collapses. Time continues to run forward in either t-time or Tc-time, from the beginning of the Universe to its end. Note also that Tc has no real values, and hence no meaning, for times before the Big Bang or after the Big Crunch.

Since the collapse applies to the whole universe, there is no escape — unless one can find a way to leave this universe completely, or modify its structure. I dealt with both those possibilities in the novel TOMORROW AND TOMORROW (Sheffield, 1997).

4.8 At the eschaton. I want to mention another aspect of the end of the universe, something that appears only in the case where it is closed. Consider the following statement:

The existence of God depends on the amount of matter in the universe.

That is proposed, as a serious physical theory, by Frank Tipler. It was the subject of a paper (Tipler, 1989) and a later book (Tipler, 1994). Both concern the "eschaton." That is the final state of all things, and it therefore includes the final state of the universe.

Tipler argues that certain types of possible universes allow a physicist to deduce (his own term is prove) the ultimate existence of a being with omnipresence, omniscience, and omnipotence. This being will have access to all the information that has ever existed, and will have the power to resurrect and re-create any person or thing that has ever lived. Such a being can reasonably be called God.

The universe that permits this must satisfy certain conditions:

1) the universe must be such that life can continue for infinite subjective time.

2) spacetime, continued into the future, must have as a boundary a particular type of termination, known as a c-boundary.

3) the necessary c-boundary must consist of a single point of space-time.

Then, and only then, according to Tipler, God with omnipresence, omniscience, and omnipotence can be shown to exist.

Conditions 2) and 3) are satisfied only if the universe is closed. It cannot be expanding forever, or even asymptotically flat, otherwise the theory does not work. The choice, open or closed, depends as we already noted on the mass-energy density of the universe.

The definition of "omnipotent" now becomes extremely interesting. Would omnipotence include the power to avoid the final singularity, by changing the universe itself to an open form?

I like to think so, and in TOMORROW AND TOMORROW I took that liberty.

When the question of missing matter and the closed or open universe was introduced, it seemed interesting but quite unrelated to the subject of religion. Tipler argues that the existence of God, including the concepts of resurrection, eternal grace, and eternal life, depends crucially on the current mass-energy density of the universe.

We already noted the surprising way in which the observation of those remote patches of haze, the galaxies, showed that the universe began a finite time ago. That was a striking conclusion: simple observations today defined the far past of the universe.

Now we have a still stranger notion to contemplate. The search for exotic particles such as "hot" neutrinos and "cold" photinos and axions will tell us about the far future of the universe; and those same measurements will have application not only to physics, but to theology.

4.9 Expansion forever. Suppose that the universe is open rather than closed. Then it will expand forever.

Freeman Dyson was the first to analyze this situation (Dyson, 1979). First, all ordinary stellar activity, even of the latest-formed and smallest suns, will end. That will be somewhat less than a million billion (say, 1014) years in the future. After that it is quiet for a while, because everything will be tied up in stellar left-overs, neutron stars and black holes and cold dwarf stars.

Then the protons in the universe begin to decay and vanish.

That requires a word of explanation. A generation ago, the proton was thought to be an eternally stable particle, quite unlike its cousin, the unstable free neutron. Then a class of theories came along that said that protons too may be unstable, but with a vastly long lifetime. If these theories are correct, the proton has a finite lifetime of at least 1032 years. In this case, as the protons decay all the stars will finally become black holes.

The effect of proton decay is slow. It takes somewhere between 1030 and 1036 years before the stellar remnants are all black holes. Note that on this time scale, everything that has happened in the universe so far is totally negligible, a tick at the very beginning. The ratio of the present age of the universe to 1036 years is like a few nanoseconds compared with the present age of the universe.

In terms of T-time, the stellar remnants collapse to form black holes between T = 19.8 and T = 25.8. The T-transformation still does pretty well in describing the open universe.

Long after the protons are all gone, the black holes go, too. Black holes evaporate, as we saw in Chapter 3. Today, the universe is far too hot for a black hole of stellar mass to be able to lose mass by radiation and particle production. In another 1064 years or so that will not be true. The ambient temperature of the expanding universe will have dropped and dropped, and the black holes will evaporate. Those smaller than the Sun in mass will go first, ones larger than the Sun will go later; but eventually all, stars, planets, moons, clouds of dust, everything, will turn to radiation.

In this scenario, the universe, some 1080 years from now, will be an expanding ocean of radiation, with scattered within it a possible sprinkling of widely-separated electron-positron pairs.

The idea of proton decay is controversial, so we must consider the alternative. Suppose that the proton is not an unstable particle. Then we have a rather different (and far longer) future for the universe of material objects.

All the stars will continue, very slowly, to change their composition to the element with the most nuclear binding energy: iron. They will be doing this after some 101600 years.

Finally (though it is not the end, because there is no end) after somewhere between 10 to the 1026 and 10 to the 1076 years, a time so long that I can find no analogy to offer a feel for it, our solid iron neutron stars will become black holes. Now our T-time scale also fails us. A t-time of 10 to the 1026 years corresponds to T = 1026, itself a number huge beyond visualization.

Is this the end of the road? No. The black holes themselves will disappear, quickly on these times scales. The whole universe, as in the previous scenario, becomes little more than pure radiation. This all-encompassing bath, feeble and far-diluted, is much too weak to permit the formation of new particles. A few electron-positron pairs, far apart in space, persist, but otherwise radiation is all.


TABLES 4.4, 4.5 and 4.6 show the calendar for the future in "normal" t-time, for the closed and open universes with the unstable and stable proton. Time is measured from today, rather than the beginning of the universe.

4.10 Life in the far future. There is something a little unsatisfactory about the discussion so far. A universe, closed or open, without anyone to observe it, feels dull and pointless.

What are the prospects for observers and conscious participants, human or otherwise? We will certainly not equate "intelligence" with "humanity," since over the time scales that we have encountered, the idea that anything like us will exist is remotely improbable.

Let us note that, on the cosmological scale, life as we know it on Earth has a respectable ancestry. Life emerged quite early in this planet's lifetime, about three and a half billion years ago, so life is now about a fourth as old as the universe itself.

Land life appeared much later, 430 million years ago for simple plants. The first land animals came along a few tens of millions of years later. Mammals have existed for maybe 225 million years, and flowering plants about a hundred million. Recognizable humans, with human intelligence, appeared a mere three or four million years ago. We are upstarts, in a universe where ordinary turtles have been around, essentially unchanged in form and function, for a couple of hundred million years. Perhaps that is why we lack the calm certainty of the tortoise.

Humans have a short past, but we could have a long future. We have already taken care of the "near-term" future. The Earth should remain habitable (unless we ourselves do something awful to it) for a few billion years. After that we can head for a dwarf star, and be comfortable there for another thirty to a hundred billion years. Dwarf stars shine dimly, but a planet or free-space colony in orbit a few million miles away from one will find more than enough energy to support a thriving civilization; and of all the stars in the universe, the inconspicuous, long-lived dwarf stars appear to form the vast majority.

Earth, of course, will be gone — unless perhaps our descendants, displaying a technology as far beyond ours as we are beyond the Stone Age, decide to take the home planet along on their travels for sentimental reasons.

If we are to consider longer time scales, beyond thirty billion years, we must distinguish between the cases of an open and a closed universe.

In the universe of the Big Crunch it seems obvious that life and intelligence cannot go on forever. The future contains a definite time at which everything in existence will be compressed to a single point of infinite pressure and temperature. If we continue to measure time in the usual way, life can exist for a finite time only. However, we have already noted that in Tc-time, even the Big Crunch is infinitely far away. Although the transformation that we introduced seemed like a mere mathematical artifice, it can be shown that there is enough time (and available energy) between now and the Big Crunch to think an infinite number of thoughts. From that point of view, if we work with subjective time, in which life survives long enough to enjoy infinite numbers of thoughts, that will be like living "forever" according to one reasonable definition. It is all a question of re-defining our time coordinates.

The open universe case has no problem with available time, but it does have a problem with available energy. In the far future our energy sources will become increasingly diluted and distant.

Dyson has also analyzed this situation (personal communication, Dyson, 1992). He has examined the possibility of continued life and intelligence for the case of an asymptotically flat spacetime, where the universe sits exactly on the boundary of the open and closed cases. I have not seen the details of his analysis, and to my knowledge they have not been published. Here, however, are his conclusions.

First, hibernation will be increasingly necessary. The fraction of time during which a thinking entity can remain "conscious" must become less, like t-1/3. Also, the thinking rate must decrease, so that "subjective time" will proceed more slowly, like t-1/3. To give an example of what this implies, one million billion years from now you will be able to remain awake for only ten years out of each million. And during those ten years, you will only be able to do as much thinking as you can do now in one hour. There will be no more "lightning flashes of wit." Instead it will all be Andrew Marvell's "vaster than empires and more slow." All thought must be "cool calculation."

The good news is that you have an indefinitely long time available, so that you can eventually think an infinitely large number of thoughts.

Curiously enough, in an ultimately flat universe an infinite number of thoughts can be thought with the use of only a finite amount of energy. That's just as well, because in such a universe free energy becomes less and less easy to come by as time goes on.


Event time, t
Gravity decouples 10-43 seconds
Inflation of universe 10-35 seconds
Nuclear matter density 0.0001 seconds
Neutrinos decouple 1 second
Electron/positron pairs vanish 30 seconds
Helium nuclei form 3.75 minutes
Atoms form 1 million years
Galaxy formation begins 1 billion years
Birth of solar system 10 billion years
Today 15 billion years

T = log(t/tN), where tN is chosen as 15 billion years

Event T
Big Bang -Infinity
Gravity decouples -60.7
Inflation of universe -52.7
Nuclear matter density -21.7
Neutrinos decouple -17.7
Electron/positron pairs vanish -16.2
Helium nuclei form -15.3
Atoms form -4.2
Galaxy formation begins -1.2
Birth of solar system -0.2
Today 0.




Tc = log(t/log(C-t))

Event Tc
Big Bang -Infinity
Gravity decouples -61.31
Inflation of universe -52.31
Nuclear matter density -22.31
Neutrinos decouple -18.31
Electron/positron pairs vanish -16.83
Helium nuclei form -15.95
Atoms form -4.81
Galaxy formation begins -1.81
Birth of solar system -0.74
Today -0.52
"Halfway" point(32.5 billion yrs) 0
Helium dissociates +15.95
Electron/positron pairs form +16.83
Neutrinos couple +18.31
Gravity couples +61.31
End point (65 billion years) +Infinity


Event t (billions
of years)
Today 0
Sun becomes red giant 5
Halfway point, expansion ceases 17
Most dwarf stars cease to shine 30
Big Crunch 50



Event t (years)
Today 0
Sun becomes red giant 5 billion
Most dwarf stars cease to shine 30 billion
All stellar activity ceases 1014
Stellar remnants become black holes 1030 - 1036
Black holes evaporate 1064
Radiation only 1080


Event t (years)
Today 0
Sun becomes red giant 5 billion
Most dwarf stars cease to shine 30 billion
All stellar activity ceases 1014
Stars are iron neutron stars 101600
Neutron stars form black holes >10 to the 1026
Radiation only 10 to the 1076

Copyright © 1999 by Charles Sheffield
Chapter I 1 2 3 4 5 6

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Baen Books 02/02/03