Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction

by Adler, Charles L.

  The implication of the accelerating universe is that it will never go through a “Big Crunch” in which the universe halts its expansion and recollapses. (I preferred Douglas Adams’s term “Gnab Gib” over “Big Crunch,” but the point is moot now.) These new discoveries invalidate narratives like Tau Zero that are predicated on a cycle of Big Bang–Big Crunch–Big Bang, and so on. Such a model was aesthetically pleasing, but it seems that the universe is just not like that.

  Among science fiction writers, Olaf Stapledon was probably the first to consider the very long-term future history of the universe. We’ve discussed Star Maker in an earlier chapter [225]. Stapledon implicitly used the idea of the open universe in Star Maker; at the end of time, the universe consisted mostly of galaxies too far away to communicate with each other, each consisting of a few remnants of the once great stellar populations. The long-term evolution of the universe has made its way into a number of science fiction books since Stapledon. Most writers have seemed to favor the “closed universe,” or cyclical universe, as Tau Zero does, as it seems to envision an infinite future. Even if humanity can’t survive forever, perhaps some form of intelligent life can evolve in the next cycle of the universe.

  Main-sequence lifetime is tied to stellar mass. For example, in about four or five billion years our Sun will move off the main sequence and swell into a red giant whose luminosity will exceed 100 times its present value. As it burns through its nuclear fuel, it will cool and contract to a white dwarf whose total luminosity will be initially about 10−3 what it is today, and will further cool over time. Higher mass stars than the Sun burn through their nuclear fuel quickly and end their lives in spectacular supernova explosions. Low-mass stars live significantly longer than high-mass ones. Luckily, most stars in the sky are low-mass M-class stars with low luminosity but very long lives. This allows the possibility (as in Asimov’s story) of moving to a new star once the Sun has evolved into a white dwarf. Stellar lifetimes are given by the approximate formula

  where M is the stellar mass and L its luminosity. From chapter 14, an M7 star has M = 0.08 and L = 0.0025, leading to a main-sequence lifetime of 3.2 × 1011 (320 billion) years. The estimate is probably an underestimate of their true lifetime. Our Sun will run through only about 10% of its hydrogen before swelling into a red giant. This is because the Sun’s core isn’t convective, that is, it isn’t stirred around by currents created by temperature gradients. This means that the core, where fusion takes place, can’t get any new material to fuse once the core’s supply is gone. M-class red dwarf stars have fully convective cores, meaning that new fuel is mixed in constantly. This increases their main-sequence lifetime above our crude estimate [47]. Stars at the lowest end of the mass range may have lifetimes near 1013 years.

  The long-term issue we face is similar to the short-term issue our civilization is facing today: resource depletion. We are rapidly using up the Earth’s reserve of fossil fuels now. In this imaginably distant future we will be using up the resources of the stars. If human civilization lasts for a few billion years, I feel confident in predicting that we will have some form of interstellar travel by then, because we will need it. Maybe we can move our entire planet to a new solar system. Both Freeman Dyson and Larry Niven have speculated on methods of doing this.

  21.5 BLACK HOLE–POWERED CIVILIZATIONS

  When all the stars die, that’s the end, right? All other sources of energy are gone, aren’t they? Well, maybe not. Black holes will be around for a long time after all the stars grow cold, and they offer the potential of providing energy in those end times (which may last for a much, much longer time than all of the other eras in the universe).

  A black hole may seem like a bad source of energy. After all, the common conception of them is that they swallow everything that enters them, and nothing can come out. However, that’s not quite true: as we saw in an earlier chapter, black holes radiate away energy (Hawking radiation), even though it is too low to detect directly.

  There are two ways to retrieve energy from things dropped into black holes:

  1. By electromagnetic radiation from the things dropped into them via heating in the accretion disk; and

  2. By gravitational radiation.

  The first process is relatively straightforward: the radius for the event horizon of a (nonrotating) black hole is determined only by the mass of the black hole:

  where Ms is the Sun’s mass. Therefore, a small black hole ten times the mass of the Sun would have an event horizon 15 km in radius. Imagine dropping a 1 kg object from a long distance away toward the black hole and abruptly stopping it three radii away. In its fall it would acquire kinetic energy approximately equal to 1/6 mc2 = 1.5 × 1016 J. I am using the Newtonian formula here, which is an approximation to the full relativistic formula. I chose three radii from the black hole for two reasons:

  1. Three radii away is the closest distance at which stable orbits are possible [235].

  2. It is also the closest radius at which I feel comfortable using Newtonian formulas to estimate the energy liberated.

  This is an enormous amount of energy; we could power all of current-day America’s energy needs using 600 kg of trash, assuming we could reclaim the energy liberated at 100% efficiency. This is essentially the method that astronomers have used to find black holes that have evolved from large stars: if a black hole is in close orbit with a normal star, gases from the star will be funneled into the black hole. Frictional heating of the gases as they fall in leads to them reaching temperatures in the millions to hundreds of millions of degrees, radiating away enormous amounts of energy in the x-ray spectrum.

  A more subtle way of generating energy using black holes is via gravitational waves. Gravitational waves are literally ripples in the fabric of space and time. It has been predicted that merging black holes are a strong source of gravitational waves. This is one of the key things the Laser Interferometer Gravitational-wave Observatory (LIGO) gravity wave detector is looking for. In principle, up to 50% of the mass-energy content of any junk dropped into a black hole can be recovered as useful energy [236]. This is a much higher efficiency than any other known source of energy.

  The science fiction idea of powering objects using black holes dates back to the 1980s, if not earlier. The earliest use I know of is in the McAndrew chronicles, a series of stories written by Charles Sheffield centering on the eponymous astrophysicist [217]. In these stories, mini-black holes are used to power spacecraft, and presumably other things as well. The discovery of Hawking radiation makes the stories obsolete, as the mini-black holes of the stories would evaporate too quickly to use. The final episode of the new Battlestar Galactica series had the Cylons in orbit around a black hole, powering their civilization by throwing trash into it.

  This is the method by which a civilization past the death of all of the stars could get energy: take your unused trash and toss it into the black hole. This leads to various baroque speculations and plot ideas. One can imagine some far-distant civilization powering their energy needs for the upcoming year by tossing a sacrificial virgin or two into the black hole. It would certainly work better than tossing them into a volcano.

  21.6 PROTONS DECAY—OR DO THEY?

  One thing that may cut short our joyous spree into the forever is the possibility that ordinary matter may softly and silently vanish away. This Boojum is the Snark of proton decay.

  The proton is one of the three building blocks of ordinary matter. Of the other two constituents, the electron is the lightest—about 1,800 times lighter than the proton. It is a stable particle. The neutron is not: the neutron is a charge-neutral particle made up of one “up” quark, with charge +2/3 of the electron charge, and two “down” quarks, each with charge −1/3. When outside the nucleus, the neutron can decay into a proton, electron, and antielectron neutrino; the decay converts one of the down quarks into an up quark, which is what turns the neutron into a proton. The proton is two ups and one down, with net charge of +1. The neutron deca
y time is long by physics standards, taking tens of seconds. Because protons aren’t elementary particles either, the possibility exists that they could decay into lighter particles as well.

  This has never been seen experimentally, but some theories of physics such as string theory predict it. If protons decay, they take a very long time to do so. The universe has been around for 13.7 billion years, so this puts a lower limit on the time it takes. If protons decayed much faster than this, we wouldn’t be around. Because it’s so hard to calculate anything using string theory, there are no very good predictions for the proton decay rate. However, experiments set the proton lifetime as being greater than 1034 years. A friend of mine once made the comment that that was a pretty good definition of forever, but it’s not good enough for our purposes!

  21.7 A GOOGOL YEARS—ALL THE BLACK HOLES EVAPORATE

  If protons don’t decay, then the ultimate lifetime of life in the universe may be set by the timescale it takes for black holes to evaporate. Black hole evaporation is a quantum mechanical phenomenon. If we attempt to put a particle inside a black hole, quantum mechanics tells us there is a small probability that one will find it outside the hole. This is because of the Heisenberg uncertainty principle. We work out the basics of this phenomenon in the web problems. The bottom line is that in 1974, Stephen Hawking showed that black holes aren’t completely black. They act as blackbody radiators, though at very low temperature. A small amount of energy leaks out, and one can even assign a temperature to them:

  where M is the mass of the black hole, and the other terms have been defined previously in this book. A black hole with the mass of the Sun would have a temperature of only 6 × 10−8 K, and larger ones would have lower temperatures. Still, the standard formulas for blackbody radiation apply, even to such exotic objects. One can show that the rate at which the black hole radiates away energy is proportional to 1/M2. A completely isolated black hole will spontaneously radiate away energy. Its mass will decrease. As its mass decreases, it will radiate away energy at a higher rate, which will cause it to decrease in size more rapidly, leading to an explosion of energy in the last few seconds of its existence. One can calculate the time the black hole will last from this formula:

  A black hole with the mass of the Sun will far outlast all of the stars. A more typical black hole with a mass ten times that of the Sun will last for 2 × 1070 years. But this is peanuts compared to the largest black holes around.

  At the center of each galaxy are ultramassive black holes whose mass can range from about a million times to several billion times the mass of our Sun. Our own galaxy has a relatively modest one with a mass of only 30 million Suns. A civilization in an artificial planet or Dyson net in orbit around this could potentially last for more than 1089 years. This dwarfs the current age of the universe by a huge margin, but we can do better.

  The largest known black holes have a mass of more than 1010 solar masses. They are billions of light-years away, of course, but we have all the time we need to get there. A 70 billion solar mass black hole would have a lifetime of 10100 years—a googol years. There are currently no known black holes with this mass. The largest, discovered in April 2011, has a mass of 21 billion solar masses. However, I’m going to assume we can find a larger one, because writing a googol years is cooler than writing 1098 years.

  To indicate how long a time this is, the current age of the universe is about 1010 years. If the age of the universe so far was represented by, say, the mass of a proton, a googol years would be represented by … what? The mass of all the grains of sand on all the world’s beaches? No. The mass of the Earth? No. The Sun? No. The mass of our universe? NO. A googol years would be represented by all of the visible mass in ten billion universes just like the one we are in right now. By the way, one great satisfaction in writing this section is that I have finally found a practical use for the term googol, which has not had much application in mathematics or physics to date, despite a fair amount of commercial (if badly spelled) success.

  I am not the first person to have made these speculations. One of the interesting things about writing this book is that certain names keep popping up. Larry Niven and Poul Anderson are the two science fiction writers whom I have turned to for inspiration many times; among scientists, Freeman Dyson is the clear standout. In a 1974 paper he did the same thing that I am doing here: he calculated how long intelligent life in the universe could last [73]. His conclusions were more optimistic than mine: he concluded that life in the universe could last indefinitely by going through cycles of hibernation and activity, using less and less energy on each active cycle. One point: because of the paper’s date, a number of ideas in it don’t reflect current ideas in cosmology. In particular, the consensus model invalidates a number of his ideas. The paper “A Dying Universe: The Long-Term Fate of Astrophysical Objects” by Fred C. Adams and Gregory Laughlin is a more up-to-date analysis of this idea; I recommend it for the science fiction writer interested in very far-out ideas, as it contains a trove of data and formulas on the subject of the eventual fate of the universe [20]. The article is slightly out of date, as it predates the supernova measurements leading to the concept of the accelerating universe. However, its last section discusses the fate of the universe with non-zero cosmological constant.

  21.8 OUR LAST BOW

  My pen halts, though I do not. Reader, you will walk no more with me. It is time we take up our lives.

  —GENE WOLFE, THE CITADEL OF THE AUTARCH

  Jack Vance, Isaac Asimov, Gene Wolfe, Neil Gaiman, and many, many others have stories set at the end of time. This usually means at the end of the Earth’s lifetime, but some have gone much, much farther than that. As I said at the outset, this book is not meant to be predictive. The same can be said for science fiction itself. I do not expect humankind to last for a googol years; even if it does, it would not exist in any form recognizably human for even a tiny fraction of that time. However, even in the most far-flung stories, humans must remain human if we (as humans) are to sympathize with (or even understand) their actions. This is perhaps the greatest limitation of the literature: the hopes and dreams of one little species don’t amount to a hill of beans in this crazy universe. Or as Neil Gaiman put it, you can have happy endings as long as you end the story early enough.

  I’m finishing this book with a quotation that speaks to me in a deep way. Let it be a metaphor for the best in science fiction as well as the best in humanity. It is from the great French mathematician Henri Poincaré:

  Geologic history shows us that life is only a short episode between two eternities of death, and that even in this episode, conscious thought has lasted and will only last a moment. Thought is only a gleam in the midst of a long night.

  But it is this gleam which is everything [192].

  ACKNOWLEDGMENTS

  Any book is the work of many people, not just the author. Much of the credit for what you are reading goes to a lot of other people. Any mistakes in it are of course mine alone.

  First and foremost, I would like to thank my editor at Princeton University Press, Vickie Kearn, whose support and enthusiasm made this project possible. I would also like to thank Quinn Fusting, Natalie Baan, and Marjorie Pannell for their invaluable help, and all of the other staff at the press and elsewhere who worked on the book.

  I would like to personally thank one of my favorite science fiction writers, Larry Niven, for letting me quote from his letter to Roger Zelazny. I found the quotation in Zelazny’s papers, held in the Azriel Rosenfeld Science Fiction Collection at the Albin O. Kuhn Library at the University of Maryland–Baltimore. I thank the staff of the library, especially the curator, Thomas Beck, for their help. Much of this book was researched and written at the Library of Congress in Washington, D.C.; I thank the staff of the main reading room for their help in retrieving books and articles, and for maintaining such a wonderful place to work.

  The science fiction writers who have influenced me are too numerous to be mentioned here b
y name in every case, but two must be singled out for special credit: Poul Anderson, to whom the book is dedicated, for his essays on writing science fiction, and Olaf Stapledon, who provides the best example of what speculative fiction can do. Among scientists, Freeman Dyson has probably seeded more ideas used in hard science fiction today than any other living physicist.

  My colleague, Mark Vagins of the University of Tokyo, had the initial idea for the section “Thrown for a Loop” in chapter 7. Mark is one of the smartest people I know, and has been a close friend since high school. It is with his kind permission that the analysis is reprinted here. Anthony Bowdoin “Bow” van Riper is an expert on science and popular culture. He gave me invaluable information on infrastructure costs associated with the Space Shuttle program. Raymond Lee, a physicist and meteorologist at the Naval Academy, measured the luminous flux of several candles for chapter 3, “Why Hogwarts Is So dark.” He also discussed subtle points concerning different types of photometric measurements and gave me advice on calculating luminosity and luminous flux from various light sources. I also thank Zeke Kisling, who gave me invaluable advice on formatting and presentation in the book.

  The anonymous reviewers for Princeton University Press pointed out several issues with the book that needed correction. I thank them for their diligent reading, and their support for the book. I also wish to thank Larry Weinstein and Paul Nahin for their reviews of the original proposal.