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

by Adler, Charles L.

  We are left with the uncomfortable feeling that the wave function of correlated particles “collapses” faster than light. Because of relativity, it collapses instantaneously in some reference frames and in reversed time order in others. However, there doesn’t seem to be any way to transmit information using it. The issue comes down to how we define “information.” Because the spin is either up or down with equal probability, Al measures a sequence of random bits on repeated runs of the experiment. He knows that if Bert performs the same measurements, Bert will measure exactly the opposite of what he measures. However, he has no way to know that Bert actually made these measurements until Bert transmits the results of his experiments to Al. This can only be done at the speed of light or slower.

  These results are interesting for two reasons. First, it made physicists define what “information” meant in a stringent way. Even though Al knows the results of Bert’s experiment if Bert makes the measurements, he has no way to tell if Bert made them. If he converts the measurements into a binary code (spin up = 1, down = 0), all he has is a random string of ones and zeros, which by definition carries no information. Second, in recent years physicists have found applications for Bell’s work. There are two main applications of such quantum-correlated measurements. One is in creating secure cryptography. Random strings of ones and zeros are ideal one-time pads for encrypting information. Quantum mechanics guarantees that such strings created by correlated measurements are both perfectly random and uninterceptible [166]. This is good, because quantum correlations are also used in quantum computing. Quantum computing is based on the fact that quantum systems are inherently massively parallel in the language of computer science. Because a particle isn’t in any quantum state until it is measured, it is effectively in all quantum states accessible to it. If we act on the particle by different combinations of physical perturbations without measuring the state, it takes all possible paths available to it. It only takes on a settled value once we measure its state. Entangled states, like Al’s and Bert’s photons, also play a key role in creating quantum computation, although the role that entanglement plays is complicated. If we could build a quantum computer, in principle it could be used to break public key encryption systems.

  Public key encryption is used today to secure information for banks, credit cards, and a large section of the global economy. It is based on the fact that it is much easier to multiply two very large primes than factor the composite number back into them using conventional computers. However, if an effective quantum computer can be built, Shor’s algorithm could be used to factor them, and all of this information becomes vulnerable. To date, the largest number factored by a quantum processor is 21, but it’s a start. All work today on quantum information and quantum computation was motivated by the philosophical and mathematical attempt to reconcile the ideas of relativity theory with quantum mechanics, a process that is far from over.

  I’m ending the section of this book on space travel in the same place we began, with a discussion of the physics of computation. Ideas concerning computation have moved from a peripheral position in physics to its very center, as physicists realized that information is a physical quantity like energy and momentum. It is fascinating to see how these ideas have become intertwined with science fiction ideas like the manned exploration of space and faster-than-light travel. Science can be much weirder than science fiction at times.

  NOTES

  1. Quantum entanglement has also been used as a justification for extrasensory perception, clairvoyance, and other psionic powers. The word “quantum” is often used to justify anything mysterious or improbable in a story.

  2. A tesseract is a four-dimensional hypercube; L’Engle may have gotten the idea of the tesseract as folded space from Heinlein’s short story “… And He Built a Crooked House,” in which an architect builds a house in the shape of a three-dimensional projection of a tesseract, which then folds itself through four dimensions because of an earthquake. I really wanted to live there when I was eleven.

  3. Kinda, sorta.… It’s not a great way to think about it. There’s an interesting paper about what things would look like as seen through a wormhole as a result of the strange behavior of light rays entering it: “Visual Appearance of a Morris-Thorne Wormhole,” by Thomas Muller [171].

  4. Sarah Vowell invented the “grandfather paradox paradox,” which investigates the moral paradox of using violence to stop his violence [244].

  5. This is a relatively conservative view of what a time machine will do. I have explicitly used the idea that causality isn’t violated by time machines. For a discussion of alternative ideas, see Matt Visser’s book, Lorentzian Wormholes [242].

  PART III

  WORLDS AND ALIENS

  CHAPTER FOURTEEN

  DESIGNING A HABITABLE PLANET

  Far too many stories merely give us a planet exactly like Earth except for having neither geography nor history.… The process of designing a world serves up innumerable story points.

  —POUL ANDERSON, “HOW TO BUILD A PLANET”

  A large number of science fiction stories take place on alien worlds, and the process of designing an alien world with alien life on it is perhaps as old as science fiction itself. Hard science fiction writers tend to pay attention to astrophysics and planetary science to make their worlds both realistic and exotic, a very difficult combination to achieve. Several examples of successful designs spring to mind, including Hal Clement’s Mesklin in the novel Mission of Gravity, a large planet whose rapid rotation spun it into an ellipsoid, with a surface gravity three times that of Earth at its poles but several hundred times Earth’s at its equators. There is also Plateau in Larry Niven’s A Gift from Earth, a Venus-like planet with one inhabitable point on it, Mt. Lookitthat, a tall mountain sticking out of most of the atmosphere [59, 175]. The balancing act one must go through to create worlds both credible and incredible simultaneously takes a good deal of work on the writer’s part, but is also part of the fun of the narrative.

  The discussion in this chapter is strongly motivated by the wonderful essay “How to Build a Planet,” by Poul Anderson. The essay was originally published in the Science Fiction Writers of America bulletin in 1966 and later expanded into one of the famous Writers Chapbooks published by Pulphouse Press. My own copy of the essay is a badly worn Xerox copy of unknown origin. I don’t remember when I originally read the essay, only that I was about ten years old and that it was my first exposure to the scientific ideas in science fiction. If I hadn’t read the essay I wouldn’t have written this book. By necessity, this chapter and that essay cover much the same ground, though in different ways. I would recommend that readers who enjoy this book try to track down a copy of the essay.

  14.1 ADLER’S MANTRA

  I’m going to stick to a discussion of Earth-like worlds in this chapter. That means worlds that are capable of supporting life as it exists on Earth, sometimes referred to as “carbon-based life.” Speculations about non-carbon-based life abound, but since I am a physicist rather than a biologist or chemist, I plan to stick closer to what I know and review the physics of life as we know it. This is still a big issue, but more manageable than if we throw the subject open entirely.

  Life as we know it on Earth requires two things at a bare minimum: an atmosphere with large quantities of oxygen in it and average planetary temperatures between the freezing and boiling points of water. The requirements for intelligent life are more restrictive, but we’ll deal with that later.

  To start with, these two basic requirements probably dictate the type of planet we need to deal with. In our own Solar System there are two basic types of planets, terrestrial and gas planets.

  • The terrestrial or rocky planets are Mercury, Venus, Earth, and Mars. Some astronomers also include Earth’s Moon as a member of this class of planets. They are the small planets closest to the Sun, and are characterized by compositions that are mostly metals and rocks, with thin to nonexistent atmospheres. The zone
of these planets more or less extends out to the asteroid belt, about 2–3 AU from the Sun, and is defined by the so-called “frost line,” the distance from the Sun where ices of various kinds form. Until recently it was thought that most solar systems would have the terrestrial planets closest to their sun because of this: as the Solar System was forming, heat from the Sun drove the volatile ices to the outer edge of the Solar System, where they formed the gas giants.

  • The gas giants are the four planets of the outer system, Jupiter, Saturn, Uranus, and Neptune. In general, they are characterized by their large sizes compared to the terrestrial planets, with masses ranging from 18 to 318 times the mass of Earth. They are also composed mostly of ice and liquids, with perhaps no solid surfaces.

  Other bodies, too small to be called planets, also occur in our Solar System, including dwarf planets, such as Pluto, and over fifty moons, mostly circling the gas giants. Several of these moons might have conditions conducive to life. This has served as the basis of several science fiction stories, the most famous being 2010: Odyssey Two by Arthur C. Clarke, which centers on the discovery of life on Jupiter’s moon Europa [57]. The movie Avatar is also set on the habitable moon of a gas giant planet, but not one in our solar system. So is the rebel base in the movie Star Wars: A New Hope.

  Until 1993, astronomers thought that the arrangement in our Solar System was typical. However, once scientists began discovering planets circling other stars, a third class of planets was discovered, called hot Jupiters, planets the size of gas giants but circling their stars at very close distances, sometimes so close that their orbital period is merely hours long! About 25% of all exoplanets discovered are hot Jupiters. To some extent this reflects instrumentation issues: it’s easier to discover large planets close to their stars. However, I think it safe to say that no astronomers would have predicted any of these odd giants before they were discovered.

  This leads me to a mantra I tell all my astronomy students the very first day of class:

  All stars are fundamentally the same; all planets are different from each other.

  This doesn’t mean that all stars have exactly the same properties or behaviors, but all of the properties of a star stem from two basic data, the star’s mass and its composition at the time of its formation. This is known as the Russell-Vogt theorem in astrophysics. However, stars are almost identical in their initial composition (mostly hydrogen with a little helium and even less of everything else), so the big determinant of how stars behave, their luminosity, their surface temperature, and their lifetimes is the stellar mass.

  On the other hand, planets are a chaotic mess. Although we can put the planets in our Solar System into two broad classes, individual differences among them are as great as their overall similarities. For example, Earth is unique among the terrestrial planets in having a large moon circling it. It is also the only known planet with plate tectonics. There are reasons to think that both these properties might be needed for life on Earth to thrive. Venus is very similar to Earth in size and overall composition, but it has a thick atmosphere with a runaway greenhouse effect, probably a result of being slightly closer to the Sun than Earth is. Mars has a long trench similar to the Grand Canyon but 3,000 miles long, and boasts the largest mountain in the Solar System, the extinct volcano Mons Olympus. Mercury has odd, extremely long cracks running along its surface called lobate scarps, which no other planet has. And so on. The differences result from the fact that many causes determine the characteristics of the planets, and it is not always possible to cleanly separate cause from characteristic.

  Here’s a short list of causes of planetary characteristics:

  • Type of star the planet circles;

  • Mean distance of the planet from the star and orbital eccentricity;

  • Planetary mass;

  • Planetary atmosphere;

  • Exact planetary composition; and

  • A history of impacts with other objects in the system (especially during formation).

  In particular, the history is important. When Poul Anderson mentioned history in his essay, he was probably thinking of the history of the societies on these worlds. In a larger sense, the exact geological history of the planet, and especially the history of planetary impacts, is very important in determining the later features of the planet. If Earth hadn’t undergone an impact with a Mars-size object in exactly the right way during its formation, we wouldn’t have the Moon. Without the Moon it is very possible that life wouldn’t have developed on Earth.

  14.2 TYPE OF STAR

  Stars are simple objects: all of their properties are determined by mass and composition. The obvious question that arises is, what are those properties? There are really only four important ones:

  • Luminosity. Luminosity is how bright the star is and is usually measured relative to our Sun. Luminosities range from about 1/1,000 (10−3) of our Sun’s to about 1 million (106) times greater than it.

  • The surface temperature of the star. The surface temperature runs from about 3,000 K to 30,000 K; our Sun is right in the middle, with a surface temperature of about 5,800 K. Surface temperature also determines the overall color of the star, which ranges from red for the cooler stars to blue-white for the hottest.

  • Main-sequence lifetime. About 90% of the stars in the sky burn hydrogen via fusion in their cores into helium. These are referred to as main-sequence stars. Our sun is a main-sequence star, and most stories feature planets orbiting main-sequence stars because post-main-sequence evolution probably ends up destroying life on the planets. Main-sequence lifetime is strongly determined by mass: bigger stars live shorter lives.

  • Old age and death. After the star eats up all the hydrogen in its core and leaves the main sequence, it evolves into a giant phase, followed by one of three possibilities. For stars with less than about eight times the mass of the sun, the star shrinks and becomes a white dwarf, while for stars between eight and twenty solar masses, the star supernovas, leaving a neutron star behind. The largest stars collapse into black holes.

  Stellar modeling is essentially a solved subject. From the 1960s on astrophysicists have developed elaborate computer codes to model the stars, and these models have been extensively compared against observation. If any of my readers are interested in the subject I highly recommend the hefty textbook An Introduction to Modern Astrophysics by Bradley W. Carroll and Dale A. Ostlie; much of the information presented here is adapted from that book [47]. Main-sequence stars are divided into spectral classes. For historical reasons the classes are (listed from hottest stars to coolest): O, B, A, F, G, K, M, L, and T. Ignoring the two final classes, they can be remembered using the mnemonic “Oh, Be A Fine Girl/Guy, Kiss Me.” The classes run in that order from the hottest, brightest, most massive stars to the dimmest, coolest, and smallest. L and T are brown dwarf stars, which radiate light mainly in the infrared region of the spectrum and use different fusion processes in their interior; I’ll ignore them, although some science fiction stories have been set on worlds circling these stellar objects.

  Table 14.1

  Stellar Properties and Spectral Class

  Source: Data from Carroll and Ostlie [47, appendix G].

  The representative properties of main-sequence stars are a useful way to start thinking about stars for science fiction stories. (The data in table 14.1 and referenced in the following discussion are from Carroll and Ostlie’s An Introduction to Modern Astrophysics [47, appendix G]).

  L is the stellar luminosity (the rate at which the star emits energy in the form of light) in units where the Sun’s luminosity is 1; M and R are the mass and radius of the star, again measured with respect to the Sun. The variable τ is the main-sequence lifetime, the amount of time it spends as a middle-aged star, burning hydrogen in its core until all the hydrogen is used up. The numbers after each class represent different subclasses: the lower the subclass: the hotter and brighter the star is.

  For any star, the following formula is very us
eful:

  This is the Stefan-Boltzmann formula for a black body emitter in normalized units. We can turn it around and write the temperature in terms of radius and luminosity:

  or write the radius in terms of temperature and luminosity:

  The star’s light is emitted in a spectral range that depends on its surface temperature: this is described by the Wien displacement formula from chapter 1, which I’ll write in the following way:

  where T is (as always) measured in Kelvin. The reason this is useful is as follows. The smallest stars have surface temperatures right around 2,900 K, so they will emit most of their light in the infrared region of the spectrum (i.e., with wavelengths around 1µm, longer than what the eye can see). Most of the visible light they emit is in the red end of the spectrum, making them red in appearance. Our Sun, however, has a temperature of 5,800 K, just about twice this, meaning that the light is emitted around a peak wavelength of 0.5 µm, right in the middle of the visible spectrum. The brightest stars have temperatures above 29,000 K, so their light will be concentrated at wavelengths of about 0.1 µm or shorter—that is, in the ultraviolet. Most of the visible light they emit will be at short wavelengths, making them appear blue or blue-white.