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

by Adler, Charles L.

  The only novel I have read that deals with this issue honestly is Building Harlequin’s Moon by Larry Niven and Brenda Cooper [185]. In this book the starship John Glenn is fleeing an Earth ruined by nanotechnology. The ship is powered using a matter-antimatter drive, but it has a problem with it and so must stop en route in another star system to generate enough antimatter to be able to proceed onward. The process of generating the antimatter takes literally hundreds of years. Antimatter generation is power-limited: the rate at which you can create it is limited by the available energy you have on hand. Under most circumstances, this will be pretty slow.

  11.3 RADIATION PROBLEMS

  Edward Purcell, the Nobel laureate physicist, pointed out in 1963 that even apart from expenses, there are some real problems with trying to get anywhere really fast [46, pp. 121–143]. Those problems have to do with exposure to radiation, both to our astronauts and (if we’re not careful) to the entire Earth as well.

  Let’s say we have collected our antimatter and we want to send a group of explorers out to explore the cosmos. Let’s say we want them to travel there and back at 99.5% of the speed of light, and maintain an acceleration of 1 g the whole time. This implies a mass ratio of 20 from table 11.1. But that’s only for the journey out. To decelerate to a stop, we need another ratio of 20; to accelerate back toward Earth, another ratio of 20; and to decelerate to a stop again, another ratio of 20. The rocket equation is merciless: the overall mass ratio must be

  R = 204 = 160,000.

  In words, for every kilogram of mass on the spacecraft we need 160,000 kg of fuel, which is to say 80,000 kg of antimatter and 80,000 kg of normal matter. For our canonical 10,000 kg payload spacecraft (which, if manned, will probably need to be bigger), we need a whopping 1.6×1019 kg of fuel with a stored energy of 1.44×1026 J, or about as much energy as our current world civilization will use in about half a million years.

  Be that as it may: the power the spacecraft will use on “takeoff” can be found from the fact that the power of a “photon rocket” such as this one is intimately related to the force the rocket exerts: if the force is F and the power P, then

  F = 2P/c.

  Since the rocket is accelerating at 1 g, and g ∼ 10 m/s2, the net force needed is 1.6×1010 N and the power required is 2.4×1018 W. This is an energy usage rate more than 100,000 times greater than our current civilization’s. The total power the Earth receives from the Sun is about an order of magnitude less than this; if we launch from anywhere near Earth’s orbit, we will likely destroy Earth’s ecology, especially if we consider that (unlike the Sun) all of the energy is being delivered in the form of high-energy gamma rays [46, p. 138]. A starship like this one clearly cannot be launched from anywhere near Earth. Purcell also pointed out that at relativistic speeds like this, the hydrogen atoms that the ship intercepts essentially have the same energy as high-energy cosmic rays, meaning that extensive shielding would be needed to protect the crew from radiation exposure.

  Purcell ended with this comment:

  Well, this is preposterous, you are saying. That is exactly my point. It is preposterous. And remember, our conclusions are forced upon us by the elementary laws of mechanics. All those people who have been talking about lebensraum in space and so on, simply haven’t made this calculation and until they do, what they say is nonsense. [46, p. 138]

  I am not sure it is nonsense, but it is clearly beyond anything our human civilization can do, and perhaps beyond what any civilization can do. Or perhaps not: in chapter 21, “A Googol Years,” I’ll discuss the issue of really advanced civilizations. In the meanwhile, however, we’ve talked about relativity a lot but haven’t gotten to the good part: the issues of time dilation. Even though the universe is huge, given access to enough energy one could explore it within one human lifetime because of the theory of relativity.

  NOTES

  1. “Corpsicles” are people placed in cryogenic suspension just before death and revived, in different bodies, by the state hundreds of years later. This sort of cryonics is popular in science fiction. Perhaps the most widely seen use of it is in the TV show Futurama.

  2. The author of this paper has suggested that because of Bremsstrahlung losses, using a ramscoop might be a good way of decelerating a ship traveling at a large fraction of the speed of light.

  3. One must take care when reading this report to distinguish between two types of efficiency: overall energy efficiency, which is relatively low, versus the efficiency of a high-energy proton striking a target in producing antiprotons, which is relatively high.

  CHAPTER TWELVE

  INTERSTELLAR TRAVEL AND RELATIVITY

  12.1 TIME ENOUGH FOR ANYTHING

  He looked me up and down and said wonderingly, “I knew intellectually you would not have changed with the years. But to see it, to realize it, is another thing, eh? ‘The Picture of Dorian Gray.’ ”

  —ROBERT A. HEINLEIN, TIME FOR THE STARS

  Robert A. Heinlein’s novel Time for the Stars is essentially one long in-joke for physicists. The central characters of the novel are Tom and Pat Bartlett, two identical twins who can communicate with each other telepathically. In the novel, telepathy has a speed much faster than light. Linked telepaths, usually pairs of identical twins, are used to maintain communications between the starship Lewis and Clark and Earth. Tom goes on the spacecraft while Pat stays home; the ship visits a number of distant star systems, exploring and finding new Earth-like worlds. On Tom’s return, nearly seventy years have elapsed on Earth, but Tom has only aged by five [113].

  I call this a physicist’s in-joke because Heinlein is illustrating what is referred to as the twin paradox of relativity: take two identical twins, fly one around the universe at nearly the speed of light, and leave the other at home. On the traveler’s return, he or she will be younger than the stay-at-home, even though the two started out the same age. This is because according to Einstein’s special theory of relativity, time runs at different rates in different reference frames.

  This is another common theme in science fiction: the fact that time slows down when one “approaches the speed of light.” It’s a subtle issue, however, and is very easy to get wrong. In fact, Heinlein made some mistakes in his book when dealing with the subject, but more on that later. First, I want to list a few of the many books written using this theme:

  • The Forever War, by Joe W. Haldeman. This story of a long-drawn-out conflict between humanity and an alien race has starships that move at speeds near light speed to travel between “collapsars” (black holes), which are used for faster-than-light travel. Alas, this doesn’t work. The hero’s girlfriend keeps herself young for him by shuttling back and forth at near light speeds between Earth and a distant colony world [107].

  • Poul Anderson’s novel, Tau Zero. In this work, mentioned in the last chapter, the crew of a doomed Bussard ramship is able to explore essentially the entire universe by traveling at speeds ever closer to the speed of light [22].

  • The Fifth Head of Cerberus, by Gene Wolfe. In this novel an anthropologist travels from Earth to the double planets of St. Croix and St. Anne. It isn’t a big part of the novel, but the anthropologist John Marsch mentions that eighty years have passed on Earth since he left it, a large part of his choice to stay rather than return home [253].

  • Larris Niven’s novel A World out of Time. The rammer Jerome Corbell travels to the galactic core and back, aging some 90 years, while three million years pass on Earth [182]. 1

  There are many, many others, and for good reason: relativity is good for the science fiction writer because it brings the stars closer to home, at least for the astronaut venturing out to them. It’s not so simple for her stay-at-home relatives. The point is that the distance between Earth and other planets in the Solar System ranges from tens of millions of kilometers to billions of kilometers. These are large distances, to be sure, but ones that can be traversed in times ranging from a few years to a decade or so by chemical propulsion. We can ima
gine sending people to the planets in times commensurate with human life. If we imagine more advanced propulsion systems, the times become that much shorter.

  Unfortunately, it seems there is no other intelligent life in the Solar System apart from humans, and no other habitable place apart from Earth. If we want to invoke the themes of contact or conflict with aliens or finding and settling Earth-like planets, the narratives must involve travel to other stars because there’s nothing like that close to us. But the stars are a lot farther away than the planets in the Solar System: the nearest star system to our Solar System, the triple star system Alpha Centauri, is 4.3 light-years away: that is, it is so far that it takes light 4.3 years to get from there to here, a distance of 40 trillion km. Other stars are much farther away. Our own galaxy, the group of 200 billion stars of which our Sun is a part, is a great spiral 100,000 light-years across. Other galaxies are distances of millions of light-years away.

  From our best knowledge of physics today, nothing can go faster than the speed of light. That means that it takes at least 4.3 years for a traveler (I’ll call him Tom) to go from Earth to Alpha Centauri and another 4.3 years to return. But if Tom travels at a speed close to that of light, he doesn’t experience 4.3 years spent on ship; it can take only a small fraction of the time. In principle, Tom can explore the universe in his lifetime as long as he is willing to come back to a world that has aged millions or billions of years in the meantime.

  12.2 WAS EINSTEIN RIGHT?

  This weird prediction—that clocks run more slowly when traveling close to light speed—has made many people question Einstein’s results.2 The weirdness isn’t limited to time dilation; there is also relativistic length contraction. A spacecraft traveling close to the speed of light shrinks in the direction of motion. The formulas are actually quite simple. Let’s say that Tom is in a spacecraft traveling along at some speed v, while Pat is standing still, watching him fly by. We’ll put Pat in a space suit floating in empty space so we don’t have to worry about the complication of gravity. Let’s say the following: Pat has a stopwatch in his hand, as does Tom. As Tom speeds by him, both start their stopwatches at the same time and Pat measures a certain amount of time on his watch (say, 10 seconds) while simultaneously watching Tom’s watch through the window of his spacecraft. If Pat measures time Δt0 go by on his watch, he will see Tom’s watch tick through less time. Letting Δt be the amount of time on Tom’s watch, the two times are related by the formula

  where the all-important “gamma factor” is

  The gamma factor is always greater than 1, meaning Pat will see less time go by on Tom’s watch than on his. Table 12.1 shows how gamma varies with velocity.

  Note that this is only really appreciable for times greater than about 10% of the speed of light. The length of Tom’s ship as measured by Pat (and the length of any object in it, including Tom) shrinks in the direction of motion by the same factor.

  Even though the gamma factor isn’t large for low speeds, it is still measurable. To quote Edward Purcell, “Personally, I believe in special relativity. If it were not reliable, some expensive machines around here would be in very deep trouble” [46, p. 134]. The time dilation effect has been measured directly, and is measured directly almost every second of every day in particle accelerators around the world. Unstable particles have characteristic lifetimes, after which they decay into other particles. For example, the muon is a particle with mass 206 times the mass of the electron. It is unstable and decays via the reaction

  It decays with a characteristic time of 2.22 µs; this is the decay time one finds for muons generated in lab experiments. However, muons generated by cosmic ray showers in Earth’s atmosphere travel at speeds over 99% of the speed of light, and measurements on these muons show that their decay lifetime is more than seven times longer than what is measured in the lab, exactly as predicted by relativity theory [233]. This is an experiment I did as a graduate student and our undergraduates at St. Mary’s College do as part of their third-year advanced lab course. Experiments with particles in particle accelerators show the same results: particle lifetimes are extended by the gamma factor, and no matter how much energy we put into the particles, they never travel faster than the speed of light. This is remarkable because in the highest-energy accelerators, particles end up traveling at speeds within 1 cm/s of light speed. Everything works out exactly as the theory of relativity says, to a precision of much better than 1%.

  Table 12.1

  Gamma Factor as a Function of Rocket Velocity

  v/c

  γ

  0

  0.1

  0.2

  0.4

  0.5

  0.6

  0.866

  0.9

  0.95

  0.99

  0.995

  0.9995

  1-δ, δ 1 √

  1

  1

  1.01

  1.02

  1.09

  1.15

  1.25

  2

  2.29

  3.2

  7.09

  10.01

  31.63

  ∞

  How about experiments done with real clocks? Yes, they have been done as well. The problems of doing such experiments are substantial: at speeds of a few hundred meters per second, a typical speed for an airplane, the gamma factor deviates from 1 by only about 10−13. To measure the effect, you would have to run the experiment for a long time, because the accuracy of atomic clocks is only about one part in 1011 or 1012; the experiments would have to run a long time because the difference between the readings on the clocks increases with time. In the 1970s tests were performed with atomic clocks carried on two airplanes that flew around the world, which were compared to clocks remaining stationary on the ground. Einstein passed with flying colors. The one subtlety here is that you have to take the rotation of the Earth into account as part of the speed of the airplane. For this reason, two planes were used: one going around the world from East to West, the other from West to East [252]. This may seem rather abstract, but today it is extremely important for our technology. Relativity lies at the cornerstone of a multi-billion-dollar industry, the global positioning system (GPS).

  GPS determines the positions of objects on the Earth by triangulation: satellites in orbit around the Earth send radio signals with time stamps on them. By comparing the time stamps to the time on the ground, it is possible to determine the distance to the satellite, which is the speed of light multiplied by the time difference between the two. Using signals from at least four satellites and their known positions, one can triangulate a position on the ground. However, the clocks on the satellites run at different rates as clocks on the ground, in keeping with the theory of relativity. There are actually two different effects: one is relativistic time dilation owing to motion and the other is an effect we haven’t considered yet, gravitational time dilation. Gravitational time dilation means that time slows down the further you are in a gravitational potential well. On the satellites, the gravitational time dilation speeds up clock rates as compared to those on the ground, and the motion effect slows them down. The gravitational effect is twice as big as the motion effect, but both must be included to calculate the total amount by which the clock rate changes. The effect is small, only about three parts in a billion, but if relativity weren’t accounted for, the GPS system would stop functioning in less than an hour [146, p. 68]. To quote from Alfred Heick’s textbook GPS Satellite Surveying,

  Relativistic effects are important in GPS surveying but fortunately can be accurately calculated.… [The difference in clock rates] corresponds to an increase in time of 38.3 µsec per day; the clocks in orbit appear to run faster.… [This effect] is corrected by adjusting the frequency of the satellite clocks in the factory before launch to 10.22999999543 MHz [from their fundamental frequency of 10.23 MHz].

  This statement says two things: first, in the dry language of an engineering handbook, it is made quite clear t
hat these relativistic effects are so commonplace that engineers routinely take them into account in a system that hundreds of millions of people use every day and that contributes billions of dollars to the world’s commerce. Second, it tells you the phenomenal accuracy of radio and microwave engineering. So the next time someone tells you that Einstein was crazy, you can quote chapter and verse back at him!

  12.3 SOME SUBTLETIES

  The problem with Heinlein’s Time for the Stars is that when the Lewis and Clark begins to get close to the speed of light, the twins have problems communicating with each other. This is Tom speaking:

  At three-quarters of the speed of light [Pat] began to complain that I was drawling while it seemed to me he was beginning to jabber. At nine-tenths of the speed of light it was close to 2 to 1, but we knew what was wrong now and I talked fast and he talked slow.

  At 99% of c it was 7 to 1 and all we could do to make ourselves understood. Later that day we fell out of touch entirely. [113, chap. 11, “Slippage”]

  This seems reasonable, but unfortunately, it violates one of the underlying principles of the special theory of relativity. Relativity is called relativity because measurements made by an observer are relative to that observer and not to anyone else. The odd thing is that Tom will measure his own time going by normally and Pat’s clock slowed down—by the same gamma factor that Tom’s clock appears slowed down as measured by Pat. This is so odd that when I was teaching this once, one of my students shook her head and stated flatly, “No, that isn’t right.” This is one of the two underlying ideas of Einstein’s special theory of relativity: you can’t tell whether you are moving at constant velocity or standing still by any measurement you can make. If it were true that Pat would measure Tom’s clock as running slow if Tom were moving and he weren’t, and Tom would measure Pat’s clock as running fast, then that would be proof that Tom was moving and Pat wasn’t. Since you can’t do that, both clocks must run slow as measured by the other.3 (How this works and makes sense would take far too long to go into in this book. If you are interested, there are good books on the subject, a few written by Einstein himself [198].) The other principle of relativity, which is what leads to the time dilation effects, is that both Tom and Pat will measure the speed of light as having the same value no matter how fast they are moving in relation to each other.