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

by Adler, Charles L.

  13.4 THE GENERAL THEORY

  If faster-than-light travel is allowed in our universe, we have to appeal to the general theory of relativity for this. The general theory of relativity expresses how space and time are “warped” (curved) in the presence of mass, and how this spatial curvature affects the trajectories of particles moving near the masses.

  One of the common metaphors used in discussing this subject is to say that space is something like a giant stretched sheet of rubber and the stars are bowling balls dropped onto the sheet. If we were to roll a marble across the sheet near the bowling ball it would be deflected from a straight line because of the depression created by the bowling ball, not because of some “attraction” between the bowling ball and the marble. The marble represents a planet or a spaceship on a trajectory flying close to the star. It’s not a great analogy because we are dealing with the curvature of time and space, not just space; clocks on the spacecraft will be slowed as they approach the star, and so will show a time difference compared to ones that are farther away.

  This deformation of space-time is what would, hypothetically, make it possible to take a “shortcut” around normal space. I’m going to be more descriptive in this chapter than in others because the mathematics one needs for understanding the general theory is very difficult. Where possible I will put the numbers in.

  One of the first tests of the general theory of relativity was the deflection of light by a large mass, in this case the Sun. In 1919 Sir Arthur Stanley Eddington photographed stars that were near the Sun during a total eclipse and found that their positions were changed from their apparent positions six months later, when they were far away from it. This is not surprising: one might expect Newton’s older theory of gravity to predict this as well. Light rays from a distant star skimming by the Sun would be attracted by the strong gravitational “pull” of the star, and would therefore be bent, making the star appear in a different place in the sky. However, Newton’s theory predicts a different value than Einstein’s theory: deflection under the Einsteinian theory is twice what Newton predicts. If anyone is interested in using this in a story, the angular deflection of a light ray skimming by the edge of a star like our Sun (in radians) is

  Here, G = 6.67×10−11 Nm2/kg2 is the universal gravitational constant, and Mmetric and Rmetric are the mass and radius of the star. The subscript “metric” means that in this equation we use the metric value for these units: for the Sun, the metric mass is 1.99×1030 kg and the metric radius is 6.85×108 m. In this formula the deflection is expressed in radians, where 1 rad ≈ 57.3 degrees. In more useful units,

  where M and R are relative to the mass and radius of the Sun (i.e., Msun = 1 and Rsun = 1 in these units), and 1 arc-second (″) is 1/3,600 of a degree. These are units we will use a lot in later chapters when discussing the subject of life in the universe.

  The deflection of light even due to a massive object like the Sun is small. This is because the amount of space-time curvature around the Sun is small. The curvature of space-time around a mass of a given radius is a function of the dimensionless parameter:

  If this number is small, the effects due to the general theory of relativity (curvature of space and time dilation) are small, but as it approaches 1 the curvature becomes severe. If it is greater than 1/2, the star disappears from the universe entirely.

  13.5 GRAVITATIONAL TIME DILATION AND BLACK HOLES

  The escape velocity of a rocket taking off from a planet of radius R is given by the formula

  If the rocket travels at less than this speed, it eventually falls back to the planet. If it has an initial speed equal to or greater than this speed, it doesn’t come back down. Nothing travels faster than the speed of light, so if we insert light speed into the equation and solve for the radius, we find the following:

  where the second expression is in units where the Sun’s mass is 1. That is, if we squeezed all of the Sun’s mass into a radius of less than 2.9 km, it would become dark: light itself couldn’t escape from its gravitational pull. This is using the Newtonian formula, but in fact, Einstein’s theory of relativity predicts the same thing: one must interpret the radius in a somewhat different way because of space-time curvature, but the formula is the same. For a good, elementary discussion of general relativity I highly recommend the book Exploring Black Holes, by Edwin F. Taylor and John Archibald Wheeler [135]. This is almost a handbook for science fiction writers trying to use black holes in their stories in some way. The level of the book is about that of a senior in college majoring in physics.

  Black holes exist: they are the product of the end stages of the evolution of massive stars, ones more than twenty times the mass of the Sun. Stars are a balance between the heat and pressure pushing outward generated by fusion at their cores and the force of gravity pulling them in. Once the fusion fires are ended, if the star is large enough nothing can prevent it from collapsing completely and effectively disappearing from our universe. What is left behind is an “event horizon,” a one-way barrier with radius given by equation (13.5). Once you cross it you cannot return to the rest of the universe. Beyond that point one must inexorably fall to the singularity at the center, where tidal forces inevitably destroy everything entering the black hole’s maw. What happens to matter once it falls to the singularity isn’t very well understood; understanding it requires a complete quantum theory of gravity, which we don’t have yet.

  Another fascinating aspect of black holes is that from the view of an outside observer, time slows down as you approach the event horizon. At the horizon, it slows to a stop. However, someone falling through the horizon to the singularity notices nothing unusual! Imagine we have two people, one (Al) in a spaceship at a very large distance away from the black hole and another (Bert) in a spacecraft closer to it. Assume both are at rest with respect to the black hole (i.e., their spaceships are firing their engines to keep from falling into it). The clock on Al’s spaceship will run faster than the one on Bert’s. If Al measures a time tA on his clock, he will see time tB run by on Bert’s clock, where

  on Bert’s clock in that time. Here, r is the distance of Bert from the center of the black hole [235]. If either observer is moving relative to the center of the black hole, the formula must be changed to account for the motion. As expected, time on Bert’s clock slows to zero as one approaches the event horizon. As I mentioned in chapter 12, this gravitational time dilation and the relativistic motion effects must be taken into account for the GPS system to work. This to me, is one of the most amazing effects of relativity. Time slowing as one approaches the event horizon has been used in a number of science fiction stories, not always accurately.

  Black holes have fascinated science fiction writers nearly as long as they have fascinated physicists. As Kip Thorne says, the “golden age” of black hole research was in the 1960s, and it is in literature from the 1960s and early 1970s that we find black holes first making their way into mainstream science fiction. It’s possible that the “black sun” of Arthur C. Clarke’s The City and the Stars has an early use of the idea. In that story the black sun, an artificially constructed star, is used to imprison the “Mad Mind,” a noncorporeal being created by the denizens of a galactic empire, which turned on them and nearly destroyed the galaxy [53]. If so, Clarke was being prescient, as the book predicted that eventually the Mad Mind would escape once the black sun failed. When the novel was written, in the late 1950s, it wasn’t known that black holes evaporate through Hawking radiation. Eventually (long, long after all the stars are dead) all of the black holes will boil away to nothingness.

  The story “He Fell into a Dark Hole” by Jerry Pournelle features the rescue of a group of spacemen stranded in orbit around a black hole, and the ship being nearly destroyed by gravitational radiation emitted by the hole. The story is set in the same universe as The Mote in God’s Eye and is scientifically accurate except for the idea that astrophysicists would have forgotten all about black holes five centuries from now. Larry Niven, o
f course, wrote several stories featuring black holes, including “Singularities Make Me Nervous,” which involves an astronaut who travels backward in time by going through a black hole.

  Black holes, occasionally called “collapsars,” have been used for FTL travel in a number of science fiction novels. Two that spring readily to mind are The Forever War by Joe W. Haldeman and Fiasco by Stanislaw Lem [107][151]. I’ve mentioned the first book already. The second records the attempt of a human expedition to make first contact with an alien race, the Quintans. In the book the ship uses a “gracer” (a gravitational laser, whatever that is) to make the collapsar oscillate, permitting the ship to dive through it and travel either faster than light or backward in time—the rather recondite language of the book doesn’t really make it clear which. Lem was a highly educated man, and so this bit probably came from work physicists had done on so-called Kerr black holes.

  The relationship between time travel, FTL travel, and black holes has to do with the Kerr solutions of the Einstein field equations for rotating black holes. At the center of every black hole is a singularity, a place where space is infinitely curved, where anything falling into it is torn apart by simultaneous squeezing and compression along different axes. For a nonrotating hole, the singularity is a point, meaning that anyone falling through the singularity will be killed, but a rapidly rotating black hole has a singularity in the form of a ring. The solutions of the Einstein equations for an object falling through the ring are very strange: they indicate that an astronaut falling through the singularity could travel along a “closed timelike curve,” which is to say into his past. They also appear to permit FTL travel as well. Unfortunately, the solutions are unstable, and any matter entering the ring singularity almost certainly destroys it, killing the traveler in the process [168]. This unfortunately scotches Niven’s and Lem’s stories and movies like The Black Hole in which people travel beyond the singularity.

  13.6 WORMHOLES AND EXOTIC MATTER

  A number of solutions of Einstein’s equations lead to odd, seemingly unphysical behavior. Kurt Gödel, the great mathematician, found equations describing a universe in constant rotation in which travel indefinitely into the past and future was possible by picking the correct trajectory to follow. Because the real universe is not in rotation, this isn’t taken very seriously, but in 1974 Frank Tipler showed that closed timelike curves appeared around an infinitely long, rapidly rotating cylinder. The paper is titled “Rotating Cylinders and the Possibility of Global Causality Violation” [237]; Tipler showed that if the outer edge of the cylinder rotated at speeds greater than half the speed of light, it could be used as a time machine.

  It is unclear whether a long but finite, rapidly spinning cylinder could do the same thing. Tipler thought so, but other physicists have pointed out some unphysical properties of this solution of Einstein’s equations. Tipler’s paper has been used in at least two science fiction stories written in the late 1970s. In the first, Larry Niven’s “Rotating Cylinders and the Possibility of Global Causality Violation,” various alien races try to build a Tipler cylinder, only to be stopped by one natural disaster after another [184]. In the final paragraph, as the scientist is telling his monarch about the device, the star their planet circles goes nova. This is the universe protecting itself from anyone building a time machine. It is a radical example of Steven Hawking’s “chronology protection hypothesis,” the idea that the universe will not permit the existence of a time machine. It was originally a science fiction idea, but modern research in gravitational physics has explored this idea in some detail. The other story is the novel The Avatar, by Poul Anderson, which is a more conventional space travel story in which the Tipler device (called “T-machine” in the text) is used for faster-than-light travel [21].

  In 1988 Kip Thorne, a Caltech physicist, was asked by his friend Carl Sagan to look over the manuscript for the novel Contact and critique it for scientific accuracy. In the first draft the heroine, Ellen Archway, traveled to a distant star by diving through a black hole. As I have said, this doesn’t work: Ellen would have been killed by the singularity at the center. To help Sagan out, Thorne investigated whether another solution of Einstein’s equations, the wormhole, was a possibility. Wormholes form a bridge (more or less a tube) between two different regions of space-time, allowing travel between two widely separated parts of the universe. This does require two different things happening, however:

  1. Space must be bent like a crumpled sheet of paper as seen through higher dimensions, or (if it is not already folded) the sufficiently advanced civilization must be able to fold space in this manner. This idea is not uncommon in science fiction: Heinlein’s novel Starman Jones posits interstellar travel using this trick. In it, the astrogators of the starships pilot their craft through “Horstian anomalies,” places where distant points in space are folded together through higher dimensions[112]. To quote Max, the protagonist of the story, showing off for his girlfriend:

  “You can’t go faster than light, not in our space. If you do, you burst out of it. But if you do it where space is folded back and congruent, you pop back in our space—but a long way off. How far off depends on how it’s folded.”

  Figure 13.1. Space-time diagram of a wormhole.

  Ships travel to near the speed of light and then reach it and exceed it at the anomaly point, “bursting through” to points light-years away. The plot centers on a ship getting lost when the astrogators make a bad calculation right before the “jump.” Heinlein is again making the error of assuming a privileged reference frame: the ship’s speed depends on who is looking at it, and no object with mass can move at, let alone exceed, the speed of light. The book, written in 1953, is still a fun read despite anachronisms such as human astrogators programming computers in binary code in 3000 CE.

  2. If space is folded or can be folded in this way, one must also be able to create and maintain the wormhole. Morris and Thorne found out that this would require the wormhole to be threaded with exotic matter.

  If one looks at standard diagrams for wormholes, they simply look like they need to be held apart by something, some sort of bands or hoops running through them to keep them from collapsing back on themselves (fig 13.1).

  What Thorne and Morris found was that these bands needed to be of a strange form of mass or energy that had a tension greater than its energy density. One can kind of rationalize this by saying that the mass of the exotic matter would tend to collapse the wormhole owing to its gravitational self-attraction, while the tension would be needed to keep it from collapsing, so the tension must be greater, but this is a gross oversimplification. Morris and Thorne called this material “exotic” because it is very different from any matter we have experience with. In an earlier chapter we discussed one example of a structure built entirely in tension, the space elevator. A wormhole is another structure entirely in tension, but a much more extreme one. Let’s define a dimensionless parameter, the “exoticity” of a given material, as

  The smaller η is, the more “normal” the material is. The value h is the “breaking height” of a structure as defined in chapter 5, the maximum length of a cable made of the material that can be supported under gravity. Carbon nanotubes, the materials with the highest strength-to-weight ratio known, and therefore the ones with the highest exoticities, have h ≈ 106 m, or an exoticity parameter of 10−10. Puting it differently, since the acceleration of gravity is just about 1 LY/yr2, the exoticity parameter can be written in a very simple form:

  It is essentially equal to its breaking length expressed in light-years. To construct a wormhole, this parameter must be greater than 1. Another way to say it is that to create a wormhole, one would need something that could support a length of itself at least one light-year long against Earth’s gravity. There is a generic proof that any means of traveling faster than light or traveling backward in time must involve exotic matter in its creation [168].

  Exotic matter reminds me of the chain gleipnir used to bind
the Fenris Wolf in Norse mythology: made from the sounds of cats’ paws, the breath of fish, the spittle of birds, the hairs of a woman’s beard, and the roots of a mountain, it was as thin as a thread but strong enough to hold the fiercest monster bound fast [62]. It seems to me that this would work as well as anything else in keeping a wormhole throat open. I therefore propose renaming exotic matter gleipnirsmalmi (“gleipnir’s metal,” about the closest I could come to the idea using an online Icelandic dictionary).

  Physicists have proposed things almost as exotic as the ingredients used in the myth to keep wormholes open. It’s relatively easy to show that ordinary solid matter cannot be exotic. One of the web problems for this book works this constraint through in detail. Exotic matter is called exotic because it violates the weak energy condition of general relativity; in some reference frame the mass density will be negative [168][169][236]. The reason is that a parallel bundle of light rays going through the wormhole will be defocused; gravitational lensing by ordinary matter focuses light, meaning that in some ways, this gleipnirsmalmi behaves almost like antigravity.3 One candidate is the Casimir vacuum energy; in quantum mechanics the vacuum isn’t really empty but is filled with particles and fields popping in and out of existence; as long as this happens in a time too short for anyone to notice, there’s no problem. The Casimir vacuum is achieved by closing off a piece of space, like putting two highly reflecting parallel plates a distance a apart from each other. Hendrik G. B. Casimir showed in 1948 that this arrangement lowered the energy of the vacuum between them, resulting in an attractive force [48]