The Physics Of Star Trek Part 1

The Physics of Star Trek.

by Lawrence Maxwell Krauss.

FOREWORD

Stephen Hawking

I was very pleased that Data decided to call Newton, Einstein, and me for a game of poker aboard the Enterprise. Here was my chance to turn the tables on the two great men of gravity, particularly Einstein, who didn't believe in chance or in G.o.d playing dice. Unfortunately, I never collected my winnings because the game had to be abandoned on account of a red alert. I contacted Paramount studios afterward to cash in my chips, but they didn't know the exchange rate.Science fiction like Star Trek is not only good fun but it also serves a serious purpose, that of expanding thehuman imagination. We may not yet be able to boldly go where no man (or woman) has gone before, but at least we can do it in the mind. We can explore how the human spirit might respond to future developments in science and we can speculate on what those developments might be. There is a two-way trade between science fiction and science. Science fiction suggests ideas that scientists incorporate into their theories, but sometimes science turns up notions that are stranger than any science fiction. Black holes are an example, greatly a.s.sisted by the inspired name that the physicist John Archibald Wheeler gave them. Had they continued with their original names of "frozen stars" or "gravitationally completely collapsed objects," there wouldn't have been half so much written about them.One thing that Star Trek and other science fiction have focused attention on is travel faster than light. Indeed, it is absolutely essential to Star Trek's story line. If the Enterprise were restricted to flying just under the speed of light, it might seem to the crew that the round trip to the center of the galaxy took only a few years, but 80,000 years would have elapsed on Earth before the s.p.a.ceship's return. So much for going back to see your family!Fortunately, Einstein's general theory of relativity allows the possibility for a way around this difficulty: one might be able to warp s.p.a.cetime and create a shortcut between the places one wanted to visit. Although there are problems of negative energy, it seems that such warping might be within our capabilities in the future. There has not been much serious scientific research along these lines, however, partly, I think, because it sounds too much like science fiction. One of the consequences of rapid interstellar travel would be that one could also travel back in time. Imagine the outcry about the waste of taxpayers' money if it were known that the National Science Foundation were supporting research on time travel. For this reason, scientists working in this field have to disguise their real interest by using technical terms like "closed timelike curves" that are code for time travel. Nevertheless, today's science fiction is often tomorrow's science fact. The physics that underlies Star Trek is surely worth investigating. To confine our attention to terrestrial matters would be to limit the human spirit.PREFACEWhy the physics of Star Trek? Gene Roddenberry's creation is, after all, science fiction, not science fact. Many of the technical wonders in the series therefore inevitably rest on notions that may be ill defined or otherwise at odds with our current understanding of the universe. I did not want to write a book that ended up merely outlining where the Star Trek writers went wrong.Yet I found that I could not get the idea of this book out of my head. I confess that it was really the transporter that seduced me. Thinking about the challenges that would have to be faced in devising such a fictional technology forces one to ponder topics ranging from computers and the information superhighway to particle physics, quantum mechanics, nuclear energy, telescope building, biological complexity, and even the possible existence of the human soul! Compound this with ideas such as warped s.p.a.ce and time travel and the whole subject became irresistible.I soon realized that what made this so fascinating to me was akin to what keeps drawing fans to Star Trek today, almost thirty years after the series first aired. This is, as the omnipotent Star Trek prankster Q put it, "charting the unknown possibilities of existence." And, as I am sure Q would have agreed, it is even good fun to imagine them.As Stephen Hawking states in the foreword to this book, science fiction like Star Trek helps expand the human imagination. Indeed, exploring the infinite possibilities the future holdsincluding a world where humanity has overcome its myopic international and racial tensions and ventured out to explore the universe in peaceis part of the continuing wonder of Star Trek. And, as I see this as central to the continuing wonder of modern physics, it is these possibilities that I have chosen to concentrate on here.Based on an informal survey I carried out while walking around my university campus the other day, the number of people in the United States who would not recognize the phrase "Beam me up, Scotty" is roughly comparable to the number of people who have never heard of ketchup. When we consider that the Smithsonian Inst.i.tution's exhibition on the starship Enterprise was the most popular display in their Air and s.p.a.ce Museummore popular than the real s.p.a.cecraft thereI think it is clear that Star Trek is a natural vehicle for many people's curiosityabout the universe. What better context to introduce some of the more remarkable ideas at the forefront of today's physics and the threshold of tomorrow's? I hope you find the ride as enjoyable as I have.Live long and prosper.THE PHYSICS OF STAR TREK

SECTION ONE.

A Cosmic Poker Game

In which the physics of inertial dampers and tractor beams paves the way for time travel, warp speed, deflector shields, wormholes, and other s.p.a.cetime oddities

CHAPTER ONE.

NEWTON Antes"No matter where you go, there you are." From a plaque on the starship Excelsior, in Star Trek VI: The Undiscovered Country, presumably borrowed from The Adventures of Buckaroo BanzaiYou are at the helm of the starship Defiant (NCC-1764), currently in orbit around the planet Iconia, near the Neutral Zone. Your mission: to rendezvous with a nearby supply vessel at the other end of this solar system in order to pick up components to repair faulty transporter primary energizing coils. There is no need to achieve warp speeds; you direct the impulse drive to be set at full power for leisurely half-light-speed travel, which should bring you to your destination in a few hours, giving you time to bring the captain's log up to date. However, as you begin to pull out of orbit, you feel an intense pressure in your chest. Your hands are leaden, and you are glued to your seat. Your mouth is fixed in an evil-looking grimace, your eyes feel like they are about to burst out of their sockets, and the blood flowing through your body refuses to rise to your head. Slowly, you lose consciousness ... and within minutes you die.What happened? It is not the first signs of spatial "interphase" drift, which will later overwhelm the ship, or an attack from a previously cloaked Romulan vessel. Rather, you have fallen prey to something far more powerful. The ingenious writers of Star Trek, on whom you depend, have not yet invented inertial dampers, which they will introduce sometime later in the series. You have been defeated by nothing more exotic than Isaac Newton's laws of motionthe very first things one can forget about high school physics.OK, I know some trekkers out there are saying to themselves, "How lame! Don't give me Newton. Tell me things I really want to know, like 'How does warp drive work?' or 'What is the flash before going to warp speedis it like a sonic boom?' or 'What is a dilithium crystal anyway?'" All I can say is that we will get there eventually. Travel in the Star Trek universe involves some of the most exotic concepts in physics. But many different aspects come together before we can really address everyone's most fundamental question about Star Trek: "Is any of this really possible, and if so, how?"To go where no one has gone beforeindeed, before we even get out of Starfleet Headquarterswe first have to confront the same peculiarities that Galileo and Newton did over three hundred years ago. The ultimate motivation will be the truly cosmic question which was at the heart of Gene Roddenberry's vision of Star Trek and which, to me, makes this whole subject worth thinking about: "What does modem science allow us to imagine about our possible future as a civilization?"Anyone who has ever been in an airplane or a fast car knows the feeling of being pushed back into the seat as the vehicle accelerates from a standstill. This phenomenon works with a vengeance aboard a starship. The fusion reactions in the impulse drive produce huge pressures, which push gases and radiation backward away from the ship at high velocity. It is the backreaction force on the enginesfrom the escaping gas and radiationthat causes the engines to "recoil" forward. The ship, being anch.o.r.ed to the engines, also recoils forward. At the helm, you are pushed forward too, by the force of the captain's seat on your body. In turn, your body pushes back on the seat.Now, here's the catch. Just as a hammer driven at high velocity toward your head will produce a force on your skull which can easily be lethal, the captain's seat will kill you if the force it applies to you is too great. Jet pilots and NASA have a name for the force exerted on your body while you undergo high accelerations (as in a plane or during a s.p.a.ce launch): G-forces. I can describe these by recourse to my aching back: As I am sitting at my computer terminal busily typing, I feel the ever-present pressure of my office chair on my b.u.t.tocksa pressure that I have learned to live with (yet, I might add, that my b.u.t.tocks are slowly reacting to in a very noncosmetic way). The force on my b.u.t.tocks results from the pull of gravity, which if given free rein would accelerate me downward into the Earth. What stops me from acceleratingindeed, from moving beyond my seatis the ground exerting an opposite upward force on my house's concrete and steel frame, which exerts an upward force on the wood floor of my second-floor study, which exerts a force on my chair, which in turn exerts a force on the part of my body in contact with it. If the Earth were twice as ma.s.sive but had the same diameter, the pressure on my b.u.t.tocks would be twice as great. The upward forces would have to compensate for the force of gravity by being twice as strong.The same factors must be taken into account in s.p.a.ce travel. If you are in the captain's seat and you issue a command for the ship to accelerate, you must take into account the force with which the seat will push you forward. If you request an acceleration twice as great, the force on you from the seat will be twice as great. The greater the acceleration, the greater the push. The only problem is that nothing can withstand the kind of force needed to accelerate to impulse speed quicklycertainly not your body.By the way, this same problem crops up in different contexts throughout Star Trekeven on Earth. At the beginning of Star Trek V: The Final Frontier, James Kirk is free-climbing while on vacation in Yosemite when he slips and falls. Spock, who has on his rocket boots, speeds to the rescue, aborting the captain's fall within a foot or two of the ground. Unfortunately, this is a case where the solution can be as bad as the problem. It is the process of stopping over a distance of a few inches which can kill you, whether or not it is the ground that does the stopping or Spock's Vulcan grip.Well before the reaction forces that will physically tear or break your body occur, other severe physiological problems set in. First and foremost, it becomes impossible for your heart to pump strongly enough to force the blood up to your head. This is why fighter pilots sometimes black out when they perform maneuvers involving rapid acceleration. Special suits have been created to force the blood up from pilots' legs to keep them conscious during acceleration. This physiological reaction remains one of the limiting factors in determining how fast the acceleration of present-day s.p.a.cecraft can be, and it is why NASA, unlike Jules Verne in his cla.s.sic From the Earth to the Moon, has never launched three men into orbit from a giant cannon.If I want to accelerate from rest to, say, 150,000 km/sec, or about half the speed of light, I have to do it gradually, so that my body will not be torn apart in the process. In order not to be pushed back into my seat with a force greater than 3G, my acceleration must be no more than three times the downward acceleration of falling objects on Earth. At this rate of acceleration, it would take some 5 million seconds, or about 2 1/2 months, to reach half light speed! This would not make for an exciting episode.To resolve this dilemma, sometime after the production of the first Const.i.tution Cla.s.s starshipthe Enterprise (NCC-1701) the Star Trek writers had to develop a response to the criticism that the accelerations aboard astarship would instantly turn the crew into "chunky salsa." 1 They came up with "inertial dampers," a kind of cosmic shock absorber and an ingenious plot device designed to get around this sticky little problem.The inertial dampers are most notable in their absence. For example, the Enterprise was nearly destroyed after losing control of the inertial dampers when the microchip life-forms known as Nanites, as part of their evolutionary process, started munching on the ship's central-computer-core memory. Indeed, almost every time the Enterprise is destroyed (usually in some renegade timeline), the destruction is preceded by loss of the inertial dampers. The results of a similar loss of control in a Romulan Warbird provided us with an explicit demonstration that Romulans bleed green.Alas, as with much of the technology in the Star Trek universe, it is much easier to describe the problem the inertial dampers address than it is to explain exactly how they might do it. The First Law of Star Trek physics surely must state that the more basic the problem to be circ.u.mvented, the more challenging the required solution must be. For the reason we have come this far, and the reason we can even postulate a Star Trek future, is that physics is a field that builds on itself. A Star Trek fix must circ.u.mvent not merely some problem in physics but every bit of physical knowledge that has been built upon this problem. Physics progresses not by revolutions, which do away with ail that went before, but rather by evolutions, which exploit the best about what is already understood. Newton's laws will continue to be as true a million years from now as they are today, no matter what we discover at the frontiers of science. If we drop a ball on Earth, it will always fall. If I sit at this desk and write from here to eternity, my b.u.t.tocks will always suffer the same consequences.Be that as it may, it would be unfair simply to leave the inertial dampers hanging without at least some concrete description of how they would have to operate. From what I have argued, they must create an artificial world inside a starship in which the reaction force that responds to the accelerating force is canceled. The objects inside the ship are "tricked" into acting as though they were not accelerating. I have described how accelerating gives you the same feeling as being pulled at by gravity. This connection, which was the basis of Einstein's general theory of relativity, is much more intimate than it may at first seem. Thus there is only one choice for the modus operandi of these gadgets: they must set up an artificial gravitational field inside the ship which "pulls" in the opposite direction to the reaction force, thereby canceling it out.Even if you buy such a possibility, other practical issues must be dealt with. For one thing, it takes some time for the inertial dampers to kick in when unexpected impulses arise. For example, when the Enterprise was b.u.mped into a causality loop by the Bozeman as the latter vessel emerged from a temporal distortion, the crew was thrown all about the bridge (even before the breach in the warp core and the failure of the dampers). I have read in the Enterprise's technical specifications that the response time for the inertial dampers is about 60 milliseconds. 2 Short as this may seem, it would be long enough to kill you if the same delay occurred during programmed periods of acceleration. To convince yourself, think how long it takes for a hammer to smash your head open, or how long it takes for the ground to kill you if you hit it after falling off of a cliff in Yosemite. Just remember that a collision at 10 miles per hour is equivalent to running full speed into a brick wall! The inertial dampers had better be pretty quick to respond. More than one trekker I know has remarked that whenever the ship is buffeted, no one ever gets thrown more than a few feet.Before leaving the familiar world of cla.s.sical physics, I can't help mentioning another technological marvel that must confront Newton's laws in order to operate: the Enterprise's tractor beamhighlighted in the rescue of the Genome colony on Moab IV, when it deflected an approaching stellar core fragment, and in a similar (but failed) attempt to save Bre'el IV by pushing an asteroidal moon back into its...o...b..t. On the face of it, the tractor beam seems simple enoughmore or less like an invisible rope or rodeven if the force exerted may be exotic. Indeed, just like a strong rope, the tractor beam often does a fine job of pulling in a shuttle craft, towing another ship, or inhibiting the escape of an enemy s.p.a.cecraft. The only problem is that when we pull something with a rope, we must be anch.o.r.ed to the ground or to something else heavy. Anyone who has ever been skating knows what happens if you are on the ice and you try to push someone away from you. You do manage to separate, but at your own expense. Without any firm grounding, you are a helpless victim of your own inertia.It was this very principle that prompted Captain Jean-Luc Picard to order Lieutenant Riker to turn off the tractor beam in the episode "The Battle"; Picard pointed out that the ship they were towing would be carried along beside them by its own momentumits inertia. By the same token, if the Enterprise were to attempt to use the tractor beam to ward off the Stargazer, the resulting force would push the Enterprise backward as effectively as it wouldpush the Stargazer forward.This phenomenon has already dramatically affected the way we work in s.p.a.ce at present. Say, for example, that you are an astronaut a.s.signed to tighten a bolt on the Hubble s.p.a.ce Telescope. If you take an electric screwdriver with you to do the job, you are in for a rude awakening after you drift over to the offending bolt. When you switch on the screwdriver as it is pressed against the bolt, you are as likely to start spinning around as the bolt is to turn. This is because the Hubble Telescope is a lot heavier than you are. When the screwdriver applies a force to the bolt, the reaction force you feel may more easily turn you than the bolt, especially if the bolt is still fairly tightly secured to the frame. Of course, if you are lucky enough, like the a.s.sa.s.sins of Chancellor Gorkon, to have gravity boots that secure you snugly to whatever you are standing on, then you can move about as efficiently as we are used to on Earth.Likewise, you can see what will happen if the Enterprise tries to pull another s.p.a.cecraft toward it. Unless the Enterprise is very much heavier, it will move toward the other object when the tractor beam turns on, rather than vice versa. In the depths of s.p.a.ce, this distinction is a meaningless semantic one. With no reference system nearby, who is to say who is pulling whom? However, if you are on a hapless planet like Moab IV in the path of a renegade star, it makes a great deal of difference whether the Enterprise pushes the star aside or the star pushes the Enterprise aside!One trekker I know claims that the way around this problem is already stated indirectly in at least one episode: if the Enterprise were to use its impulse engines at the same time that it turned its tractor beam on, it could, by applying an opposing force with its own engines, compensate for any recoil it might feel when it pushed or pulled on something. This trekker claims that somewhere it is stated that the tractor beam requires the impulse drive to be operational in order to work. I, however, have never noticed any instructions from Kirk or Picard to turn on the impulse engines at the same time the tractor beam is used. And in fact, for a society capable of designing and building inertial dampers, I don't think such a brute force solution would be necessary. Reminded of Geordi LaForge's need for a warp field to attempt to push back the moon at Bre'el IV, I think a careful, if presently unattainable, manipulation of s.p.a.ce and time would do the trick equally well. To understand why, we need to engage the inertial dampers and accelerate to the modern world of curved s.p.a.ce and time.

CHAPTER TWO.

EINSTEIN RaisesThere once was a lady named Bright, Who traveled much faster than light. She departed one day, in a relative way, And returned on the previous night. Anonymous"Time, the final frontier"or so, perhaps, each Star Trek episode should begin. Thirty years ago, in the cla.s.sic episode "Tomorrow Is Yesterday," the round-trip time travels of the Enterprise began. (Actually, at the end of an earlier episode, "The Naked Time," the Enterprise is thrown back in time three days but it is only a one-way trip.) The starship is kicked back to twentieth-century Earth as a result of a close encounter with a "black star" (the term "black hole" having not yet permeated the popular culture). Nowadays exotica like wormholes and "quantum singularities" regularly spice up episodes of Star Trek: Voyager, the latest series. Thanks to Albert Einstein and those who have followed in his footsteps, the very fabric of s.p.a.cetime is filled with drama.While every one of us is a time traveler, the cosmic pathos that elevates human history to the level of tragedy arises precisely because we seem doomed to travel in only one directioninto the future. What wouldn't any of us give to travel into the past, relive glories, correct wrongs, meet our heroes, perhaps even avert disasters, or simply revisit youth with the wisdom of age? The possibilities of s.p.a.ce travel beckon us every time we gaze up at the stars, yet we seem to be permanent captives in the present. The question that motivates not only dramatic license but a surprising amount of modern theoretical physics research can be simply put: Are we or are we notprisoners on a cosmic temporal freight train that cannot jump the tracks?The origins of the modern genre we call science fiction are closely tied to the issue of time travel. Mark Twain's early cla.s.sic A Connecticut Yankee in King Arthur's Court is more a work of fiction than science fiction, in spite of the fact that the whole piece revolves around the time-travel adventures of a hapless American in medieval England. (Perhaps Twain did not dwell longer on the scientific aspects of time travel because of the promise he made to Picard aboard the Enterprise not to reveal his glimpse of the future once he returned to the nineteenth century by jumping through a temporal rift on Devidia II, in the episode "Time's Arrow.") But H. G. Wells's remarkable work The Time Machine completed the transition to the paradigm that Star Trek has followed. Wells was a graduate of the Imperial College of Science and Technology, in London, and scientific language permeates his discussions, as it does the discussions of the Enterprise crew.Surely among the most creative and compelling episodes in the Star Trek series are those involving time travel. Ihave counted no less than twenty-two episodes in the first two series which deal with this theme, and so do three of the Star Trek movies and a number of the episodes of Voyager and Deep s.p.a.ce Nine that have appeared as of this writing.Perhaps the most fascinating aspect of time travel as far as Star Trek is concerned is that there is no stronger potential for violation of the Prime Directive. The crews of Starfleet are admonished not to interfere with the present normal historical development of any alien society they visit. Yet by traveling back in time it is possible to remove the present altogether. Indeed, it is possible to remove history altogether!A famous paradox is to be found in both science fiction and physics: What happens if you go back in time and kill your mother before you were born? You must then cease to exist. But if you cease to exist, you could not have gone back and killed your mother. But if you didn't kill your mother, then you have not ceased to exist. Put another way: if you exist, then you cannot exist, while if you don't exist, you must exist.There are other, less obvious but equally dramatic and perplexing questions that crop up the moment you think about time travel. For example, at the resolution of "Time's Arrow," Picard ingeniously sends a message from the nineteenth to the twenty-fourth century by tapping binary code into Data's severed head, which he knows will be discovered almost five hundred years later and reattached to Data's body. As we watch, he taps the message, and then we cut to LaForge in the twenty-fourth century, as he succeeds in reattaching Data's head. To the viewer these events seem contemporaneous, but they are not; once Picard has tapped the message into Data's head, it lies there for half a millennium. But if I were carefully examining Data's head in the twenty-fourth century and Picard had not yet traveled back in time to change the future, would I see such a message? One might argue that if Picard hasn't traveled back in time yet, there can have been no effect on Data's head. Yet the actions that change Data's programming were performed in the nineteenth century regardless of when Picard traveled back in time to perform them. Thus they have already happened, even if Picard has not yet left! In this way, a cause in the nineteenth century (Picard tapping) can produce an effect in the twenty-fourth century (Data's circuitry change) before the cause in the twenty-fourth century (Picard leaving the ship) produces the effect in the nineteenth century (Picard's arrival in the cave where Data's head is located) which allowed the original cause (Picard tapping) to take place at all.Actually, if the above plot line is confusing, it is nothing compared to the Mother of all time paradoxes, which arises in the final episode of Star Trek: The Next Generation, when Picard sets off a chain of events that will travel back in time and destroy not just his own ancestry but all life on Earth. Specifically, a "subs.p.a.ce temporal distortion" involving "ant.i.time" threatens to grow backward in time, eventually engulfing the amino acid protoplasm on the nascent Earth before the first proteins, which will be the building blocks of life, can form. This is the ultimate case of an effect producing a cause. The temporal distortion is apparently created in the future. If, in the distant past, the subs.p.a.ce temporal distortion was able to destroy the first life on Earth, then life on Earth could never have evolved to establish a civilization capable of creating the distortion in the future!The standard resolution of these paradoxes, at least among many physicists, is to argue a priori that such possibilities must not be allowed in a sensible universe, such as the one we presumably live in. However, the problem is that Einstein's equations of general relativity not only do not directly forbid such possibilities, they encourage them.Within thirty years of the development of the equations of general relativity, an explicit solution in which time travel could occur was developed by the famous mathematician Kurt Gdel, who worked at the Inst.i.tute for Advanced Study in Princeton along with Einstein. In Star Trek language, this solution allowed the creation of a "temporal causality loop," such as the one the Enterprise got caught in after being hit by the Bozeman. The dryer terminology of modern physics labels this a "closed timelike curve." In either case, what it implies is that you can travel on a round-trip and return to your starting point in both s.p.a.ce and time! Gdel's solution involved a universe that, unlike the one we happen to live in, is not expanding but instead is spinning uniformly. In such a universe, it turns out that one could in principle go back in time merely by traveling in a large circle in s.p.a.ce. While such a hypothetical universe is dramatically different than the one in which we live, the mere fact that this solution exists at all indicates clearly that time travel is possible within the context of general relativity.There is a maxim about the universe which I always tell my students: That which is not explicitly forbidden is guaranteed to occur. Or, as Data said in the episode "Parallels," referring to the laws of quantum mechanics, "All things which can occur, do occur." This is the spirit with which I think one should approach the physics of Star Trek. We must consider the distinction not between what is practical and what is not, but between what is possible and what is not.This fact was not, of course, lost on Einstein himself, who wrote, "Kurt Gdel's [time machine solution raises] the problem [that] disturbed me already at the time of the building up of the general theory of relativity, without my having succeeded in clarifying it.... It will be interesting to weigh whether these [solutions] are not to be excluded on physical grounds." 1The challenge to physicists ever since has been to determine what if any "physical grounds" exist that would rule out the possibility of time travel, which the form of the equations of general relativity appears to foreshadow. To discuss such things will require us to travel beyond the cla.s.sical world of general relativity to a murky domain where quantum mechanics must affect even the nature of s.p.a.ce and time. On the way, we, like the Enterprise, will encounter black holes and wormholes. But first we ourselves must travel back in time to the latter half of the nineteenth century.The marriage of s.p.a.ce and time that heralded the modern era began with the marriage, in 1864, of electricity and magnetism. This remarkable intellectual achievement, based on the c.u.mulative efforts of great physicists such as Andr-Marie Ampre, Charles-Augustin de Coulomb, and Michael Faraday, was capped by the brilliant British physicist James Clerk Maxwell. He discovered that the laws of electricity and magnetism not only displayed an intimate relationship with one another but together implied the existence of "electromagnetic waves," which should travel throughout s.p.a.ce at a speed that one could calculate based on the known properties of electricity and magnetism. The speed turned out to be identical to the speed of light, which had previously been measured.Now, since the time of Newton there had been a debate about whether light was a wavethat is, a traveling disturbance in some background mediumor a particle, which travels regardless of the presence of a background medium. The observation of Maxwell that electromagnetic waves must exist and that their speed was identical to that of light ended the debate: light was an electromagnetic wave.Any wave is just a traveling disturbance. Well, if light is an electromagnetic disturbance, then what is the medium that is being disturbed as the wave travels? This became the hot topic for investigation at the end of the nineteenth century. The proposed medium had had a name since Aristotle. It was called the aether, and had thus far escaped any attempts at direct detection. In 1887, however, Albert A. Michelson and Edward Morley, working at the inst.i.tutions that later merged in 1967 to form my present home, Case Western Reserve University, performed an experiment guaranteed to detect not the aether but the aether's effects: Since the aether was presumed to fill all of s.p.a.ce, the Earth was presumed to be in motion through it. Light traveling in different directions with respect to the Earth's motion through the aether ought therefore to show variations in speed. This experiment has since become recognized as one of the most significant of the last century, even though Michelson and Morley never observed the effect they were searching for. In fact, it is precisely because they failed to observe the effect of the Earth's motion through the aether that we remember their names today. (A. A. Michelson actually went on to become the first American n.o.bel laureate in physics for his experimental investigations into the speed of light, and I feel privileged to hold a position today which he held more than a hundred years ago. Edward Morley continued as a renowned chemist and determined the atomic weight of helium, among other things.)The nondiscovery of the aether did send minor ripples of shock throughout the physics community, but, like many watershed discoveries, its implications were fully appreciated only by a few individuals who had already begun to recognize several paradoxes a.s.sociated with the theory of electromagnetism. Around this time, a young high school student who had been eight years old at the time of the Michelson-Morley experiment independently began to try to confront these paradoxes directly. By the time he was twenty-six, in the year 1905, Albert Einstein had solved the problem. But as also often occurs whenever great leaps are made in physics, Einstein's results created more questions than they answered.Einstein's solution, forming the heart of his special theory of relativity, was based on a simple but apparently impossible fact: the only way in which Maxwell's theory of electromagnetism could be self-consistent would be if the observed speed of light was independent of the observer's speed relative to the light. The problem, however, is that this completely defies common sense. If a probe is released from the Enterprise when the latter is traveling at impulse speed, an observer on a planet below will see the probe whiz past at a much higher speed than would a crew member looking out an observation window on the Enterprise. However, Einstein recognized that Maxwell's theory would be self-consistent only if light waves behaved differentlythat is, if their speed as measured by both observers remained identical, independent of the relative motion of the observers. Thus, if I shoot a phaser beam out the front of the Enterprise, and it travels away from the ship at the speed of light toward the bridge of a Romulan Warbird, which itself is approaching the Enterprise at an impulse speed of 3/4 the speed of light, those on the enemy bridge will observe the beam to be heading toward them just at the speed of light and not at 13/4 times the speed of light. This sort of thing has confused some trekkers, who imagine that if the Enterprise is moving at near light speed and another ship is moving in the opposite direction at near light speed, the light from the Enterprise will never catch up with the other ship (and therefore the Enterprise will not be visible to it). Instead, those on the other ship will see the light from the Enterprise approaching at the speed of light.This realization alone was not what made Einstein's a household name. More important was the fact that he was willing to explore the implications of this realization, all of which on the surface seem absurd. In our normal experience, it is time and s.p.a.ce that are absolute, while speed is a relative thing: how fast something is perceived to be moving depends upon how fast you yourself are moving. But as one approaches light speed, it is speed that becomes an absolute quant.i.ty, and therefore s.p.a.ce and time must become relative!This comes about because speed is literally defined as distance traveled during some specific time. Thus, the only way observers in relative motion can measure a single light ray to traverse the same distancesay, 300 million metersrelative to each of them in, say, one second is if each of their "seconds" is different or each of their "meters" is different! It turns out that in special relativity, the "worst of both worlds" happensthat is, seconds and meters both become relative quant.i.ties.From the simple fact that the speed of light is measured to be the same for all observers, regardless of their relative motion, Einstein obtained the four following consequences for s.p.a.ce, time, and matter:(a) Events that occur for one observer at the same time in two different places need not be simultaneous to another observer moving with respect to the first. Each person's "now" is unique to themselves. "Before" and "after" are relative for distant events.(b) All clocks on starships that are moving relative to me will appear to me to be ticking more slowly than my clock. Time is measured to slow down for objects in motion.(c) All yardsticks on starships that are moving relative to me will appear shorter than they would if they were standing still in my frame. Objects, including starships, are measured to contract if they are moving.(d)All ma.s.sive objects get heavier the faster they travel. As they approach the speed of light, they become infinitely heavy. Thus, only ma.s.sless objects, like light, can actually travel at the speed of light.This is not the place to review all of the wonderful apparent paradoxes that relativity introduces into the world. Suffice it to say that, like it or not, consequences (a) through (d) are truethat is, they have been tested. Atomic clocks have been carried aloft in high-speed aircraft and have been observed to be behind their terrestrial counterparts upon their return. In high-energy physics laboratories around the world, the consequences of the special theory of relativity are the daily bread and b.u.t.ter of experiment. Unstable elementary particles are made tomove near the speed of light, and their lifetimes are measured to increase by huge factors. When electrons, which at rest are 2000 times less ma.s.sive than protons, are accelerated to near light speed, they are measured to carry a momentum equivalent to that of their heavier cousins. Indeed, an electron accelerated to .999999999999999999999999999999999999999999999999999999 99999999 times the speed of light would hit you with the same impact as a Mack truck traveling at normal speed.Of course, the reason all these implications of the relativity of s.p.a.ce and time are so hard for us to accept at face value is that we happen to live and move at speeds far smaller than the speed of light. Each of the above effects becomes noticeable only when one is moving at "rel-ativistic" speeds. For example, even at half the speed of light, clocks would slow and yardsticks would shrink by only about 15 percent. On NASA's s.p.a.ce shuttle, which moves at about 5 miles per second around the Earth, clocks tick less than one ten-millionth of a percent slower than their counterparts on Earth.However, in the high-speed world of the Enterprise or any other starship, relativity would have to be confronted on a daily basis. Indeed, in managing a Federation, one can imagine the difficulties of synchronizing clocks across a large segment of the galaxy when a great many of these clocks are moving at close to light speed. As a result, Starfleet apparently has a rule that normal impulse operations for starships are to be limited to a velocity of 0.25 c that is, 1/4 light speed, or a mere 75,000 km/sec. 2Even with such a rule, clocks on ships traveling at this speed will slow by slightly over 3 percent compared with clocks at Starfleet Command. This means that in a month of travel, clocks will have slowed by almost one day. If the Enterprise were to return to Starfleet Command after such a trip, it would be Friday on the ship but Sat.u.r.day back home. I suppose the inconvenience would not be any worse than resetting your clocks after crossing the international date line when traveling to the Orient, except in this case the crew would actually be one. day younger after the round-trip, whereas on a round-trip to the Orient you gain one day going in one direction and lose one going in the other.You can now see how important warp drive is to the Enterprise. Not only is it designed to avoid the ultimate speed limitthe speed of lightand so allow practical travel across the galaxy, but it is also designed to avoid the problems of time dilation, which result when the ship is traveling close to light speed.I cannot overemphasize how significant these facts are. The fact that clocks slow down as one approaches the speed of light has been taken by science fiction writers (and indeed by all those who have dreamed of traveling to the stars) as opening the possibility that one might cross the vast distances between the stars in a human lifetimeat least a human lifetime for those aboard the s.p.a.ceship. At close to the speed of light, a journey to, say, the center of our galaxy would take more than 25,000 years of Earth time. For those aboard the s.p.a.ceship, if it were moving sufficiently close to light speed, the trip might take less than 10 yearsa long time, but not impossibly so. Nevertheless, while this might make individual voyages of discovery possible, it would make the task of running a Federation of civilizations scattered throughout the galaxy impossible. As the writers of Star Trek have correctly surmised, the fact that a 10-year journey for the Enterprise would correspond to a 25,000-year period for Starfleet Command would wreak havoc on any command operation that hoped to organize and control the movements of many such craft. Thus it is absolutely essential that (a) light speed be avoided, in order not to put the Federation out of synchronization, and (b) faster-than-light speed be realized, in order to move practically about the galaxy.The kicker is that, in the context of special relativity alone, the latter possibility cannot be realized. Physics becomes full of impossibilities if super light speed is allowed. Not least among the problems is that because objects get more ma.s.sive as they approach the speed of light, it takes progressively more and more energy to accelerate them by a smaller and smaller amount. As in the myth of the Greek hero Sisyphus, who was condemned to push a boulder uphill for all eternity only to be continually thwarted near the very top, all the energy in the universe would not be sufficient to allow us to push even a speck of dust, much less a starship, past this ultimate speed limit.By the same token, not just light but all ma.s.sless radiation must travel at the speed of light. This means that the many types of beings of "pure energy" encountered by the Enterprise, and later by the Voyager, would have difficulty existing as shown. In the first place, they wouldn't be able to sit still. Light cannot be slowed down, let alone stopped in empty s.p.a.ce. In the second place, any form of intelligent-energy being (such as the "photonic"energy beings in the Voyager series; the energy beings in the Beta Renna cloud, in The Next Generation; the Zetarians, in the original series; and the Dal'Rok, in Deep s.p.a.ce Nine), which is constrained to travel at the speed of light, would have clocks that are infinitely slowed compared to our own. The entire history of the universe would pa.s.s by in a single instant. If energy beings could experience anything, they would experience everything at once! Needless to say, before they could actually interact with corporeal beings the corporeal beings would be long dead.Speaking of time, I think it is time to introduce the Picard Maneuver. Jean-Luc became famous for introducing this tactic while stationed aboard the Stargazer. Even though it involves warp travel, or super light speed, which I have argued is impossible in the context of special relativity alone, it does so for just an instant and it fits in nicely with the discussions here. In the Picard Maneuver, in order to confuse an attacking enemy vessel, one's own ship is accelerated to warp speed for an instant. It then appears to be in two places at once. This is because, traveling faster than the speed of light for a moment, it overtakes the light rays that left it the instant before the warp drive was initiated. While this is a brilliant stategyand it appears to be completely consistent as far as it goes (that is, ignoring the issue of whether it is possible to achieve warp speed)I think you can see that it opens a veritable Pandora's can of worms. In the first place, it begs a question that has been raised by many trekkers over the years: How can the Enterprise bridge crew "see" objects approaching them at warp speed? Just as surely as the Stargazer overtook its own image, so too will all objects traveling at warp speed; one shouldn't be able to see the moving image of a warp-speed object until long after it has arrived. One can only a.s.sume that when Kirk, Picard, or Janeway orders up an image on the viewscreen, the result is an image a.s.sembled by some sort of long-range "subs.p.a.ce" (that is, super-light-speed communication) sensors. Even ignoring this apparent oversight, the Star Trek universe would be an interesting and a barely navigable one, full of ghost images of objects that long ago arrived where they were going at warp speed.Moving back to the sub-light-speed world: We are not through with Einstein yet. His famous relation betweenma.s.s and energy, E=mc 2 , which is a consequence of special relativity, presents a further challenge to s.p.a.ce travel at impulse speeds. As I have described it in chapter 1, a rocket is a device that propels material backward in order to move forward. As you might imagine, the faster the material is propelled backward, the larger will be the forward impulse the rocket will receive. Material cannot be propelled backward any faster than the speed of light. Even propelling it at light speed is not so easy: the only way to get propellant moving backward at light speed is to make the fuel out of matter and antimatter, which (as I describe in a later chapter) can completely annihilate to produce pure radiation moving at the speed of light.However, while the warp drive aboard the Enterprise uses such fuel, the impulse drive does not. It is poweredinstead by nuclear fusionthe same nuclear reaction that powers the Sun by turning hydrogen into helium. In fusion reactions, about 1 percent of the available ma.s.s is converted into energy. With this much available energy, the helium atoms that are produced can come streaming out the back of the rocket at about an eighth of the speed of light. Using this exhaust velocity for the propellant, we then can calculate the amount of fuel the Enterprise needs in order to accelerate to, say, half the speed of light. The calculation is not difficult, but I will just give the answer here. It may surprise you. Each time the Enterprise accelerates to half the speed of light, it must burn 81 TIMES ITS ENTIRE Ma.s.s in hydrogen fuel. Given that a Galaxy Cla.s.s starship such as Picard's Enterprise-D would weigh in excess of 4 million metric tons, 3 this means that over 300 million metric tons of fuel would need to be used each time the impulse drive is used to accelerate the ship to half light speed! If one used a matter-antimatter propulsion system for the impulse drive, things would be a little better. In this case, one would have to burn merely twice the entire ma.s.s of the Enterprise in fuel for each such acceleration.It gets worse. The calculation I described above is correct for a single acceleration. To bring the ship to a stop at its destination would require the same factor of 81 times its ma.s.s in fuel. This means that just to go somewhere at half light speed and stop again would require fuel in the amount of 81x81= 6561 TIMES THE ENTIRE SHIP'S Ma.s.s! Moreover, say that one wanted to achieve the acceleration to half the speed of light in a few hours (we will a.s.sume, of course, that the inertial dampers are doing their job of shielding the crew and ship from the tremendous G-forces that would otherwise ensue). The power radiated as propellant by the engines would then be about 10 22 wattsor about a billion times the total average power presently produced and used by all humanactivities on Earth!Now, you may suggest (as a bright colleague of mine did the other day when I presented him with this argument) that there is a subtle loophole. The argument hinges on the requirement that you carry your fuel along with the rocket. What if, however, you harvest your fuel as you go along? After all, hydrogen is the most abundant elementin the universe. Can you not sweep it up as you move through the galaxy? Well, the average density of matter in our galaxy is about one hydrogen atom per cubic centimeter. To sweep up just one gram of hydrogen per second, even moving at a good fraction of the speed of light, would require you to deploy collection panels with a diameter of over 25 miles. And even turning all this matter into energy for propulsion would provide only about a hundred- millionth of the needed propulsion power!To paraphrase the words of the n.o.bel prizewinning physicist Edward Purcell, whose arguments I have adapted and extended here:If this sounds preposterous to you, you are right. Its preposterousness follows from the elementary laws of cla.s.sical mechanics and special relativity. The arguments presented here are as inescapable as the fact that a ball will fall when you drop it at the Earth's surface. Rocket-propelled s.p.a.ce travel through the galaxy at near light speed is not physically practical, now or ever!So, do I end the book here? Do we send back our Star Trek memorabilia and ask for a refund? Well, we are still not done with Einstein. His final, perhaps greatest discovery holds out a glimmer of hope after all.Fast rewind back to 1908: Einstein's discovery of the relativity of s.p.a.ce and time heralds one of those "Aha!" experiences that every now and then forever change our picture of the universe. It was in the fall of 1908 that the mathematical physicist Hermann Minkowski wrote these famous words: "Henceforth, s.p.a.ce by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."What Minkowski realized is that even though s.p.a.ce and time are relative for observers in relative motionyour clock can tick slower than mine, and my distances can be different from yoursif s.p.a.ce and time are instead merged as part of a four-dimensional whole (three dimensions of s.p.a.ce and one of time), an "absolute" objective reality suddenly reappears.The leap of insight Minkowski had can be explained by recourse to a world in which everyone has monocular vision and thus no direct depth perception. If you were to close one eye, so that your depth perception was reduced, and I were to hold a ruler up for you to see, and I then told someone else, who was observing from a different angle, to close one eye too, the ruler I was holding up would appear to the other observer to be a different length than it would appear to be to youas the following bird's-eye view shows.Each observer in the example above, without the direct ability to discern depth, will label "length" (L or L') to be the two-dimensional projection onto his or her plane of vision of the actual three-dimensional length of the ruler. Now, because we know that s.p.a.ce has three dimensions, we are not fooled by this trick. We know that viewing something from a different angle does not change its real length, even if it changes its apparent length. Minkowski showed that the same idea can explain the various paradoxes of relativity, if we now instead suppose that our perception of s.p.a.ce is merely a three-dimensional slice of what is actually a four-dimensional manifold in which s.p.a.ce and time are joined. Two different observers in relative motion perceive different three-dimensional slices of the underlying four-dimensional s.p.a.ce in much the same way that the two rotated observers pictured here view different two-dimensional slices of a three-dimensional s.p.a.ce.Minkowski imagined that the spatial distance measured by two observers in relative motion is a projection of an underlying four-dimensional s.p.a.cetime distance onto the three-dimensional s.p.a.ce that they can sense; and, similarly, that the temporal "distance" between two events is a projection of the four-dimensional s.p.a.cetime distance onto their own timeline. Just as rotating something in three dimensions can mix up width and depth, so relative motion in four-dimensional s.p.a.ce can mix up different observers' notions of "s.p.a.ce" and "time." Finally, just as the length of an object does not change when we rotate it in s.p.a.ce, the four-dimensional s.p.a.cetime distance between two events is absoluteindependent of how different observers in relative motion a.s.sign "spatial" and "temporal" distances.So the crazy invariance of the speed of light for all observers provided a key clue to unravel the true nature of the four-dimensional universe of s.p.a.cetime in which we actually live. Light displays the hidden connection between s.p.a.ce and time. Indeed, the speed of light defines the connection.It is here that Einstein returned to save the day for Star Trek. Once Minkowski had shown that s.p.a.cetime in special relativity was like a four-dimensional sheet of paper, Einstein spent the better part of the next decade flexing his mathematical muscles until he was able to bend that sheet, which in turn allows us to bend the rules of the game. As you may have guessed, light was again the key.

CHAPTER THREE.

Shows His Hand "How little do you mortals understand time. Must you be so linear, Jean-Luc?"Q to Picard, in "All Good Things... .The planet Vulcan, home to Spock, actually has a venerable history in twentieth-century physics. A great puzzle in astrophysics in the early part of this century was the fact that the perihelion of Mercurythe point of its closest approach to the Sunwas precess-ing around the Sun each Mercurian year by a very small amount in a way that was not consistent with Newtonian gravity. It was suggested that a new planet existed inside Mercury's...o...b..t which could perturb it in such a way as to fix the problem. (In fact, the same solution to an anomaly in the orbit of Ura.n.u.s had earlier led to the discovery of the planet Neptune.) The name given to the hypothetical planet was Vulcan.Alas, the mystery planet Vulcan is not there. Instead, Einstein proposed that the flat s.p.a.ce of Newton and Minkowski had to be given up for the curved s.p.a.cetime of general relativity. In this curved s.p.a.ce, Mercury's...o...b..t would deviate slightly from that predicted by Newton, explaining the observed discrepancy. While this removed the need for the planet Vulcan, it introduced possibilities that are much more exciting. Along with curved s.p.a.ce come black holes, wormholes, and perhaps even warp speeds and time travel.Indeed, long before the Star Trek writers conjured up warp fields, Einstein warped s.p.a.cetime, and, like the Star Trek writers, he was armed with nothing other than his imagination. Instead of imagining twenty-second-century starship technology, however, Einstein imagined an elevator. He was undoubtedly a great physicist, but he probably never would have sold a screenplay.Nonetheless, his arguments remain intact when translated aboard the Enterprise. Because light is the thread that weaves together s.p.a.ce and time, the trajectories of light rays give us a map of s.p.a.cetime just as surely as warp and weft threads elucidate the patterns of a tapestry. Light generally travels in straight lines. But what if a Romulan commander aboard a nearby Warbird shoots a phaser beam at Picard as he sits on the bridge of his captain's yacht Calypso, having just engaged the impulse drive (we will a.s.sume the inertial dampers are turned off for this example)? Picard would accelerate forward, narrowly missing the brunt of the phaser blast. When viewed in Picard's frame of reference, things would look like the figure at the top of the following page.So, for Picard, the trajectory of the phaser ray would be curved. What else would Picard notice? Well, recalling the argument in the first chapter, as long as the inertial dampers are turned off, he would be thrust back in his seat. In fact, I also noted there that if Picard was being accelerated forward at the same rate as gravity causes things to accelerate downward at the Earth's surface, he would feel exactly the same force pushing him back against his seat that he would feel pushing him down if he were standing on Earth. In fact, Einstein argued that Picard (or his equivalent in a rising elevator) would never be able to perform any experiment that could tell the difference between the reaction force due to his acceleration and the pull of gravity from some nearby heavy object outside the ship. Because of this, Einstein boldly went where no physicist had gone before, and reasoned that whatever phenomena an accelerating observer experienced would be identical to the phenomena an observer in a gravitational field experienced.Our example implies the following: Since Picard observes the phaser ray bending when he is accelerating away from it, the ray must also bend in a gravitational field. But if light rays map out s.p.a.cetime, then s.p.a.cetime must bend in a gravitational field. Finally, since matter produces a gravitational field, then matter must bend s.p.a.cetime!Now, you may argue that since light has energy, and ma.s.s and energy are related by Einstein's famous equation, then the fact that light bends in a gravitational field is no big surpriseand certainly doesn't seem to imply that we have to believe that s.p.a.cetime itself need be curved. After all, the paths that matter follows bend too (try throwing a ball in the air). Galileo could have shown, had he known about such objects, that the trajectories of baseb.a.l.l.s and Pathfinder missiles bend, but he never would have mentioned curved s.p.a.ce.Well, it turns out that you can calculate how much a light ray should bend if light behaved the same way a baseball does, and then you can go ahead and measure this bending, as Sir Arthur Stanley Eddington did in 1919 when he led an expedition to observe the apparent position of stars on the sky very near the Sun during a solar eclipse. Remarkably, you would find, as Eddington did, that light bends exactly twice as much as Galileo might have predicted if it behaved like a baseball in flat s.p.a.ce. As you may have guessed, this factor of 2 is just what Einstein predicted if s.p.a.cetime was curved in the vicinity of the Sun and light (or the planet Mercury, for that matter) was locally traveling in a straight line in this curved s.p.a.ce! Suddenly, Einstein's was a household name.Curved s.p.a.ce opens up a whole universe of possibilities, if you will excuse the pun. Suddenly we, and the Enterprise, are freed from the shackles of the kind of linear thinking imposed on us in the context of special relativity, which Q, for one, seemed to so abhor. One can do many things on a curved manifold which are impossible on a flat one. For example, it is possible to keep traveling in the same direction and yet return to where you beganpeople who travel around the world do it all the time.The central premise of Einstein's general relativity is simple to state in words: the curvature of s.p.a.cetime isdirectly determined by the distribution of matter and energy contained within it. Einstein's equations, in fact, provide simply the strict mathematical relation between curvature on the one hand and matter and energy on the other:What makes the theory so devilishly difficult to work with is this simple feedback loop: The curvature of s.p.a.cetime is determined by the distribution of matter and energy in the universe, but this distribution is in turn governed by the curvature of s.p.a.ce. It is like the chicken and the egg. Which was there first? Matter acts as the source of curvature, which in turn determines how matter evolves, which in turn alters the curvature, and so on.Indeed, this may be perhaps the most important single aspect of general relativity as far as Star Trek is concerned. The complexity of the theory means that we still have not yet fully understood all its consequences; therefore we cannot rule out various exotic possibilities. It is these exotic possibilities that are the grist of Star Trek's mill. In fact, we shall see that all these possibilities rely on one great unknown that permeates everything, from wormholes and black holes to time machines.The first implication of the fact that s.p.a.cetime need not be flat which will be important to the adventures of the Enterprise is that time itself becomes an even more dynamic quant.i.ty than it was in special relativity. Time can flow at different rates for different observers even if they are not moving relative to each other. Think of the ticks of a clock as the ticks on a ruler made of rubber. If I were to stretch or bend the ruler, the s.p.a.cing between the ticks would differ from point to point. If this s.p.a.cing represents the ticks of a clock, then clocks located in different places can tick at different rates. In general relativity, the only way to "bend" the ruler is for a gravitational field to be present, which in turn requires the presence of matter.To translate this into more pragmatic terms: if I put a heavy iron ball near a clock, it should change the rate at which the clock ticks. Or more practical still, if I sleep with my alarm clock tucked next to my body's rest ma.s.s, I will be awakened a little later than I would otherwise, at least as far as the rest of the world is concerned.A famous experiment done in the physics laboratories at Harvard University in 1960 first demonstrated that time can depend on where you are. Robert Pound and George Rebka showed that the frequency of gamma radiation measured at its source, in the bas.e.m.e.nt of the building, differed from the frequency of the radiation when it was received 74 feet higher, on the building's roof (with the detectors having been carefully calibrated so that any observed difference would not be detector-related). The shift was an incredibly small amount about 1 part in a million billion. If each cycle of the gamma-ray wave is like the tick of an atomic clock, this experiment implies that a clock in the bas.e.m.e.nt will appear to be running more slowly than an equivalent atomic clock on the roof. Time slows on the lower floor because this is closer to the Earth than the roof is, so the gravitational field, and hence the s.p.a.cetime curvature, is larger there. As small as this effect was, it was precisely the value predicted by general relativity, a.s.suming that s.p.a.cetime is curved near the Earth.The second implication of curved s.p.a.ce is perhaps even more exciting as far as s.p.a.ce travel is concerned. If s.p.a.ce is curved, then a straight line need not be the shortest distance between two points. Here's an example. Consider a circle on a piece of paper. Normally, the shortest distance between two points A and B located on opposite sidesof the circle is given by the line connecting them through the center of the circle:If, instead, one were to travel around the circle to get from A to B, the journey would be about 1 1/2 times as long. However, let me draw this circle on a rubber sheet, and distort the central region:Now, when viewed in our three-dimensional perspective, it is clear that the journey from A to B taken through the center of the region will be much longer than that taken by going around the circle. Note that if we took a snapshot of this from above, so we would have only a two-dimensional perspective, the line from A to B through the center would look like a straight line. More relevant perhaps, if a tiny bug (or two-dimensional beings, of the type encountered by the Enterprise) were to follow the trajectory from A to B through the center by crawling along the surface of the sheet, this trajectory would appear to be straight. The bug would be amazed to find that the straight line through the center between A and B was no longer the shortest distance between these two points. If the bug were intelligent, it would be forced to the conclusion that the two-dimensional s.p.a.ce it lived in was curved. Only by viewing the embedding of this sheet in the underlying three-dimensional s.p.a.ce can we observe the curvature directly.Now, remember that we live within a four-dimensional s.p.a.cetime that can be curved, and we can no more perceive the curvature of this s.p.a.ce directly than the bug crawling on the surface of the sheet can detect the curvature of the sheet. I think you know where I am heading: If, in curved s.p.a.ce, the shortest distance between two points need not be a straight line, then it might be possible to traverse what appears along the line of sight to be a huge distance, by finding instead a shorter route through curved s.p.a.cetime.These proper