CHAPTER TWO

EINSTEIN Raises

There 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 classic 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 spacetime 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 space 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 not

prisoners 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 classic 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. I

have 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 Space 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 “subspace temporal distortion” involving “antitime” 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 subspace 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 Gšdel, who worked at the Institute 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 space and time! Gšdel'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 space. 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 Gšdel'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.” 1

The 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 classical world of general relativity to a murky domain where quantum mechanics must affect even the nature of space 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 space and time that heralded the modern era began with the marriage, in 1864, of electricity and magnetism. This remarkable intellectual achievement, based on the cumulative efforts of great physicists such as AndrŽ-Marie Amp�re, 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 space 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 institutions 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 space, 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 Nobel 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 associated 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 space 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 quantity, and therefore space 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 quantities.

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 space, 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 massive objects get heavier the faster they travel. As they approach the speed of light, they become infinitely heavy. Thus, only massless 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 butter of experiment. Unstable elementary particles are made to

move near the speed of light, and their lifetimes are measured to increase by huge factors. When electrons, which at rest are 2000 times less massive 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 space 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 space 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. 2

Even 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 Saturday 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 spaceship. 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 spaceship, 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 massive 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 massless 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 space. 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 Space 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 pass 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 assume that when Kirk, Picard, or Janeway orders up an image on the viewscreen, the result is an image assembled by some sort of long-range “subspace” (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 between

mass and energy, E=mc 2 , which is a consequence of special relativity, presents a further challenge to space 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 powered

instead by nuclear fusionthe same nuclear reaction that powers the Sun by turning hydrogen into helium. In fusion reactions, about 1 percent of the available mass 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 MASS in hydrogen fuel. Given that a Galaxy Class 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 mass 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 mass 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 MASS! Moreover, say that one wanted to achieve the acceleration to half the speed of light in a few hours (we will assume, 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 human

activities 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 element

in 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 Nobel 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 classical 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 space 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 space 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, space 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 space and time are relative for observers in relative motionyour clock can tick slower than mine, and my distances can be different from yoursif space and time are instead merged as part of a four-dimensional whole (three dimensions of space 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 space 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 space is merely a three-dimensional slice of what is actually a four-dimensional manifold in which space and time are joined. Two different observers in relative motion perceive different three-dimensional slices of the underlying four-dimensional space in much the same way that the two rotated observers pictured here view different two-dimensional slices of a three-dimensional space.

Minkowski imagined that the spatial distance measured by two observers in relative motion is a projection of an underlying four-dimensional spacetime distance onto the three-dimensional space that they can sense; and, similarly, that the temporal “distance” between two events is a projection of the four-dimensional spacetime distance onto their own timeline. Just as rotating something in three dimensions can mix up width and depth, so relative motion in four-dimensional space can mix up different observers' notions of “space” and “time.” Finally, just as the length of an object does not change when we rotate it in space, the four-dimensional spacetime distance between two events is absoluteindependent of how different observers in relative motion assign “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 spacetime in which we actually live. Light displays the hidden connection between space 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 spacetime 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.