From Eternity to Here: The Quest for the Ultimate Theory of Time

by Sean M. Carroll

  But a real black hole, as predicted by general relativity, is a lot more dramatic than that. It is a true region of no return—once you enter, there is no possibility of leaving, no matter what technological marvels you have at your disposal. That’s because general relativity, unlike Newtonian gravity or special relativity, allows spacetime to curve. At every event in spacetime we find light cones that divide space into the past, future, and places we can’t reach. But unlike in special relativity, the light cones are not fixed in a rigid alignment; they can tilt and stretch as spacetime curves under the influence of matter and energy. In the vicinity of a massive object, light cones tilt toward the object, in accordance with the tendency of things to be pulled in by the gravitational field. A black hole is a region of spacetime where the light cones have tilted so much that you would have to move faster than the speed of light to escape. Despite the similarity of language, that’s an enormously stronger statement than “the escape velocity is larger than the speed of light.” The boundary defining the black hole, separating places where you still have a chance to escape from places where you are doomed to plunge ever inward, is the event horizon.

  Figure 19: Light cones tilt in the vicinity of a black hole. The event horizon, demarcating the edge of the black hole, is the place where they tip over so far that nothing can escape without moving faster than light.

  There may be any number of ways that black holes could form in the real world, but the standard scenario is the collapse of a sufficiently massive star. In the late 1960s, Roger Penrose and Stephen Hawking proved a remarkable feature of general relativity: If the gravitational field becomes sufficiently strong, a singularity must be formed.74 You might think that’s only sensible, since gravity becomes stronger and stronger and pulls matter into a single point. But in Newtonian gravity, for example, it’s not true. You can get a singularity if you try hard enough, but the generic result of squeezing matter together is that it will reach some point of maximum density. But in general relativity, the density and spacetime curvature increase without limit, until they form a singularity of infinite curvature. Such a singularity lies inside every black hole.

  It would be wrong to think of the singularity as residing at the “center” of the black hole. If we look carefully at the representation of spacetime near a black hole shown in Figure 19, we see that the future light cones inside the event horizon keep tipping toward the singularity. But that light cone defines what the observer at that event would call “the future.” Like the Big Bang singularity in the past, the black hole singularity in the future is a moment of time, not a place in space. Once you are inside the event horizon, you have absolutely no choice but to continue on to the grim destiny of the singularity, because it lies ahead of you in time, not in some direction in space. You can no more avoid hitting the singularity than you can avoid hitting tomorrow.

  When you actually do fall through the event horizon, you might not even notice. There is no barrier there, no sheet of energy that you pass through to indicate that you’ve entered a black hole. There is simply a diminution of your possible future life choices; the option of “returning to the outside universe” is no longer available, and “crashing into the singularity” is your only remaining prospect. In fact, if you knew how massive the black hole was, you could calculate precisely how long it will take (according to a clock carried along with you) before you reach the singularity and cease to exist; for a black hole with the mass of the Sun, it would be about one-millionth of a second. You might try to delay this nasty fate, for example, by firing rockets to keep yourself away from the singularity, but it would only be counterproductive. According to relativity, unaccelerated motion maximizes the time between two events. By struggling, you only hasten your doom.75

  There is a definite moment on your infalling path when you cross the event horizon. If we imagine that you had been sending a constant stream of radio updates to your friends outside, they will never be able to receive anything sent after that time. They do not, however, see you wink out of existence; instead, they receive your signals at longer and longer intervals, increasingly redshifted to longer and longer wavelengths. Your final moment before crossing the horizon is (in principle) frozen in time from the point of view of an external observer, although it becomes increasingly dimmer and redder as time passes.

  Figure 20: As an object approaches an event horizon, to a distant observer it appears to slow down and become increasingly redshifted. The moment on the object’s world line when it crosses the horizon is the last moment it can be seen from the outside.

  WHITE HOLES: BLACK HOLES RUN BACKWARD

  If you think a bit about this black-hole story, you’ll notice something intriguing: time asymmetry. We have been casually tossing around terminology that assumes a directionality to time; we say “once you pass the event horizon you can never leave,” but not “once you leave the event horizon you can never return.” That’s not because we have been carelessly slipping into temporally asymmetric language; it’s because the notion of a black hole is intrinsically time-asymmetric. The singularity is unambiguously in your future, not in your past.

  The time asymmetry here isn’t part of the underlying physical theory. General relativity is perfectly time-symmetric; for every specific spacetime that solves Einstein’s equation, there is another solution that is identical except that the direction of time is reversed. A black hole is a particular solution to Einstein’s equation, but there are equivalent solutions that run the other way: white holes.

  The description of a white hole is precisely the same as that of a black hole, if we simply reverse the tenses of all words that refer to time. There is a singularity in the past, from which light cones emerge. The event horizon lies to the future of the singularity, and the external world lies to the future of that. The horizon represents a place past which, once you exit, you can never return to the white-hole region.

  Figure 21: The spacetime of a white hole is a time-reversed version of a black hole.

  So why do we hear about black holes in the universe all the time, and hardly ever hear about white holes? For one thing, notice that we can’t “make” a white hole. Since we are in the external world, the singularity and event horizon associated with a white hole are necessarily in our past. So it’s not a matter of wondering what we would do to create a white hole; if we’re going to find one, it will already have been out there in the universe from the beginning.

  But in fact, thinking slightly more carefully, we should be suspicious of that word make. Why, in a world governed by reversible laws of physics, do we think of ourselves as “making” things that persist into the future, but not things that extend into the past? It’s the same reason why we believe in free will: A low-entropy boundary condition in the past dramatically fixes what possibly could have happened, while the absence of any corresponding future boundary condition leaves what can yet happen relatively open.

  So when we ask, “Why does it seem relatively straightforward to make a black hole, while white holes are something that we would have to find already existing in the universe?” the answer should immediately suggest itself: because a black hole tends to have more entropy than the things from which you would make it. Actually calculating what the entropy is turns out to be a tricky business involving Hawking radiation, as we’ll see in Chapter Twelve. But the key point is that black holes have a lot of entropy. Black holes turn out to provide the strongest connection we have between gravitation and entropy—the two crucial ingredients in an ultimate explanation of the arrow of time.

  6

  LOOPING THROUGH TIME

  You see, my son, here time changes into space.

  —Richard Wagner, Parsifal

  Everyone knows what a time machine looks like: something like a steampunk sled with a red velvet chair, flashing lights, and a giant spinning wheel on the back. For those of a younger generation, a souped-up stainless-steel sports car is an acceptable substitute; our British readers might thin
k of a 1950s-style London police box.76 Details of operation vary from model to model, but when one actually travels in time, the machine ostentatiously dematerializes, presumably to be re-formed many millennia in the past or future.

  That’s not how it would really work. And not because time travel is impossible and the whole thing is just silly; whether or not time travel is possible is more of an open question than you might suspect. I’ve emphasized that time is kind of like space. It follows that, if you did stumble across a working time machine in the laboratory of some mad inventor, it would simply look like a “space machine”—an ordinary vehicle of some sort, designed to move you from one place to another. If you want to visualize a time machine, think of launching a rocket ship, not disappearing in a puff of smoke.

  So what is actually entailed in traveling through time? There are two cases of possible interest: traveling to the future and traveling to the past. To the future is easy: Just keep sitting in your chair. Every hour, you will move one hour into the future. “But,” you say, “that’s boring. I want to move far into the future, really quickly, a lot faster than one hour per hour. I want to visit the twenty-fourth century before lunchtime.” But we know it’s impossible to move faster than one hour per hour, relative to a clock that travels along with you. You might be able to trick yourself, by going to sleep or entering suspended animation, but time will still be passing.

  On the other hand, you could distort the total amount of time experienced along your world line as compared to the world lines of other people. In a Newtonian universe even that wouldn’t be possible, as time is universal and every world line connecting the same two events experiences the same elapsed time. But in special relativity we can affect the elapsed time by moving through space. Unaccelerated motion gives us the longest time between events; so if you want to get to the future quickly (from your perspective), you just have to move on a highly non-straight path through spacetime. You could zip out on a rocket near the speed of light and then return; or, given sufficient fuel, you could just fly around in circles at ultra-high speed, never straying very far from your starting point in space. When you stopped and exited your spaceship, besides being dizzy you would have “traveled into the future”; more accurately, you would have experienced less time along your world line than was experienced by anyone who stayed behind. Traveling to the future is easy, and getting there more quickly is just a technology problem, not something that conflicts with the fundamental laws of physics.

  But you might want to come back, and that’s where the challenge lies. The problems with traveling through time arise when we contemplate traveling to the past.

  CHEATING SPACETIME

  Lessons learned from watching Superman movies notwithstanding, traveling backward in time is not a matter of reversing the rotation of the Earth. Spacetime itself has to cooperate. Unless, of course, you cheat, by moving faster than the speed of light.

  In a Newtonian universe, traveling backward in time is simply out of the question. World lines extend through a spacetime that is uniquely divided into three-dimensional moments of equal time, and the one unbreakable rule is that they must never double back and return to the past. In special relativity, things aren’t much better. Defining “moments of equal time” across the universe is highly arbitrary, but at every event we are faced with the restrictions enforced by light cones. If we are made of ordinary stuff, confined to move from each event forward into the interior of its light cone, there is no hope of traveling backward in time; in a spacetime diagram, we are doomed to march relentlessly upward.

  Things would be a little bit more interesting if we were made of non-ordinary stuff. In particular, if we were made of tachyons—particles that always move faster than light. Sadly, we are not made of tachyons, and there are good reasons to believe that tachyons don’t even exist. Unlike ordinary particles, tachyons are forced to always travel outside the light cone. In special relativity, whenever we move outside the light cone, from somebody’s point of view we’re also moving backward in time. More important, the light cones are the only structure defined on spacetime in relativity; there is no separate notion of “space at one moment of time.” So if a particle can start at some event where you are momentarily located and move outside your light cone (faster than light), it can necessarily move into the past from your point of view. There’s nothing to stop it.

  Tachyons, therefore, can apparently do something scary and unpredictable: “start” from an event on the world line of some ordinary (slower-than-light) object, defined by some position in space and some moment in time, and travel on a path that takes them to a previous point on the same world line. If you had a flashlight that emitted tachyons, you could (in principle) construct an elaborate series of mirrors by which you could send signals in Morse code to yourself in the past. You could warn your earlier self not to eat the shrimp in that restaurant that one time, or to go on that date with the weirdo from the office, or to sink your life savings into Pets.com stock.

  Figure 22: If tachyons could exist, they could be emitted by ordinary objects and zip around to be absorbed in the past. At every event along its trajectory, the tachyon moves outside the light cone.

  Clearly, the possibility of travel backward in time raises the possibility of paradoxes, which is unsettling. There is a cheap way out: Notice that tachyons don’t seem to exist, and declare that they are simply incompatible with the laws of physics.77 That is both fruitful and accurate, at least as far as special relativity is concerned. When curved spacetime gets into the game, things become more interesting.

  CIRCLES IN TIME

  For those of us who are not made of tachyons, our trajectories through spacetime are limited by the speed of light. From whatever event defines our current location, we necessarily move “forward in time,” toward some other event inside our light cone—in technical jargon, we move on a timelike path through spacetime. This is a local requirement, referring only to features of our neighborhood of the universe. But in general relativity, spacetime is curved. That means light cones in our neighborhood may be tilted with respect to those far away. It’s that kind of tilting that leads to black holes.

  But imagine that, instead of light cones tilting inward toward a singularity and creating a black hole, we had a spacetime in which light cones tilted around a circle, as shown in Figure 23. Clearly this would require some sort of extremely powerful gravitational field, but we’re letting our imaginations roam. If spacetime were curved in that way, a remarkable thing would happen: We could travel on a timelike path, always moving forward into our future light cone, and yet eventually meet ourselves at some moment in our past. That is, our world line could describe a closed loop in space, so that it intersected itself, bringing us at one moment in our lives face-to-face with ourselves at some other moment.

  Figure 23: In curved spacetime, we can imagine light cones tilting around in a circle, creating closed timelike curves.

  Such a world line—always moving forward in time from a local perspective, but managing to intersect itself in the past—is a closed timelike curve, or CTC. That’s what we really mean when we talk about a “time machine” in general relativity. To actually move along the closed timelike curve would involve ordinary travel through spacetime, on a spaceship or something even more mundane—perhaps even sitting “motionless” in your chair. It’s the curvature of spacetime itself that brings you into contact with your own past. This is a central feature of general relativity that will become important later on when we return to the origin of the universe and the problem of entropy: Spacetime is not stuck there once and for all, but can change (and even come into or pop out of existence) in response to the effects of matter and energy.

  It’s not difficult to find spacetimes in general relativity that include closed timelike curves. All the way back in 1949, mathematician and logician Kurt Gödel found a solution to Einstein’s equation that described a “spinning” universe, which contains closed timelike curves passi
ng through every event. Gödel was friends with Einstein in his later years at the Institute for Advanced Study in Princeton, and the idea for his solution arose in part from conversations between the two men.78 In 1963, New Zealand mathematician Roy Kerr found the exact solution for a rotating black hole; interestingly, the singularity takes the form of a rapidly spinning ring, the vicinity of which is covered by closed timelike curves.79 And in 1974, Frank Tipler showed that an infinitely long, rotating cylinder of matter would create closed timelike curves around it, if the cylinder were sufficiently dense and rotating sufficiently rapidly.80

  But you don’t have to work nearly so hard to construct a spacetime with closed timelike curves. Consider good old garden-variety flat spacetime, familiar from special relativity. But now imagine that the timelike direction (as defined by some particular unaccelerated observer) is a circle, rather than stretching on forever. In such a universe, something that moved forward in time would find itself coming back to the same moment in the universe’s history over and over. In the Harold Ramis movie Groundhog Day, Bill Murray’s character keeps waking up every morning to experience the exact same situations he had experienced the day before. The circular-time universe we’re imagining here is something like that, with two important exceptions: First, it would be truly identical every day, including the actions of the protagonist himself. And second, there would not be any escape. In particular, winning the love of Andie MacDowell would not save you.