The Greatest Story Ever Told—So Far

by Lawrence M. Krauss

  Van Gogh died fifteen years before Einstein developed his ideas on space and time, but his paintings make it clear that our perceptions of the world are subjective. Picasso may have had the chutzpah to claim that he painted what he saw, even as he produced representations of disjointed people with body parts pointing in different directions, but van Gogh’s masterpieces demonstrate that the world can look very different to different people.

  So too, Einstein explicitly argued, for the first time as far as I know in the history of physics, that “here” and “now” are observer-dependent concepts and not universal ones.

  His argument was simple, based on the equally simple fact that we cannot be in two places at once.

  We are accustomed to feeling that we share the same reality with those around us because we appear to share the same experiences as we look about together. But that is an illusion created by the fast speed of light.

  When I observe something happening now, say, a car crash down the street or two lovers kissing under a lamppost as I walk nearby, neither of these events happened now, but rather then. The light that enters my eye was reflected off the car or the people just a little bit earlier.

  Similarly when I take a photo of a beautiful landscape, as I just did in Northern Ireland where I began writing this chapter, the scene I captured is not a scene merely spread out in space, but rather in space and time. The light from the distant pillared cliffs at Giant’s Causeway perhaps a kilometer away left those cliffs well before (perhaps thirty-millionths of a second before) the light from the people in the foreground scrambling over the hexagonal lava pods left to reach my camera at the same time.

  With this realization, Einstein asked himself what two events that one observer views as happening at the same time in two different locations would look like for another observer moving with respect to the first observer while the observations were being made. The example he considered involved a train, because he lived in Switzerland at a time when a train was leaving about every five minutes for somewhere in the country from virtually any other place in the country.

  Imagine the picture shown below in which lightning hits two points beside either end of a train that are equidistant from observer A, who is at rest with respect to those points, and observer B on a moving train, who passes by A at the instant A later determines the lightning bolts struck:

  A little while later A will see both lightning flashes reaching him at the same time. B, however, will have moved during this time. Therefore the light wave bringing the information that a flash occurred on the right will already have passed B, and the light bringing the information about the flash on the left will not yet have reached him.

  B sees the light coming from either end of his train, and indeed the flash at the front end occurs before the flash at the rear end. Since he measures the light as traveling toward him at speed c, and since he is in the middle of his train, he concludes therefore that the right-hand flash must have occurred before the left-hand flash.

  Who is right here? Einstein had the temerity to suggest that both observers were right. If the speed of light were like other speeds, then B would of course see one wave before the other, but he would see them traveling toward him at different speeds (the one he was moving toward would be faster and the one from which he was moving away would be slower), and he would therefore infer that the events happened at the same time. But because both light rays are measured by B to be traveling toward him at the same speed, c, the reality he infers is completely different.

  As Einstein pointed out, when defining what we mean by different physical quantities, measurement is everything. Imagining a reality that is independent of measurement might be an interesting philosophical exercise, but from a scientific perspective it is a sterile line of inquiry. If both A and B are located at the same place at the same time, they must both measure the same thing at that instant, but if they are in remote locations, almost all bets are off. Every measurement that B can make tells him that the event at the forward end of his train happened before the next, while every measurement that A makes tells him the events were simultaneous. Since neither A nor B can be at both places at the same time, their measurement of time at remote locations depends upon remote observations, and if those remote observations are built on interpreting what light from those events reveals, they will differ on their determination of which remote events are simultaneous, and they will both be correct.

  Here and now is only universal for here and now, not there and then.

  • • •

  I wrote “almost all” bets are off for a reason. For as strange as the example I just gave might seem, it can actually be far stranger. Another observer, C, traveling on a train moving in the opposite direction from B on a third track beside A and B will infer that the event on the left side (the forward part of his train) occurred before the event on the right-hand side. In other words, the order of the events seen by the two observers B and C will be completely reversed. One person’s “before” will be the other’s “after.”

  This presents a big apparent problem. In the world in which most of us believe we live, causes happen before effects. But if “before” and “after” can be observer dependent, then what happens to cause and effect?

  Remarkably, the universe has a sort of built-in catch-22, which ends up ensuring that while we need to keep an open mind about reality, we don’t have to keep it so open that our brains fall out, as the publisher of the New York Times used to say. In this case, Einstein demonstrated that a reversal of the time ordering of distant events brought about by the constancy of light is only possible if the events are far enough apart so that a light ray will take longer to travel between them than the inferred time difference between the two events. Then, if nothing can travel faster than light (which turns out to be another consequence of Einstein’s effort to coordinate Galileo and Maxwell), no signal from one event could ever arrive in time to affect the other, so one event could not be the cause of the other.

  But what about two different events that occur some time apart at the same place. Will different observers disagree about them? To analyze this situation Einstein imagined an idealized clock on a train. The ticks of the clock occur each time a light ray sent from a clock on one side of the train reflects off a mirror located on the other side and returns to the clock on the original side of the train (see below).

  Let us say each round-trip (tick) is a millionth of a second. Now consider an observer on the ground watching the same round-trip. Because the train is moving, the light ray travels on the trajectory shown below, with the clock and mirror having moved between the time of emission and reception.

  Clearly this light ray traverses a greater distance relative to the observer on the ground than it does relative to the clock on the train. However, the light ray is measured to be traveling at the same speed, c. Thus, the round-trip takes longer. As a result, the one-millionth-of-a-second click of the clock on the train is observed on the ground to take, say, two-millionths of a second. The clock on the train is therefore ticking at half the rate of a clock on the ground. Time has slowed down for the clock on the train.

  Stranger still, the effect is completely reciprocal. Someone aboard the train will observe a clock on the ground as ticking at half the rate of their clock on the train, as the figure would look identical for someone on the train watching a light travel between mirrors placed on the ground.

  This may make it seem like the slowing of clocks is merely an illusion, but once again, measurement equals reality, although in this case a little more subtly than for the case of simultaneity. To compare clocks later to see which, if any, of the observers’ clocks has really slowed down, at least one of the observers will have to return to join the other. That observer will have to change his or her uniform motion, either by slowing down and reversing, or by speeding up from (apparent) rest and catching up with the other observer.

  This makes the two observers no longer equivalent. It turns o
ut that the observer who does the accelerating or the decelerating will find, when she arrives back at the starting position, that she has actually aged far less than her counterpart, who has been in uniform motion during the whole time.

  This sounds like science fiction, and indeed it has provided the fodder for a great deal of science fiction, both good and bad, because in principle it allows for precisely the kind of space travel around the galaxy that is envisaged in so many movies. There are a few rather significant glitches, however. While it does make it possible in principle for a spacecraft to travel around the galaxy in a single human lifetime, so that Jean-Luc Picard could have his Star Trek adventures, those back at Star Fleet command would have a hard time exerting command and control over any sort of federation. The mission of ships such as the USS Enterprise might be five years long for the crew on board, but each round-trip from Earth to the center of the galaxy of a ship at near light speed would take sixty thousand years or so as experienced by society back home. To make matters worse, it would take more fuel than there is mass in the galaxy to power a single such voyage, at least using conventional rockets of the type now in use.

  Nevertheless, science fiction woes aside, “time dilation”—as the relativistic slowing of clocks is called with regard to moving objects—is very much real, and very much experienced every day here on Earth. At high-energy particle accelerators such as the Large Hadron Collider, for example, we regularly accelerate elementary particles to speeds of 99.9999 percent of the speed of light and rely on the effects of relativity when exploring what happens.

  But even closer to home, relativistic time dilation has an impact. We on Earth are all bombarded every day by cosmic rays from space. If you had a Geiger counter and stood out in a field, the counter would click at a regular rate every few seconds, as it recorded the impact of high-energy particles called muons. These particles are produced where high-energy protons in cosmic rays smash into the atmosphere, producing a shower of other, lighter particles—including muons—which are unstable, with a lifetime of about one-millionth of a second, and decay into electrons (and my favorite particles, neutrinos).

  If it weren’t for time dilation, we would never detect these muon cosmic rays on Earth. Because a muon traveling at close to the speed of light for a millionth of a second would cover about three hundred meters before decaying. But the muons raining down on Earth make it twenty kilometers, or about twelve and a half miles or so, from the upper atmosphere, in which they are produced, down to our Geiger counter. This is possible only if the muons’ internal “clocks” (which prompt them to decay after one-millionth of a second or so) are ticking slowly relative to our clocks on Earth, ten to one hundred times more slowly than they would be if they were produced at rest here in a laboratory on Earth.

  • • •

  The last implication of Einstein’s realization that the speed of light must be constant for all observers appears even more paradoxical than the others—in part because it involves changing the physical behavior of objects we can see and touch. But it also will help carry us back to our beginnings to glimpse a new world beyond the confines of our normal earthbound imagination.

  The result is simply stated, even if the consequences may take some time to digest. When I am carrying an object such as a ruler, and moving fast compared to you, my ruler will be measured by you to be smaller than it is for me. I might measure it to be 10 cm, say:

  But to you, it might appear to be merely 6 cm:

  Surely, this is an illusion, you might say, because how could the same object have two different lengths? The atoms can’t be compressed together for you, but not for me.

  Once again, we return to the question of what is “real.” If every measurement you can perform on my ruler tells you it is 6 cm long, then it is 6 cm long. “Length” is not an abstract quantity but requires a measurement. Since measurement is observer dependent, so is length. To see this is possible while illuminating another of relativity’s slippery catch-22s, consider one of my favorite examples.

  Say I have a car that is twelve feet long, and you have a garage that is eight feet deep. My car will clearly not fit in your garage:

  But, relativity implies that if I am driving fast, you will measure my car to be only, say, six feet long, and so it should fit in your garage, at least while the car is moving:

  However, let’s view this from my vantage point. For me, my car is twelve feet long, and your garage is moving toward me fast, and it now is measured by me to be not eight feet deep, but rather four feet deep:

  Thus, my car clearly cannot fit in your garage.

  So which is true? Clearly my car cannot both be inside the garage and not inside the garage. Or can it?

  Let’s first consider your vantage point, and imagine that you have fixed big doors on the front of your garage and the back of your garage. So that I don’t get killed while driving into it, you perform the following. You have the back door closed but open the front door so my car can drive in. When it is inside, you close the front door:

  However, you then quickly open the back door before the front of my car crashes, letting me safely drive out the back:

  Thus, you have demonstrated that my car was inside your garage, which of course it was, because it is small enough to fit in it.

  However, remember that, for me, the time ordering of distant events can be different. Here is what I will observe.

  I will see your tiny garage heading toward me, and I will see you open the front door of the garage in time for the front of my car to pass through.

  I will then see you kindly open the back door before I crash:

  After that, and after the back of my car is inside the garage, I will see you close the front door of your garage:

  As will be clear to me, my car was never inside your garage with both doors closed at the same time because that is impossible. Your garage is too small.

  “Reality” for each of us is simply based on what we can measure. In my frame the car is bigger than the garage. In your frame the garage is bigger than my car. Period. The point is that we can only be in one place at one time, and reality where we are is unambiguous. But what we infer about the real world in other places is based on remote measurements, which are observer dependent.

  But the virtue of careful measurement does not stop there.

  The new reality that Einstein unveiled, based as it was on the empirical validity of Galileo’s law, and Maxwell’s remarkable unification of electricity and magnetism, appears on its face to replace any last vestige of objective reality with subjective measurement. As Plato reminds us, however, the job of the natural philosopher is to probe deeper than this.

  It is said that fortune favors the prepared mind. In some sense, Plato’s cave prepared our minds for Einstein’s relativity, though it remained for Einstein’s former mathematics professor Hermann Minkowski to complete the task.

  Minkowski was a brilliant mathematician, eventually holding a chair at the University of Göttingen. But in Zurich, where he was one of Einstein’s professors, he was a brilliant mathematician whose classes Einstein skipped, because while he was a student, Einstein appeared to have a great disdain for the significance of pure mathematics. Time would change that view.

  Recall that the prisoners in Plato’s cave also saw from shadows on their wall that length apparently had no objective constancy. The shadow of a ruler might at one time look like this, at 10 cm:

  and, at another time like this, at 6 cm:

  The similarity with the example I presented when discussing relativity is intentional. In the case of Plato’s cave dwellers, however, we recognized that this length contraction occurred because the cave dwellers were merely seeing two-dimensional shadows of an underlying three-dimensional object. Viewed from above, it can easily be seen that the shorter shadow projected on the wall results because the ruler has been rotated at an angle to the wall:

  And as another Greek philosopher, Pythagoras, taught us, when seen this w
ay, the length of the ruler is fixed, but the projections onto the wall and a line perpendicular to the wall always combine together to give the same length, as shown below:

  This yields the famous Pythagorean theorem, L2 = x2 + y2, which high school students have been subjected to for as long as high schools have taught geometry. In three dimensions, this becomes L2 = x2 + y2 + z2.

  Two years after Einstein wrote his first paper on relativity, Minkowski recognized that perhaps the unexpected implications of the constancy of the speed of light, and the new relations between space and time unveiled by Einstein, might also reflect a deeper connection between the two. Knowing that a photograph, which we usually picture as a two-dimensional representation of three-dimensional space, is really an image spread out in both space and time, Minkowski reasoned that observers who were moving relative to each other might be observing different three-dimensional slices of a four-dimensional universe, one in which space and time are treated on an equal footing.

  If we return to the ruler example in the case of relativity, where the ruler of the moving observer is measured to be shorter by the other observer than it would be in the frame in which it is at rest, we should also remember that for this observer the ruler is also “spread out” in time—events at either end that are simultaneous to the observer at rest with respect to the ruler are not simultaneous for the second observer.

  Minkowski recognized that one could accommodate this fact, and all the others, by considering that the different three-dimensional perspectives probed by each observer were in some sense different “rotated” projections of a four-dimensional “space-time,” where there exists an invariant four-dimensional space-time “length” that would be the same for all observers. The four-dimensional space, which we now call Minkowski space, is a little different from its 3-D counterpart, in that time as a fourth dimension is treated slightly differently from the three dimensions of space, x, y, and z. The four-dimensional “space-time length,” which we can label as S, is not written, in analogy to the three-dimensional length, which we denoted by L, above, as