Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone)
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But the universe isn’t so sensible. Maxwell, Einstein, and experiment all tell us that both observers will measure the speed of both light rays relative to themselves to be exactly the same, and, as a result, each observer is forced to a different conclusion about the simultaneity of the two events. It is important here not to think that one observer is right, and one is wrong. They are both right. There is not a single experiment either person can do to change her own perception of the events or to prove the other person wrong. If they could, then one of them would be able to prove that she was at rest while the other person was moving. But that is the whole point. There is no absolute rest frame with respect to the speed of light. All observers are equivalent. So that means that whether or not distant events are simultaneous depends upon who is doing the observing. There is no absolute “now.” “Now” means something unambiguous only right where you are. Anything you conclude about “now” elsewhere is simply an inference, and it is unique to you. To put another way, “now” is relative.
It is also important not to think that any sense of “now” is therefore completely arbitrary. It is just as constrained after Einstein’s gedanken experiment as it was before. Each observer can base a consistent reality on what she sees, and she can count on the fact that events never precede causes, and so on, even if it turns out that for one observer one event may happen before another, while for another observer precisely the opposite may be true. It turns out that the mathematics of relativity happily only allow this reversal in temporal ordering for events witnessed by different observers whenever the events are sufficiently remote in space and close in time, so that one event cannot have been the cause of the other. Put another way, if a signal can travel between the events in the time between them, then all observers will end up agreeing about which happened first, even if the observers might disagree about how much time had elapsed between them. But just in case you were beginning to think things might be sensible after all, consider the following: The same type of reasoning that led Einstein to recognize that simultaneity was relative led him to recognize that measures of length and time themselves were also relative. For example, let us return to our train example. When the lightning struck simultaneously (for the observer on the train), let us say it scorched the tracks at the same time. Thus, that observer can come back later and measure the distance between the scorch marks to determine the length of the train. But the observer who was on the ground at the time will call foul. She will insist that because one lightning bolt hit before the other, and during the time between the two events, the train was moving, that, the scorch marks on the ground represent a distance that is longer than the actual size of the moving train. In short, the observer on the ground who sees the train moving past will insist that the train is shorter than will an observer on the train, who is at rest with respect to it. So far so good. Moving objects are measured to be contracted along their direction of motion. In fact, this contraction is precisely that calculated earlier by Lorentz (which was dubbed the “Lorentz contraction” by Poincaré) when he tried to make sense of the Michelson-Morley experiment. But here the resemblance ends. In Lorentz’s worldview, where there was an ether and a universal rest frame, moving objects could be contracted relative to those standing still. But in Einstein’s universe, which happens to be the one we live in, all motion is relative. There is no universal rest frame and no ether. So, for a person on the moving train, it is the person on the ground who can be said to be moving past, in the opposite direction. And exactly the same type of reasoning as given above will convince you that the person on the train will measure the lengths of objects at rest with respect to the person on the ground to be shorter than will the person on the ground!
Thus, each observer will measure the length of objects at rest in the other person’s frame of reference to be shorter. The Lorentz contraction is not absolute; it is relative.
Once again, the relative nature of the Lorentz contraction should not lull you into assuming that it is not real. It is as real as the nose on my face, whose size will, of course, depend upon who is viewing it. This is illustrated by my very favorite paradox from relativity. Thankfully for you, it is the last one I will attack your brain with here.
Say I have a fast sports car—a really fast one, which can travel at a large fraction of the speed of light, where the mysterious effects of relativity become more apparent. After all, if you consider the gedanken experiments I have discussed above, clearly the discrepancies about length and time between observers are related to how far the train could have traveled during the time the light rays crossed it. To have observable effects, one needs either very large trains or very fast ones. Well, say I am moving past you at a very large fraction of the speed of light. My car will therefore be measured by you to be shorter than I will measure it to be. Now, say you have a garage with two doors, one at either end, into which I am driving. If my car is ten feet long to me, say it would be measured to be six feet long by you. Say your garage is eight feet in depth. Then, for you it should certainly be possible to quickly close the front door of your garage after my car has entered and continues speeding along, completely enclosing it within the garage. You would then hopefully run very quickly to open the door at the rear of the garage so that my speeding car would not run into it. Relativity tells us that this is certainly possible, at least in principle. But now there is a problem. In my reference frame, it is your moving garage that is shorter. To me, it appears to be only five feet long, and there is no way that my car will fit within it!
Am I doomed to crash? Well, if I do hit a door, both observers would have to agree that such an event happened. (After all, they can come back together afterward and see the tangled mess, if both people are still alive.) So, if one observer sees me making it through the garage safely, then I must have done so. Rather, I will insist that my car and I were never entirely within the garage, because I will measure the order in time of the remote events, including opening and closing the garage doors, to be different than will the observer on the ground. I will insist that, for example, the rear door of the garage was opened before its front door was closed. Thus, as I sped through, the front of my car exited the back of the garage before its rear end passed through the front of the garage. The point is that each observer’s reality is real. For you, my car was completely inside the garage. For me, it never was. There is no experiment you can perform that will prove me wrong, and vice versa. At the same time, it is clear that the contraction, while real, is still very much in the eye of the beholder. Or, as Einstein would say, measurements of length are relative.
A similar relativity occurs for the slowing of clocks. If I am moving very fast relative to you, you will measure my clocks to be running slowly. I will appear to you to age more slowly if you watch me recede into the distance. But I will in turn measure your clocks to be running slowly as, well, you will appear to me to age more slowly.
At this point a conventional reaction to the implications of relativity is to throw up one’s hands and decide that the world has no order in it whatsoever and that there exist no absolutes. Everything is relative, so anything goes! Indeed, this was the reaction of many artists and writers in the early part of the twentieth century to the results of relativity, as I shall soon discuss. But even if it feels justified, this is not the correct response. Hermann Minkowski had been one of Einstein’s mathematics teachers in Zurich—in fact, one of the few whose lectures Einstein actually enjoyed. In 1902 Minkowski moved to the University of Göttingen, where one of the most renowned mathematicians of his time, David Hilbert, was located. Interestingly, Hilbert would later help Einstein provide the mathematical tools that would change our picture of space and time in profoundly new ways. But well before that, the first Göttingen mathematician to have had such an impact was Minkowski.
In 1908 in Cologne Minkowski gave a lecture entitled “Space and Time,” which created a tremendous stir and has since been recognized as a watershed moment in our underst
anding of physical reality. The epigraph of this chapter is from that lecture, which began with words that are both enticing and particularly significant for a mathematician to have uttered: “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical.” In his speech, and in the more technical paper that accompanied it, Minkowski delivered exactly what he had promised. By the time he was finished, space and time could no longer sensibly be individually discussed, and only a union of the two, which we now call space-time, was understood to retain any independent reality.
The seeds of Minkowski’s realization lie in the example I presented involving Einstein’s long train. Recall that simultaneous lightning bolts for an observer on a moving train provide an ideal method for her to measure the length of the train. She merely has to later disembark, return to the scene of the lightning strikes, and measure the distance between the scorch marks on the tracks.
Now, also recall that an observer on the ground will contest this measurement, arguing that the two lightning bolts were not simultaneous and therefore the scorch marks represent events that happened at two different times at either end of a moving train. Thus, the distance between the scorch marks must represent a larger distance than the true length of the train. Let us then consider what this implies by thinking in terms of what, precisely, is meant by a measurement. The observer on the train measures an interval in space. That is, after all, what a measurement of distance is. For the observer on the ground, however, this same measurement involves an interval in space and time.Seen from this perspective, perhaps it is not surprising that the individual distance and time difference measurements for the two observers differ. To visualize this a little more dramatically, let us imagine two observers in Plato’s cave. One of them sees the following shadow on the cave wall, in the morning:
Later in the day, the other observer sees this one:
Has the person whose shadow they have seen at different times of day changed in height? No, of course not. Rather, the sun is higher in the sky, and the length of the shadow on the back wall of the cave will change accordingly.
Let’s simplify the issue. Imagine the cave dwellers are viewing the shadow of a transparent ruler:
Now suddenly the shadow changes:
The shadow-ruler has inexplicably changed in length. How was this possible? Simple: The original ruler was rotated with respect to the light source. As seen from above, the two different situations appear as follows:
The length of the original ruler has certainly not changed by this rotation, but the projection of this three-dimensional object onto the twodimensional wall at the back of the cave has. Physicists in this cave-dwelling society may initially be baffled by the fact that the lengths of shadowobjects are apparently not absolute. But eventually someone would intuit that the objects being observed behave as two-dimensional projections of three-dimensional objects that can be rotated perpendicular to the wall. Mathematically, there is a quantity that is absolute and doesn’t change under such rotations—namely, the length of the original ruler. If this ruler has a length L, while the length of the shadow-ruler (i.e., the projection of L on the cave wall) is X, then a cave mathematician, who, for the sake of argument we might call Pythagoras, might suggest that there is a quantity, L, whose value does not change, and that is given by the relation L 2 = X 2 + Y 2, where X is the projection of the ruler on the cave wall and Y is the projection of L perpendicular to the cave wall:
By now you don’t have to be Einstein to see where we are heading. What Hermann Minkowski realized is that there is a similarity (but just a similarity) between this scenario and what occurs, according to relativity, for observers in relative motion measuring the same object. Recall that the speed of light in empty space, c, is measured to be the same by all observers. Say one observer measures the distance traveled by a light ray in some time t to have a value d. Since distance traveled is determined by the speed of the light ray times the time it travels, this observer thus finds d = ct. Any other observer moving with respect to this observer may in general measure a different length d ' and time t ' , but they must find d ' = ct ' if they are to determine the same speed relative to them for this light ray. Thus, at least for a light ray, different observers in relative motion will measure distances and times such that the combination d 2 − c 2 t 2 = d '2 −c 2 t '2 = 0 for any light ray. While this is manifestly true for a light ray, it turns out that this combination will be measured to be the same by all observers for any two “space-time” events measured to be separated by a distance d and time t for any one of them, so that d 2 − c 2 t 2 = d '2 −c 2 t '2 for all events separated in space and time even if the combination is not zero (i.e., the two distances and times are not for points connected by the trajectory of a light ray). This will be true even though the separate observers will in general arrive at different separate measurements of d and t. Minkowski realized that this particular combination of distance and time, which Einstein recognized remains invariant between observers in relative motion, is strikingly analogous to the way the different length projections of a ruler can be combined to always produce the same value—namely the length of the ruler itself—regardless of its orientation. Except for the weird minus sign (i.e., d 2 − c 2 t 2 instead of d 2 + c 2 t 2 ), which we will discuss shortly, the combination is the same.
Thus, the exotic results of Einstein’s relativity can be understood by analogy to the two-dimensional cave example. In the latter case, different observations of the same object appeared inconsistent because each presented a different two-dimensional projection of the same threedimensional object. In our universe, different observers in relative motion are simply presented with different three-dimensional slices of an underlying four-dimensional universe where space and time are tied together. Minkowski called the mathematical combination d 2 − c 2 t 2 the “spacetime distance” between the events, to distinguish it from the threedimensional, purely spatial distances we are used to. Just as rotations in regular space can change projections, so, too, can relative motion change the separate time and space intervals measured by different observers, while the space-time distance is preserved. Indeed, motion reproduces certain aspects that are reminiscent of rotations. As Einstein’s train example makes clear, one man’s space interval can be another man’s time interval. With this unveiling of what we now call “Minkowski space,”
Minkowski delivered on the promise of his Cologne lecture. Our Plato’s cave illustration merely makes literal his metaphorical exclamation that heretofore space by itself and time by itself would fade away into mere shadows. From 1908 onward, three-dimensional space and the seemingly distinct and unrelated one-dimensional progression of time became inextricably linked together. What had begun with tentative inklings in basement laboratories filled with compasses and currents had blossomed into a whole different perspective of our universe to be explored and understood. This four-dimensional space that we discovered we occupy, however, differs dramatically from the world that Edward Abbott envisaged in the plaintive pleas of his Flatland hero. The weird relative minus sign between the spatial part and the time part of space-time distance (remember that for normal spatial separations, the square of total distance between two points is the sum of the squares of the individual projections, with no minus signs) changes everything, so that time and space are tied together in a way that is quite unlike the way up and sideways are tied together. We cannot walk into time as we can apparently walk into space, nor, as far as we yet know, can we back up. Time travel is so exotic compared to motion in space that entire movies and (fictional) books have been written to consider this possibility. The minus sign fundamentally seems to distinguish between spacetime intervals that are “timelike” compared to those that are “spacelike.” (Minkowski himself coined this terminology.)
Physics was thus left at the brink. A fourth dimension had been discovered, but not the one that Abbott had imagined. But pe
ople most often hear what they want to hear, and consequently they often tend to interpret the new results of science in terms that justify their previous expectations. Thus, the feature that makes Minkowski space special, while profound, was overshadowed by the newfound freedom of action offered by Einstein’s special relativity, and the promise of a “fourth dimension.” But Einstein was not yet finished with space and time.
C H A P T E R 5
DISTURBING THE UNIVERSE
What is derived from experience has only comparative universality, namely, that which is obtained through induction. We should therefore only be able to say that, so far as hitherto observed, no space has been found which has more than three dimensions.
—Immanuel Kant, Critique of Pure Reason
For Kant, space existed in the mind, as a backdrop for all of our experience. From his perspective Euclid’s fundamental axioms of geometry were a priori necessary features of a universe in which thinking beings could live. Kant felt that these axioms were not derived from experience or experiment, for if they were, they would merely be provisional, not absolute.
Well, Kant was correct in at least one respect: The postulates of Euclid, in particular his famous fifth postulate—that there is only one line that can be drawn through any point that does not intersect with (i.e., is parallel with) a given line—cannot be derived from fundamental principles or from experience. That is not the case, however, because they are intrinsic to our existence. It is, rather, because they are not universally true. On a sphere, for example, lines of longitude are parallel, but they all meet at the North and South poles.
Such, it seems to me, is the limitation of much of philosophy: It is often subsumed as empirical knowledge supplants pure thought. The irony in this statement is that Einstein’s most significant contribution to human knowledge comes as close as any major development I know of in the history of physics to something akin to pure thought. I refer to Einstein’s general theory of relativity, which he developed in the decade after his formulation of special relativity in 1905. The term general here refers to the fact that special relativity applied to observers in constant relative motion. What general relativity did was to extend these considerations to accelerating observers. Remarkably, in the process, it turned out to be a new theory of gravity!