Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone)

by Lawrence M. Krauss

  It is said that a picture is worth a thousand words. But what if, as Pawlowski stressed, words fail completely? Charles Hinton spent much of his life attempting to teach others how to develop a visual intuition about four-dimensional space, as he believed he had himself done. Recall that the heart of his technique, which was reflected in essentially every other subsequent effort, including A. Square’s, was to display different threedimensional projections of a four-dimensional object as it is “rotated” in the fourth dimension. Just as one can color the six faces of a cube and display the different colors that result when one rotates it by ninety degrees in order to help visualize both the nature of the cube and precisely what is meant by the set of rotations in three dimensions, one might hope to build up a similar understanding of four-dimensional space by considering the different three-dimensional projections of a tesseract, for example. The similarity between Hinton’s approach to the tesseract and Picasso’s approach to his models is striking. But is there more to it than a simple spatial operation? Certainly, Picasso never claimed there was. His famous statement, “I paint objects as I think them, not as I see them,” was more a reflection of his protest against the confines of standard perspective than a claim to be interpreting higher dimensions. Just as Van Gogh fought against the tyranny of color, one might say that Picasso and his contemporaries Braque, Gris, Metzinger, Weber, and Duchamp were struggling to free us from the tyranny of space. Yet, at the same time the ultimate goals of the mathematicians and the artists were similar: to compel us to use our minds to liberate ourselves from the confines of our own experience. Picasso was a product of the intellectual ferment of those heady times after the turn of the century, and this was also reflected in the cubist revolution, in which he was a leading figure. The circle of artists and writers at the Bateau Lavoir in Montmartre, where cubism had its origins, discussed many of the exciting ideas of the day, including extra dimensions. While cubism was born out of a sense of questioning of the traditional views of the world, if the existence of an extra dimension could provide validation for its attempt to extract a new, hidden reality in nature, all the better. Certainly those authors who chose to write about cubism—notably Jean Metzinger and Guillaume Apollinaire—as well as related French literary figures like Jarry—and ultimately the artist perhaps most closely associated in the modern mind with this aspect of the movement, Marcel Duchamp, all explicitly described a relationship between cubist art and four dimensions, with the analogies being alternately poetic and explicit. Witness Duchamp, in a later interview, discussing his motivation in creating one of his most famous pieces representing a higher-dimensional reality, The Bride Stripped Bare by Her Bachelors, Even (the Large Glass), created between 1915 and 1923:

  What we were interested in at the time was the fourth dimension. . . . Do you remember someone called, I think, Povolowski?

  He was a publisher, in the rue Bonaparte. . . . He had written some articles in a magazine popularizing the fourth dimension. . . . In any case, at the time I had tried to read things by Povolowski, who explained measurements, straight lines, curves, etc. That was working in my head while I worked, although I almost never put any calculations into the Large Glass. Simply, I thought of the idea of a projection, of an invisible fourth dimension, something you couldn’t see with your eyes.

  Notes, however, for Large Glass do contain substantial references to mathematical discussions of a fourth dimension, including the writings of Poincaré. While Duchamp claimed only a passing knowledge of these ideas, observations he made in these notes, such as, “Poincaré’s explanation about n-dim’l continuums by means of the Dedekind cut of the n-1 continuum is not in error,” demonstrate the depth of his interest in the topic. Interestingly, in spite of his truly meticulous efforts to methodically attempt to portray projections of a fourth dimension—efforts that made him more than any other artist an explicit student of this mathematics—

  Duchamp later disavowed them. “It wasn’t for love of science that I did this,” he said. “On the contrary, it was rather in order to discredit it, mildly, lightly, unimportantly. But irony was present.”

  For Duchamp, then, as well as for his cubist, and literary contemporanes, reacting against a three-dimensional Euclidean world was subversive and thus attractive. I use the terms three-dimensional and Euclidean here in spite of the fact that there is nothing about the four-dimensional space-time of Minkowski or, for that matter, the four-dimensional projections of Hinton and others that is remotely non-Euclidean. These spaces are quite flat. Having to go beyond Euclid to consider a possible curvature of space is essentially never explicit, except perhaps in Duchamp’s piece, Stoppages, and in the later distorted landscapes of Salvador Dali. Yet, in the literature of cubism non-Euclideanism was rampant. Indeed, in one of the first essays on cubism, “Du Cubisme” (1912), by Albert Gleizes and Jean Metzinger, the authors state explicitly: “If we wished to tie the painters’ space to a particular geometry, we should have to refer it to the non-Euclidean scholars.”

  Somehow what was occurring, one might argue, was a rebellion against perspective, one of the hallmarks of our three-dimensional world. Certainly a curvature of space, causing light rays to travel on curved paths, is one way to distort perspectives, but another is to imagine viewing many different three-dimensional perspectives simultaneously, which was the preferred method of the cubists. Duchamp, one of the most mathematically literate of the emerging school, employed both non-Euclidean themes and multiple perspectives. This ultimately allowed him to go even further in his art, becoming perhaps the first of the modern conceptual artists. While the liberation achieved by abandoning three-dimensional perspective was intoxicating, it may have been inspired, at least for some, by an incorrect understanding of the developments in science at that time. I have no idea if Einstein, a notorious antiauthoritarian, coined the word rel- ativity with malice aforethought, but the term carries a great deal of intellectual baggage, and has encouraged, and continues to encourage, the incorrect notion that it somehow does away with all absolutes, making truth itself relative and observer dependent. And if special relativity, which demonstrated that space and time are tied together into a four-dimensional space-time, had everything to do with absolutes, it also has virtually nothing to do with the non-Euclidean ideas that so fascinated many of the writers and artists of the time, who may have seemed in retrospect to have been inspired by it.

  One must remember also that Einstein was not yet the household name he would become in 1919, following the observations of the bending of light from distant stars which confirmed the predictions of general relativity. There is no doubt that with the passing of time his perceived impact on his cultural contemporaries may be viewed as being more significant than it actually was. In any case, as I have argued, the facets of a fourth dimension that most fascinated artists and writers alike actually had little to do with the actual ideas contained in special relativity, but were at best culled and adapted from what they perceived the theory might contain, based on preexisting cultural fascinations. In spite of the confusions regarding the nature of the four-dimensional universe implied by relativity, and about the relations between nonEuclidean geometry and the geometry of extra dimensions, the almost accidental prescience about these concepts in the literary and artistic worlds at the beginning of the twentieth century was remarkable. I have often found (for example, when I have in other books compared science fiction and science) that the confluence of ideas and language among different disciplines is simply due to the fact that when creative people think about similar problems, even from totally different vantage points, they sometimes come up with similar ways of approaching them.

  An even more remarkable coincidence, perhaps, lies in the fact, as I shall next describe, that the first concrete scientific proposal for the existence of extra spatial dimensions arose not by generalizing the notions of the space-time of Minkowski, but rather by attempting to extend Einstein’s general theory of relativity, building, as fortuitousl
y envisaged by many of the cubist artists, a bridge between curved space and extra dimensions that has been central to the scientific pursuit of extra dimensions into the twenty-first century.

  So, once again, life imitates art.

  C H A P T E R 8

  THE FIRST HIDDEN UNIVERSE: AN EXTRA DIMENSION TO PHYSICS

  We are such stuff

  As dreams are made on, and our little life

  Is rounded with a sleep

  —William Shakespeare, The Tempest

  It is one thing for a writer to dream up a new hypothetical universe in which to stage a drama, and quite another to propose that such a universe might really exist. This requires a different kind of chutzpa—the kind that arises following a period of such great success building new pictures of reality that one becomes emboldened in one’s predictions. I first experienced this kind of hubris when I was a graduate student at MIT in 1980. This was a heady era in particle physics and an exciting time to be a student. In less than a decade physicists had gone from clearly understanding only one of the four known forces in nature (i.e., electromagnetism) in a way that was consistent with quantum mechanics and relativity to understanding in detail all the known forces except for gravity.

  It was easy to feel that we were witnessing the emergence of an astonishing new picture of the natural world. A year earlier, Sheldon Glashow and Steven Weinberg (two faculty members at nearby Harvard, where I took most of my graduate courses) had won the Nobel Prize (along with Abdus Salam) for their development in the 1960s—confirmed by experiments in the 1970s—of a theory that unified two of the four forces in nature: the electromagnetic and weak forces. The latter is the force that is responsible for many nuclear reactions that turn protons into neutrons and vice versa, and is an integral part of the process of “nuclear fusion” that powers the sun. Shortly after that a graduate student at Harvard, David Politzer, had discovered contemporaneously with a Princeton graduate student, Frank Wilczek, and his advisor David Gross, a key mathematical characteristic of a theory that was soon recognized to describe the third nongravitational force in nature, the so-called strong force between quarks, the fundamental building blocks of protons and neutrons. The theory in question, called quantum chromodynamics (QCD), provided predictions about the interactions between quarks that were previously unthinkable, and that were ultimately verified to be in agreement with experiment, leading to a Nobel Prize thirty years later for this trio.

  Everywhere we turned, it seemed that the new tools of elementary particle physics—based on combining special relativity, quantum mechanics, and Maxwell’s electromagnetism—were opening up doors. Emboldened by their success, physicists began to seriously consider whether they might soon be able to unify not just two forces in nature, but perhaps three or maybe even all four, within a single theoretical mathematical framework, the holy grail of “Grand Unification.”

  I will return to grand unification and its predictions later in this book, but took the liberty of jumping ahead chronologically here to present a brief contemporary perspective on how the excitement of discovery can be contagious and can breed the kind of confidence that allows one to address problems one would never have had the boldness to even consider otherwise. A comparable situation occurred in the second decade of the twentieth century, following the development of special and then general relativity by Einstein.

  Remarkably, just as the discoveries by Faraday, Maxwell, Oersted, Ampère, and others about the relations between electricity and magnetism led Einstein and Minkowski to propose the existence of an underlying four-dimensional space-time continuum, and just as the mathematical form of electromagnetism provided the key that allowed the physicists mentioned above to solve the mysteries surrounding the strong and weak interactions, so, too, did electromagnetism play a central role in the first serious scientific proposal that other dimensions, beyond the four we experience, might actually exist. This proposal, like grand unification some sixty years later, was motivated by a desire to unify the forces of nature, and, as would be true of grand unification, the specific mathematical form of electromagnetism provided the direction. However, unlike the case of grand unification, the direct trigger was the remarkable discovery by Einstein that the force we feel as gravity could instead be understood in terms of the curvature of space-time.

  In retrospect, it is perhaps not surprising that the advent of general relativity led physicists to consider the possibility that extra dimensions might allow for a unification of what were then the two known forces in nature: gravity and electromagnetism. Einstein’s theory implied that local observers could interpret the forces they felt as either due to gravity or the effects of acceleration, depending upon their frame of reference; similarly, Maxwell’s relations between electric forces and magnetic forces also imply that observers can interpret the forces they feel as either electric or magnetically induced, depending upon their own state of motion. If gravity, then, could be interpreted as being due to an underlying local curvature of three-dimensional space, then could electromagnetism be somehow due to some other sort of underlying local curvature? And since curvature in an observable three-dimensional space resulted in gravity, could curvature in some unperceived new dimension be responsible for the extra force of electromagnetism?

  The Finnish physicist Gunnar Nordström actually developed the first physical theory that incorporated an extra dimension in 1914, slightly before Einstein’s fully developed general relativity appeared. His version of unification was in spirit the opposite of the approach outlined above, as he tried to derive gravity from electromagnetism, rather than vice versa. Nordström had in fact developed his own theory of gravity, which attempted to generalize special relativity, just as general relativity would successfully do several years later. In Nordström’s theory, the universe was five-dimensional, with one extra spatial dimension, and Maxwell’s electromagnetism was a force felt in every one of the dimensions. But if, for some reason, all the electromagnetic fields were independent of the extra spatial dimension (i.e., the fields were of a constant fixed magnitude in that extra dimension, but could vary in strength over the three spatial dimensions we are used to), then those of us sensitive to only the three dimensions in which electromagnetic fields could vary would measure not only electromagnetism, but an additional remnant of the fourth spatial dimension. That additional remnant was precisely Nordström’s gravitational field. Of course, once Einstein’s general relativity was unveiled, interest in Nordström’s ideas waned—especially interest by Einstein, who was known to have had a less than cordial relationship with Nordström. In fact, in all the subsequent proposals involving extradimensional unifications in physics up through the early 1980s, there is not a single reference to Nordström. Such was, I suppose, the danger of competing with Einstein, at least where gravity was concerned.

  The person generally credited with introducing the idea of extra dimensions into mainstream physics was the German mathematician Theodor Kaluza, in a beautiful paper entitled “On the Unity Problem in Physics,” in which he argued that searching for a unified worldview was “one of the great favorite ideas of the human spirit.”

  Kaluza also proposed a five-dimensional universe, with four spatial dimensions plus time. He was motivated in his efforts by an earlier proposal by Hermann Weyl to unify electromagnetism and gravity in a purely geometric manner, as Einstein had done for gravity alone. Thus, instead of considering an electromagnetic field as fundamental, Kaluza imagined only a gravitational field, described by a five-dimensional version of general relativity (i.e., his theory described the curvature of four spatial dimensions in terms of a gravitational field that operated in four spatial dimensions plus time).

  TThe fundamental quantity that determines the nature of gravity in Einstein’s general relativity is something called the metric. This is actually a set of quantities that tell you at any point in space exactly how physical distances between nearby points are related to any local coordinate system (e.g., x, y, and z coo
rdinates that describe length, width, and height) that a local observer may set up. If space is flat, then the relation between physical distances and coordinates such as x, y, and z is generally simple. In two dimensions, for example, the square of the physical distance between two points separated by coordinates x and y is, as Pythagoras taught us, simply x2 + y2.

  But on a curved space such as the surface of a sphere, the relation between physical distances and coordinates can get strange. If one maps out points on this surface by latitudes and longitudes, for example, as one does on Earth, then near the poles, where the longitudes draw closer together, the physical distances between them are very different than they are near the equator. Thus, on a map in which latitudes and longitudes are represented by perpendicular coordinate grids, Greenland looks huge. It turns out that all of the geometric information about the sphere is precisely encoded in the metric quantities that describe the changing relation between distances as a function of latitude and longitude, and that tells us how to find out the actual size of Greenland from the difference in longitude between one side of it and the other. In a five-dimensional space, more quantities are needed to describe all the possible coordinates for any given point. If one does the mathematics, it turns out that there are five more quantities needed at every point to completely specify the geometry of such a five-dimensional space. Kaluza the mathematician argued as follows: Imagine a fivedimensional space that has one dimension that is periodic, such as a circle, so that when you travel in this direction, you return to your starting point. A simple example of this in two dimensions is a cylinder. Further imagine that the other four dimensions in the five-dimensional space are just like the four dimensions of space and time that we experience. The force we feel as gravity is related to the geometry of these four dimensions, described completely by the metric quantities I described earlier. Now, imagine that all the metric quantities that describe the distances between nearby points along the four-dimensional slices of five-dimensional space do not change as you move around the circular fifth spatial dimension (as would also be the case for a cylinder in two dimensions). This is the same as saying that all metric quantities that describe the five-dimensional space (there are a total of fifteen of them at any point) are independent of this circular fifth dimension.