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

  We who can only move around in three spatial dimensions sense gravity in a way that depends upon ten of the fifteen quantities that vary from point to point in our four-dimensional slice of this five-dimensional “cylinder.” So what do the other quantities determine? Kaluza was able to show that four of the extra five quantities satisfy equations that are precisely those discovered by Maxwell to describe the electric and magnetic fields. In this way, the two known forces in nature appeared to be unified in a beautiful and remarkable way, thereby suggesting that what we measure as electromagnetic fields might be merely a remnant of an underlying curvature in an invisible fifth dimension. This is a truly amazing possibility that sounds almost too ideal not to be true. So, why did Einstein vacillate for almost two years before finally sponsoring its publication after receiving Kaluza’s manuscript in 1919?

  Well, in the first place the astute reader may have noticed that I spoke of “four of the extra five quantities” that describe the geometry of the fivedimensional universe. What about the extra quantity? It turns out that Kaluza essentially ignored it, for no good reason. If one does not do this, then it turns out that the theory one arrives at in four dimensions is not quite electromagnetism plus general relativity. There is an extra term, which changes the nature of gravity. In modern language this could be described as being due to the existence of an extra massless particle in nature, which we have not observed. We shall return to this issue later. The other question that Kaluza’s work completely begs is one that resembles that question young children are required to ask on the Jewish holiday of Passover: “Why is this day different from all other days?” In this case, one would ask: “Why is the fifth dimension different from all other dimensions?” To this, Kaluza provided no concrete answer. Such was the luxury, perhaps, of being a mathematician.

  To be fair, it is worth noting that Kaluza himself introduced a fifth dimension as a purely mathematical convenience, and did not necessarily ascribe any physical significance to it. Indeed, in his analysis he was apologetic in tone, calling the decision to introduce such a possibility a “strongly alienating decision.” He was driven to do so by a mathematical similarity pointed out by Weyl between the way in which electromagnetic fields could be written and the way in which a certain mathematical quantity, called a “connection tensor” or just “connection,” based on the metric in a curved space, could be written. Because this quantity in our fourdimensional space-time is used to describe the effects of gravity, he was forced, as he put it, to consider an extra dimension that would allow additional pieces of the connection to be interpreted as electromagnetic fields in space-time. This would only be the case as long as the extra dimension itself was rather impotent, with all physical quantities (i.e., the metric) being independent of the circular fifth dimension.

  Nevertheless, Kaluza was not immune to the seductions of mathematical beauty. He found the remarkable connection between the mathematical form of electromagnetism and general relativity too compelling to resist, and in the conclusion of his paper he wrote hopefully: Even in the face of all the physical and epistemological difficulties which we have seen piling up against the conception presented here, it is still hard to believe that all these relations in their virtually unsurpassed formal unity, should amount to the mere alluring play of a capricious accident. Should more than an empty formalism be found to reside behind these presumed connections, we would then face a new triumph of Einstein’s general relativity, whose appropriate application to a five-dimensional world is our main concern here.

  It would fall to later investigators to begin to ascribe possible physical meaning to Kaluza’s fifth dimension, to attempt to explain why it might be invisible, and to explore the other possible physical consequences of this idea.

  The first person to seriously take up this task was physicist Oskar Klein (the son of Sweden’s first rabbi), who in 1926 independently discovered the mathematical relations earlier demonstrated by Kaluza. (Somewhat later, even Einstein himself became sufficiently enamored by the idea that in 1938 he and colleague Peter Bergmann essentially reproduced Klein’s ideas in a paper that represents Einstein’s own continuing search for a unified theory of all interactions.) Klein, who studied with one of the fathers of quantum mechanics, Niels Bohr, was motivated in his investigations to try to understand the underlying nature of various strange phenomena predicted in quantum theory, where particles can sometimes act like waves, and probability appears to replace certainty in physical predictions. Indeed, the developments associated with this possible unification of electromagnetism and gravity were taking place even as the top theoretical physicists of the time were wrestling with the implications and mathematics of the emerging quantum mechanical understanding of atomic phenomena. The strange nature of atomic spectra—the discrete set of colors of light emitted by different gases as you heat them up—and the nature of the radiation emitted by so-called black bodies (i.e., objects that very nearly absorb all colors of radiation equally and thus appear black) as you heat them were considered to be much more urgent problems than the more esoteric possible unification of two theories that on their own held up remarkably well. Between 1913 and 1918, Niels Bohr had developed the first quantum theory of atomic spectra by developing a series of rather ad hoc and unusual rules to “explain” the energy levels of hydrogen. It was not until 1925–26—coincident with Klein’s work on extra dimensions—that Werner Heisenberg and Erwin Schrödinger independently developed selfconsistent formulations of quantum mechanics, which also implied a host of associated “spooky” phenomena, to use a phrase of Einstein’s, who never fully bought into the whole picture.

  Still, having a solid mathematical formulation of the rules of quantum mechanics and having a full physical understanding of the theory are two different things. Unlike both special relativity and general relativity, where comprehending the mathematics can provide one with a more or less complete physical picture, quantum mechanics defies all classical intuition.

  For example, in the quantum world, subatomic particles such as electrons behave at the same time as if they are both waves and particles. That is, while individual electrons may seem like particles, they can nevertheless do things baseballs never do, such as being partially transmitted through and reflected by objects simultaneously. While the equations of quantum mechanics are themselves completely deterministic, the results of experiments are not. Rather, the equations allow one to calculate the probability that an experiment will yield a certain result. In 1927 Werner Heisenberg discovered one of the most frustrating aspects of the newly developed quantum theory, which has become known as the Heisenberg uncertainty principle, and which stated that, independent of one’s measuring apparatus, there were certain combinations of physical quantities, such as a particle’s position and velocity, that could never be known with an accuracy beyond some fundamental limit, no matter how long or hard one tried. Einstein was not the only one who was repelled by the thought of inherent indeterminacy in our knowledge of the physical world. Perhaps this uncertainty arose because of our lack of experimental knowledge of some

  “hidden variables” that, if we had access to them, would allow precise and arbitrarily accurate predictions of experimental phenomena. Klein thus rediscovered Kaluza’s five-dimensional unification scheme, but his motivation was somewhat different than Kaluza’s. Klein, the student of Bohr, hoped that this higher-dimensional framework might explain the basis of weird quantum mechanical phenomena, like the uncertainty principle, which he thought might be understood somehow as being due to our experiencing only a four-dimensional projection of a five-dimensional universe. This was the scientific equivalent, in a very loose sense, of explaining weird paranormal phenomena by means of the agency of an invisible fourth dimension (the difference, of course, being that weird quantum phenomena actually have been experimentally shown to exist!). It is also somewhat ironic that Klein’s motivation for rediscovering Kaluza’s model came from quantum mechanics, because this was precisely what
Kaluza worried about as possibly killing the whole idea. As he somewhat poetically stated at the end of his 1919 paper: “In any case, every Ansatz (i.e., postulate) which claims universal validity is threatened by the sphinx of modern physics, quantum theory.”

  In any case, not only did Klein reproduce Kaluza’s mathematics, but because he took the possible physical existence of a fifth dimension more seriously, as a physicist rather than a mathematician, he was able to examine the physical consequences, in particular for quantum theory and also for electromagnetism, of such a fifth dimension. He also addressed the question of why it might not be observable.

  His solution, which was later reproduced by Einstein and Bergmann, was to argue that this extra dimension was curled in a small circle—so small, in fact, that it could not be probed with existing experiments. In this scenario one could imagine the four dimensions of space as follows: “Above” every single point in our visible three-dimensional space a small circle “sticks out” into the fourth dimension. If one suppresses one dimension and represents our three-dimensional universe as a plane, the extra dimension could therefore be pictured by lining up an infinite number of infinitely long soda straws side by side. At each point in the plane one could travel in a circle around the side of the soda straw lying on the plane, returning back to where one started.

  In fact, this analogy of the soda straw is useful from another point of view. Seen from a distance, a straw looks as if it has no thickness—as if it were a simple one-dimensional line. However, upon closer examination, one sees that the straw is actually a cylinder: a two-dimensional object (two-dimensional because one can move up and down along the length of the straw, or travel in a perpendicular direction around the side of the straw). If the diameter of the straw was small enough—say, the size of a human hair—one might not be able to perceive its thickness in the second dimension without a microscope. If it was really small, even a microscope might not reveal this second dimension. And so it could be with our universe: An extra curled-up dimension lying above every point in space would be invisible if it was curled up on a subatomic scale. While I have presented this example by appealing to our classical intuition, Klein’s argument actually relied instead on the wavelike nature of elementary particles arising out of quantum mechanics. It is well known that waves are not significantly disturbed by obstacles that are much smaller than their wavelength. A water wave in the ocean, for example, moves around a small pebble without any problem, but a large rock will protect the water behind it from the disturbance produced by an oncoming wave. The French physicist Louis de Broglie had shown in 1924 that quantum mechanics implied that a “wavelength” could be ascribed to every particle, that would be inversely proportional to the particle’s “momentum” (which in turn depends upon its mass times its velocity). The higher the momentum, the smaller the wavelength. Indeed, this is why objects that are much more massive than atoms tend to behave classically: Their quantum mechanical wavelengths are so small as to be invisible, so that these objects behave, for all intents and purposes, as if they were simply particles, like billiard balls.

  In order for an experiment to probe some scale, the wavelengths of the particles that one sends in as probes—be they the elementary particles associated with electromagnetic radiation called photons, or some other particles, such as electrons—must be smaller than the scale that one wishes to explore. (Otherwise, the incoming wave will not be disturbed by the object one wishes to probe.) This in turn means that the momentum, and thus the energy imparted to our particle probes, must be larger than a certain amount.

  As a result, Klein, and later Einstein and Bergmann, argued that if the radius of the fifth dimension was smaller than a certain amount, then in order to send particles into this extra dimension to even resolve it one would need more energy than was then currently available in existing experiments. Because of this property, the fifth dimension could exist, yet remain effectively invisible in all existing experiments.

  At the same time as providing this physical mechanism to keep the fifth dimension phenomenologically viable, Klein argued that the existence of an extra curled-up dimension might explain why all electric charges come in integral multiples of the charge on an electron (i.e., why we have never discovered any object with a charge equal to, say, 1.33 or minus 2.4 times the charge on an electron). Every object has a charge equal to . . .−3,−2,−1,0,1,2,3 . . . times an electron’s charge.

  Remember that in the Kaluza theory, electromagnetism is a four-dimensional remnant of what one would, if one had five-dimensional sensibilities, feel as part of a five-dimensional gravitational field. Also remember that general relativity provides a relation between the underlying energy of objects moving through space with the curvature of space they thus produce.

  All of this together implies that if some particle can move in the direction of the circular fifth dimension, it will have an impact upon the geometry of the fifth dimension. To this, Klein added one last feature of the quantum world—namely, that every particle also has a wavelike character. For particles whose motion in the fifth dimension is fast enough so that their quantum-mechanical wavelength is small enough to allow them to “fit” within it, then some familiar features of wave phemonena will take over. Now, on a vibrating string only certain wavelengths are allowed, which explains why longer strings, when plucked, produce lower notes than shorter strings. On a vibrating string, only certain harmonics can survive—waves whose wavelength has a specific relationship to the length of the string (that is, is equal to the length of the string, half the length of the string, one-third the length of the string, etc.). (For those of you who are getting excited at the mention of the word string, you may calm down. This has nothing to do with superstring theory, which we shall get to later.)

  Now, if this held true for particles moving around the circular extra dimension, since a particle’s quantum mechanical wavelength is determined by its velocity, then only particles with certain fixed velocities would be able to propagate all the way around the extra dimension. A fixed set of velocities implies a fixed set of energies associated with the particle. But since energy affects geometry in general relativity, then if this theory applies in the full five-dimensional space, it means that the geometry of the fourth spatial dimension will be affected in specific, discrete ways by the presence of such particles.

  Remarkably, in the Kaluza theory the effect of this change in the geometry of the fourth spatial dimension would be measured in our three-dimensional space as the existence of an electric field. Since the energies are only allowed in discrete values, the resulting electric fields, which arise from electric charges that we would view as emanating from the location in our three-dimensional space where these particles start their voyages around the extra dimension, must also come in discrete steps. Thus, all charged particles would have electric charges that are discrete multiples of some basic charge. In this way, Klein proposed that an extra dimension could explain not only the existence of both gravity and electromagnetism, but also the nature of all charged objects we measure in our universe!

  With so much going for it, one might wonder why the Kaluza-Klein theory (as it is now known) did not become the next big thing in physics in the 1920s and ’30s, and why these physicists are not now household names, like Einstein. There are a number of reasons. First, it became clear in these decades that the laws of quantum mechanics developed by Schrödinger, Heisenberg, and later by Dirac and others, while weird in the extreme, were nevertheless perfectly consistent with all experiments and, moreover, were inconsistent with the existence of extra “hidden variables” that might somehow lead to the apparent probabilistic nature of the theory. Thus, there was no apparent need for extra dimensions in which to hide these variables.

  More important, however, was the fact that while Klein’s explanation for why the fifth dimension was hidden was ingenious, it was also clearly incomplete. Namely, why would the fourth spatial dimension curl up in a circle while the other three s
patial dimensions did not? Compounding this issue was the residual problem of that one extra quantity related to the five-dimensional metric that Kaluza, and later Klein, continued to ignore. It was recognized clearly by the 1940s that this extra quantity would affect the nature of gravity, so that the residual theory in four dimensions would no longer precisely be described by general relativity. Finally, and most important of all, perhaps, was the fact that the world of physics was continuing to undergo revolutionary changes. Starting in 1930, with the discovery of the neutron, the subatomic world began to become far more complex and interesting. In short order, antimatter was discovered, as was a then new force in nature, now known as the weak force, responsible for radioactive decays. Any unification of merely gravity and electromagnetism would thus fall far short of a complete description of nature. Consequently, the majority of the physics community—rightly, I would argue—began to concentrate on trying to understand this host of new experimental phenomena, and left speculations about unobserved extra dimensions aside. It would take almost half a century before events would once again drive physicists to reconsider the possibility of a new hidden universe lying just out of sight.