69 pages • 2-hour read
The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory
A modern alternative to SparkNotes and CliffsNotes, SuperSummary offers high-quality Study Guides with detailed chapter summaries and analysis of major themes, characters, and more.
Part 4, Chapters 10-12Chapter Summaries & Analyses
Part 4: “String Theory and the Fabric of Spacetime”
Part 4, Chapter 10 Summary: “Quantum Geometry”
Chapter 10 demonstrates why the complexities of string theory required a new kind of geometry. Greene first explains the previous form of geometry, which Einstein used in developing relativity theory. Because classical geometry, called Euclidean geometry (after the Greek mathematician Euclid) works with flat planes, it was insufficient for Einstein’s needs. In 1854, mathematician Georg Riemann “broke the chains of flat-space Euclidean thought” (231) to develop a geometry that worked on curved surfaces. Reimann geometry was integral to Einstein’s development of general relativity. Reimann geometry describes the curvature of spacetime by relying on the “distorted distance relations between points” (233). However, string theory proposes that particles are not points but strings with a tiny but non-zero length dimension. Therefore, a new geometry, called quantum geometry, is necessary for string theory equations.
According to current theory, the universe began with a big bang from a single tiny point, from which it continues to expand. The standard model of physics, using quantum mechanics, argues that the universe both begins and ends (first expanding and then contracting) in a single zero-size point: The entire mass of the universe condenses to this point with no spatial dimensions. However, string theory argues that “the universe cannot be squeezed to a size shorter than the Planck length in any of its spatial dimensions” (236). This is because string theory holds that the smallest possible size is a single string, which roughly equals a Planck length. However, it is more complex than that. The theory is based on the idea that loops of string can lay flat on a dimension or wrap around a curved dimension. Whereas the previous discussion of strings focused on their vibrational energy, this ability to wrap around a dimension adds a second component: winding energy. To describe this, Greene returns to the garden hose analogy. Consider the garden hose as a universe in the process of contraction back down to its beginning stage: As the radius of the circular (clockwise) dimension of the hose shrinks down to the Planck length and then, according to general relativity, continues to shrink smaller still, “string theory insists upon a radical reinterpretation of what actually happens” (239):
[A]ll physical processes in the Garden-hose universe in which the radius of the circular dimension is shorter than the Planck length and is decreasing are absolutely identical to physical processes in which the circular dimension is longer than the Planck length and increasing! (239).
Thus, string theory prevents any attempt to shrink to a size smaller than Planck length, leading not to “complete cosmic collapse” but to “cosmic bounce” (239). Theorists then determined that a string’s vibrational energy is inversely proportional to the radius of a circular dimension, whereas winding energy is directly proportional to the radius. A large radius results in large winding energies and small vibrational energies; a small radius results in small winding energies and large vibrational energies. Because the physical properties of matter depend on the total energy of a string rather than the specific divisions of energy between vibration and winding, “there is no physical distinction between these geometrically distinct forms” (241). A “fat” garden hose universe and a “thin” garden hose universe have the same physical laws. Furthermore, because these two universes are indistinguishable, “any experiment done in one such universe has a corresponding experiment that can be done in the other, leading to exactly the same results” (246).
This works even as theorists expand beyond the two-dimensional garden hose universe to the three-dimension universe and beyond. String theory argues that distances can be measured by strings that are wound or strings that are not wound, leading to two different but inversely proportional measurements. One version dictates that the “universe is large and expanding,” and the second dictates that it is “small and contracting” (251), but both measurements are equally valid. This proves indispensable to working equations in string theory. As one method of measuring distance becomes smaller and more difficult, the other becomes larger and easier; if one always switches to the easier mode, the “minimal value encountered is the Planck length” (252). Therefore, sub-Planck-length distances become inaccessible.
Based on this quantum geometry, a group of physicists suggested that “it might be possible for two different Calabi-Yau shapes, chosen for the extra curled-up dimensions in string theory, to give rise to identical physics” (255). According to Kaluza-Klein, the physical properties of particles depend on the number of holes in the extra dimensions and the way strings vibrate in those dimensions. The exact arrangement of those holes is unimportant. In 1987, as a postdoctoral fellow, Greene and his colleague Ronen Plesser demonstrated that the holes of a Calabi-Yau shape could be manipulated into a new shape and retain the same physical properties. They had discovered mirror symmetry, which claims that “particular pairs of Calabi-Yau spaces, pairs that were previously thought to be completely unrelated, are now intimately connected by string theory” (259). Most importantly, just as with the two interchangeable notions of distance, mirror pair Calabi-Yau shapes can be interchanged as needed to make equations easier to manage so that “calculations of almost unimaginable difficulty could be accomplished by using the mirror perspective, with a few pages of algebra and a desktop computer” (261).
Part 4, Chapter 11 Summary: “Tearing the Fabric of Space”
Physicists have long speculated about the concept of tears in the fabric of spacetime. According to Einstein’s relativity theory, tears in spacetime are not possible because “were the fabric of space to develop such irregularities [tears], the equations of general relativity would break down, signaling some or other variety of cosmic catastrophe” (263), implying that a tear might be possible, but only once, as no universe would exist afterward. This has not stopped physicists from imagining what a tear in space would look like. The two most concept images are wormholes and black holes.
Wormholes are hypothetical bridges or tunnels that “provide a shortcut from one region of the universe to another” (264). A wormhole would essentially “create a new region of space” (265) between two regions of space. No strong evidence suggests that wormholes exist. Black holes, on the other hand, are possible. Some physicists suggest that black holes are punctures or tears through spacetime.
String theorists have shown how tears in space such as black holes are not only possible but likely. Mathematician Shing-Tung Yau devised a math procedure in which certain Calabi-Yau shapes could be punctured along their surface, and then the resultant tears “sewn up” (266) into new configurations. This series of manipulations, called a flop transition, is complex but suggests that spatial dimensions may tear and then repair themselves, allowing for tears in spacetime that do not result in cosmic catastrophe. Physicist Paul Aspinwall, a schoolmate of Greene’s, continued this line of thought by asking what such a flop transition tear would look like in the Calabi-Yau shape’s mirror symmetry partner.
Mathematician Victor Batyrev found Greene and Plesser’s work on mirror symmetry intriguing and built on their work, though as a mathematician Batyrev had vastly different approaches and skills. Using his own techniques, he was able to refine Greene and Plesser’s work, which he called “black magic” (271), into more conventional and manageable math. Greene, Aspinwall, and another colleague, David Morrison, realized that they could use this new math to address the question of flop transitions on mirror pairs. In 1992, Greene, Aspinwall, and Morrison worked together at the Institute for Advanced Study to attempt to do so. In the process, Greene and Morrison realized that their respective approaches (Greene as a physicist and Morrison as a mathematician) were often at odds, and they needed to give each other crash courses in their specialties to proceed. Aspinwall was the computer programming specialist who needed to take the equations Greene and Morrison devised and transcribe them for a computer to understand and compute.
Edward Witten was at the Institute during this time as well and became intrigued by their work. He began working on the same problem, approaching it from an entirely different direction. It became a friendly race to see who would have results first. Eventually, Greene and his group devised a set of equations to plug into the computer program. This is the essential argument: A Calabi-Yau shape and its mirror partner can have vastly different shapes and still display the same physical properties. Usually, the mathematics involved are extremely complex in one partner and relatively straightforward in the other. Greene and his group posited that a tear in one Calabi-Yau shape might imply catastrophe in one pair, while being manageable in its mirror partner. If the tear is not catastrophic in the mirror, then it is not catastrophic in the original either.
The complex equations they plugged into the computer program showed this was true. Even better, they completed their equations just days before Witten reached the same conclusions using an entirely different method. Together, they had discovered a process they called topology-changing transitions. Greene explains that “the fabric is tearing, but it does so in a fairly mild way. It’s more like the handiwork of a moth on wool than that of a deep knee bend on shrunken trousers” (280). Both Witten’s work and the work of Greene and his team showed that the physical properties “such as the number of families of string vibrations and the types of particles within each family” (280) remain completely unaffected by the tearing process.
This raises two more questions. The first is whether similar tears can also happen in the three extended spatial dimensions of normal space. Greene, Witten, and the others analyzed the math to determine that this appears possible. The second question is whether such tears have happened in the past and whether they could happen now. Again, the answer is yes. It is entirely possible that such tears happened in the past, but humans simply lack the equipment necessary to observe the aftereffects. Furthermore, Greene posits that “the universe could currently be in the midst of a spatial rupture. It if were occurring slowly enough, we would not even know it was happening” (281-82). The absence of direct observational evidence of such a tear is likely a good thing.
Part 4, Chapter 12 Summary: “Beyond String Theory: In Search of M-Theory”
Greene argues that a unifying theory of physics would be most convincing if it could show that “things are the way they are because they have to be that way” (283). This kind of inevitability would mean that “there are no choices [...] the universe could not have been different” (283). Although physicists had early hope for string theory, subsequent work proved that it falls short for two reasons. First, there are five competing versions of string theory. Second, these five versions each offer many different possible solutions to the equations involved.
Edward Witten suggested a solution to the first problem in 1995, thus starting the second superstring revolution. He suggested that the five versions are complementary pieces of one overarching framework. Witten and subsequent theorists argued that the five versions of string theory are “like the starfish’s five arms” (illustrated in Figures 12.1-2 in the book, pages 286-87), connected into one organism or structure that encompasses them all. This all-encompassing structure is called M-theory.
The second problem arises from the fact that physicists have managed to devise only “approximate versions of the equations” (285). The mathematical formulas of string theory are currently too complex and require details that cannot yet be determined (because of insufficient technology). This requires that theorists approximate equations, leading to imprecise results. This method of approximating equations is called perturbation theory, which works by ignoring the fine details of a system to devise rough answers to questions, which are then continually refined by slowly adding the details back in. Figures 12.4-6 (292-93) help illustrate the importance of this method. However, some refinements differ so greatly from the rough estimates that they cause a “failure of perturbation theory” (289). For perturbation theory to work, the initial “ballpark estimate” (291) must be relatively close to the final data. The problem with string theory is that theorists are not yet sure if they are even “in the ballpark” (291) to begin with.
Theorists argue that the “physical processes of string theory are built up from the basic interactions between vibrating strings” (291), yet the frenzied processes of quantum mechanics and the complex interactions between strings leads to an infinite number of string loops and infinite answers in their calculations. Therefore, exact calculations become an impossible task, and “instead, string theorists have cast these calculations into a perturbative framework based on the expectation that a reasonable ballpark estimate is given by the zero-loop processes” (294).
To trust this perturbative framework, they must trust the ballpark estimate. The estimate is based on “likelihood that quantum fluctuations will cause a single string to split into two strings” (294), which is called the string coupling constant. If the estimate is close, then the resulting data is useful. Unfortunately, theorists at present do not know the value of the string coupling constant. Physicists can “be sure that conclusions based on a perturbative framework are justified only if the string coupling constant” (295) is small (less than 1). The next breakthrough in string theory will likely require a nonperturbative approach, which has not yet been devised.
Witten’s breakthrough in proposing that the five versions of string theory are parts of a whole was based on the concept of duality, in which “theoretical models that appear to be different [...] nevertheless can be shown to describe exactly the same physics” (298). It is yet another kind of symmetry. Witten suggested that the five versions of string theory exist in dual pairs, requiring a sixth yet undetermined version. Just as mirror symmetry between paired Calabi-Yau shapes leads to identical physical properties despite apparent differences, the five string theories are similarly paired by inverse string coupling constants. For instance, Witten showed that when Type I string theory has a large string coupling constant, its properties “exactly agree with known properties of Heterotic-O string theory” (304) when its coupling constant is small. They are duality pairs, and because perturbative methods work when the constant is small, theorists can determine the physical properties for Heterotic-O easily, thereby determining the properties of Type I simultaneously. Although theorists cannot yet prove this duality, if true it would provide an invaluable tool for analyzing theories with large coupling constants by simply using perturbative theory on its small constant pair instead.
To extend this concept beyond the Type I/Heterotic-O pair, however, Witten required a shift in thinking about the dimensions of strings. In quantum mechanics, the basic units of the universe were viewed as zero-dimensional point particles. String theory shifted to one-dimensional strings. Additionally, string theory required 10 dimensions (nine spatial and one time dimension). However, some formulas suggested that string theory requires 11 dimensions (10 spatial and one time dimension). Witten posited that strings were not one-dimensional but two-dimensional, possessing both length and width, and that this width constituted the tenth spatial dimension. The long-held assumption among theorists that strings must be one-dimension only, “having only length but no thickness” (311) was a blind spot. And, in fact, “Type IIA and Heterotic-E ‘strings’ are, fundamentally, two-dimensional membranes living in an eleven-dimensional universe” (311). The string theory of this eleven-dimensional universe is called M-theory.
Greene argues that by applying all dualities discussed (including the previous concept of mirror symmetry), physicists can “pass from any one theory to any other, so long as we also include the unifying central region of M-theory” (314). These conclusions still leave questions, however, such as how theorists would ensure that two-dimensional membranes are the “fundamental ingredient of string theory” (316). Greene states that they do not yet know for certain. Some properties of symmetry now suggest that in addition to two-dimensional membranes, multidimensional ingredients (each indicated by a number and the suffix “-brane” such as: three-brane, four-brane, et cetera) may exist. However, due to the inverse proportionality of string coupling theory, and the energy-mass transfer of E=mc2, most of these multidimensional branes would be so massive and require so much energy to form that they likely have not existed since the big bang. The next task for string theorists is to definitively prove that any aspect of M-theory accurately describes the universe, which requires devising exact (rather than approximate) equations. This, according to Greene, is “the program for unification in the twenty-first century” (319).
Part 4, Chapters 10-12 Analysis
In Part 4, Greene turns his attention to what string theory suggests about the fabric of spacetime, the complex realities of the universe, and the moments of the big bang. To explain how string theory comes to these conclusions, he first establishes a development that arises from string theory: quantum geometry, the topic of Chapter 10. Quantum geometry, along with being an enormous shift in mathematics and physics, once again highlights the limits of intuition. Quantum geometry defies the human ability to visualize and conceptualize shapes and spaces based on intuition and the three-dimensional observable universe. Greene once again relies on the analogy of the garden hose universe to explain the abstract concepts of quantum geometry and help readers visualize them. As before, the ideas and information established in previous chapters become increasingly important as Greene expands into more complex issues. In this way, he neatly scaffolds knowledge to make it easier for readers to absorb.
After establishing the new field of quantum geometry, Greene then shifts to two other major developments: what quantum geometry has to say about tears in spacetime, and a new proposal for a unifying theory. Once again, the ideas established here become vital building blocks for understanding the last two chapters of Part 4, which discuss black holes and the big bang. Before Greene can address these topics, however, he must deal with two major developments that arise from string theory. Additionally, these chapters build on knowledge discussed in Part 3. In particular, the discussion of tears in spacetime relies on Greene’s explanation of Calabi-Yau shapes in Chapter 8.
To explain the developments discussed in Chapter 11, Greene returns to storytelling elements. Specifically, he tells the story of his own research along with colleagues Morrison and Aspinwall in 1992, as they examined the possibility of “space-tearing flop transitions” (270). Throughout this chapter, Greene maintains his narrative structure, revealing clues and conclusions as they happened in real time, rather than summarizing the scientific concepts all at once. The mystery of spacetime tears and their effects on Calabi-Yau shapes, and therefore on string theory, unfold with a genuine sense of anticipation, which allows readers to experience the excitement just as the researchers did, while also giving readers time to assimilate each new piece of information before the next emerges. This narrative structure appears again in Chapter 13.
Chapter 12 then returns to the moment first referenced at the beginning of Part 3, when Greene briefly mentions in Chapter 6 that Witten initiated the second superstring revolution with a “breathtaking lecture” in 1995. Greene does not initially explain what this lecture was about or how it reignited interest in string theory that had waned after 1986. However, Chapter 12 explains this in detail. Greene attempts to convey the sense of shock and wonder that rippled through the physics community following Witten’s lecture. The excitement is palpable. For Greene and others, M-theory represents the closest current progress toward a true unifying theory of physics, the answer to their search for unification. Here, again, Greene focuses on the theme of unification, especially in his call for a theory that relies on inevitability. He states that confirming the “inevitability in the structure of the universe would take us a long way toward coming to grips with some of the deepest questions of the ages” (283), bringing theorists ever closer to unification. Chapter 12 is the book’s longest chapter, partly because the concepts of M-theory are so complex. Also contributing to this length is Greene’s in-depth explanation of perturbation theory, the mathematical principles by which quantum mechanics and string theory must be devised for lack of more precise technology with which to complete equations. Greene glosses over the incredibly complex mathematics of perturbation theory, but he neatly addresses the basic ideas through analogy as well as figures and diagrams. Likewise, Greene explains the concept of duality in M-theory through analogy and metaphor, like the image of a starfish.
Greene’s explanation of M-theory gestures again toward intuition as it relates to the assumptions physicists make about the universe. Witten argues in his case for M-theory that theorists have long assumed strings must be one-dimensional (having length but no width), and this assumption has limited their ability to uncover the true potential of string theory. This and other assumptions are grounded in human intuition about how the universe works, once again highlighting the theme of The Limitations of Intuition. Once theorists realize the limits of their own intuition and let go of assumptions, the implications of string theory begin to make more sense.
Unlock all 69 pages of this Study Guide
Get in-depth, chapter-by-chapter summaries and analysis from our literary experts.
- Grasp challenging concepts with clear, comprehensive explanations
- Revisit key plot points and ideas without rereading the book
- Share impressive insights in classes and book clubs