69 pages • 2-hour read
The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory
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Part 4, Chapter 13-Part 5Chapter Summaries & Analyses
Part 4: “String Theory and the Fabric of Spacetime” - Part 5: “Unification in the Twenty-First Century”
Part 4, Chapter 13 Summary: “Black Holes: A String/M-Theory Perspective”
In the late 1960s, a group of physicists posited that black holes were similar to elementary particles: miniscule points identifiable by their mass, spin, and force charge, just like matter particles. However, this is where the math breaks down: The tiny size and spin indicate that quantum mechanics are required, but the enormous mass requires relativity, which is how the two theories are most incompatible. String theory offers a route for moving past this obstacle, though the reasons are convoluted.
First, Greene returns to the idea of loops of string circling around a two-dimensional curled space, as discussed in Chapter 11. Physicists know from the equation that the loop of string protects a collapsing two-dimensional space from catastrophe. Now theorists asked what would happen if a three-dimensional sphere within a six-dimensional Calabi-Yau space collapsed. As a one-dimensional loop of string cannot surround it, early indications suggested that such a collapse would “yield a cataclysmic result” and “the workings of the universe would grind to a halt if such a collapse were to occur” (323).
However, as discussed in Chapter 12, newer versions of string theory (and M-theory) argue that along with one-dimensional strings, two-, three-, and even higher dimensional branes exist. In 1995, physicist Andrew Strominger suggested that such a “three-brane” would be able to completely wrap a three-dimensional sphere, thus canceling out the cataclysmic results. It was not initially apparent why this was important. Then Greene and Morrison applied this concept to their earlier work on tearing and repairing spaces in topology-changing transitions. They argued that “after the collapse of a three-dimensional sphere inside a Calabi-Yau space, [the] space could tear and subsequently repair itself by growing a two-dimensional sphere” (327) leading to drastic changes to the shape and configuration of the Calabi-Yau space. Thus, the shape could transform itself into a new shape without altering its physical properties. Together, Strominger, Greene, and Morrison published a paper on their findings that string theory allowed not only for the minor tears previously discussed in topology-changing transitions but also for much larger tears, which they called conifold transitions.
Greene then applied this to black holes. They had shown that conifold transitions have no pronounced effect on physics despite the “drastic rips” (329). However, they do have two important consequences. First, the three-brane smears around the three-dimensional sphere, causing a “gravitational field that looks like that of a black hole” (330). Furthermore, Strominger showed that the mass of this three-brane (in other words, the mass of a black hole) is proportional to the volume (size) of the sphere it covers so that when the sphere collapses to a point, the corresponding three-brane, or black hole, is massless—an idea that on its face makes little sense. Greene explains that initially massive black holes eventually shrink down so far that they simply become a single vibrating string. Thus, string theory shows that black holes and elementary particles are merely two sides of the same coin. The success of this discovery has inspired string theorists to investigate whether any other stubborn mysteries of physics “might also succumb to the power of string theory” (333). One such mystery is the idea of “black hole entropy” (333). Entropy is a component of thermodynamics and a “measure of disorder or randomness” (333) in a system. A chaotic and completely disordered system has high entropy, and a very organized and orderly system has low entropy.
In 1970, a physics graduate student named Jacob Bekenstein suggested that black holes have entropy. He argued that as black holes swallow all the matter around them, they swallow the entropy associated with that matter. However, the second law of thermodynamics states that the total entropy of a system always increases; therefore, the entropy swallowed by the black hole cannot disappear. The black hole itself must embody that total entropy.
The physics community initially dismissed Bekenstein’s suggestion. Stephen Hawking, renowned for his work on black holes, was at first “willing to accept that if matter carrying entropy is dropped into a black hole, this entropy is lost, plain and simple” (336) with no regard for the second law of thermodynamics. However, in 1974, Hawking showed, through a complex set of calculations that combined certain elements of quantum mechanics and relativity, that “black holes glow” (337). Black holes emit radiation (like light or heat) exactly in accordance with Bekenstein’s application of entropy and thermodynamics. Furthermore, according to Hawking, the “gravitational laws of black hole physics are nothing but a rewriting of the laws of thermodynamics in an extremely exotic gravitational context” (337). Hawking’s findings demonstrated that black holes possess large amounts of entropy that grow larger the more massive a black hole is. Following this logic, in 1996, Strominger and Vafa used string theory to show that black holes could hypothetically be constructed from a series of specific branes bound together “according to a precise mathematical blueprint” (339), thus arguing that constituents of string theory (strings and branes) are responsible for the construction and entropy of black holes.
One unresolved mystery is the question of what happens to spacetime at the center of a black hole. According to general relativity, enormous amounts of mass and energy are crushed together at the center, tearing the fabric of spacetime to devastating effect. One possible consequence of this is that time ceases to exist at the center of a black hole. Another possibility is that the center of a black hole may attach to “a gateway to another universe” (344).
Part 4, Chapter 14 Summary: “Reflections on Cosmology”
The accepted image of origins of the universe is called the standard model of cosmology, which states that the universe began approximately 15 billion years ago with an “enormously energetic, singular event, which spewed forth all of space and all of matter” (346), routinely called the big bang. At the smallest measurement of time after the big bang (approximately 10-43 seconds after), the temperature of the universe was about 1032 Kelvin. The universe continued to expand and cool, until after about three minutes, the universe cooled enough to allow the formation of hydrogen and helium (347). Little else happened until the temperature dropped enough for other “electrically neutral atoms” (347) to form. About another billion years after that, atoms calmed down enough to start forming recognizable clumps of matter like galaxies, stars, and planets.
This was hypothetical until the discovery of cosmic background radiation in the 1960s. Physicists had theorized that with the expansion and cooling of the universe, an “almost uniform bath of primordial photons, which […] have cooled to a mere handful of degrees above absolute zero” (348) should exist. Theorists confirmed that the universe is filled with a kind of microwave radiation, whose temperature is about 2.7 degrees above absolute zero, in accordance with the big bang theory. Additional proof of the big bang comes from mathematical predictions of how much helium should exist because of the big bang. Scientists are therefore convinced that the big bang theory is valid. The question is what happened in the minuscule moments of time just after it occurred. Because the universe is hotter, smaller, and denser the farther back in time physicists go, quantum mechanics and relativity become increasingly necessary to understand its properties. However, here relativity and quantum mechanics eventually fail. Greene states that the book’s central message is that this is the moment when string theory becomes vital.
One issue is the horizon problem. Studies have shown that no matter what direction one looks in the sky, the cosmic background radiation is the same temperature. One might intuitively assume this is because these locations in the universe were once in contact at the moments just after the big bang. But the standard model of cosmology does not support this, in part because information (including heat) can only travel at light speed. The regions of space farther from the center should be cooling faster than those closer to the center, and measurable differences in temperature should depend on where one looks.
Physicist Alan Guth resolves this problem in 1979. The big bang relies on the idea that the expansion rate of the universe slows due to the gravitational force of the universe resisting outward movement (like a ball thrown in the air slowing down just as it reaches its peak height and gravity pulls it back down). However, Guth proposed that the horizon problem could be solved by assuming that the expansion of the universe gets faster as it proceeds. He suggested that “the very early universe undergoes a brief period of enormously fast expansion—a period during which it ‘inflates’ in size at an unheralded exponential expansion rate” (355), which allows the separated regions the necessary time to receive the same information and reach a uniform temperature (355). Within this inflationary framework, the timeline of the standard model is refined slightly: in the time of about 10-36 to 10-34 seconds after the big bang the universe expanded by a “colossal factor of at least 1030” (356). This leaves a sliver of time between the instant of the big bang to the Planck time yet unexplored. Here, string theory modifies the standard cosmological model in three ways. First, “string theory implies that the universe has what amounts to a smallest possible size” (357). Second, the concept of small-radius/large-radius duality discussed in previous chapters has profound influences on the cosmological timeline. Third, any cosmological model must address the development of more than four dimensions, as posited by string theory.
String theory argues that as one “run[s] the clock backward in time toward the beginning” (358) the temperature of the universe rises until the universe reaches the size of about the Planck length, at which point the “temperature hits a maximum and begins to decrease” (358). This is based on the concept that dimensions with a large radius and small radius are inversely proportional, and therefore demonstrate identical physical properties. The universe compresses to the Planck length and then bounces back, once again increasing in size. Because the temperature decreases as the universe expands, “the futile attempt to squeeze the universe to sub-Planck size means that the temperature stops rising, hits a maximum, and then begins to decrease” (358).
Physicists Robert Brandenberger and Cumrun Vafa propose that at the moment of the big bang all the dimensions were curled up together in a “Planck-sized nugget” (358). The looped strings around curled dimensions keep them contained; however, when a looped string and its antistring partner (a string that wraps in the opposite direction) come into contact, they annihilate each other, creating a single unwrapped string that allows the curled dimension to expand. Brandenberger and Vafa showed that the more curled dimensions there are, the higher the chances that a string and antistring will touch. However, as the released dimensions expand, it becomes less likely for more strings to become entangled and annihilate each other, thus preventing more curled dimensions from breaking loose. Therefore, during the first stage of expansion, three curled dimensions uncurled and expanded into the spatial dimensions humans are familiar with, while the others remained curled up.
Greene states that cosmology may be “the closest we may ever come to understanding why [the universe] began” (364) and not merely how. The study of cosmology therefore is of utmost importance to those in search of a unifying theory of physics. However, there are still too many unknowns. Greene explains with another analogy: When one tosses a ball in the air, the laws of gravity determine how that ball will behave; but one cannot accurately predict where the ball will land without the additional information of the ball’s velocity. One must know its initial conditions. Similar limitations exist in cosmology. Physicists and astronomers do not know the “initial conditions of the universe” (365).
The anthropic principle suggests that it may be asking too much to ever know. This principle argues that if the properties of the universe were different than they are, human beings would not have evolved to observe it. Therefore, the universe could have formed in any number of configurations; there simply would not have been any human beings alive to know the difference. For instance, humans could merely be “one of an enormous number of island universes scattered across a grand cosmological archipelago” (366), in a concept called multiverse theory. Similarly, each black hole could in essence be a “seed for a new universe that erupts into existence through a big bang-like explosion but is forever hidden from our view by the black hole’s event horizon” (369). The ultimate unifying theory, should it be found, may be able to describe the properties of a “wealth of universes, most of which have no relevance to the one we inhabit” (370).
Part 5, Chapter 15 Summary: “Prospects”
Having explored the possible implications string theory has on the largest questions of physics, and some of the current obstacles to proving these suppositions, Greene concludes by suggesting some prospects for the future of string theory research. In particular, he focuses on five central questions that string theorists must face and attempt to address in their “pursuit of the ultimate theory” (374).
First, they must determine the “fundamental principle underlying string theory” (374). One overarching concept within physics is the principle of symmetry, as examined throughout the book. Many issues arising from relativity, quantum mechanics, and string theory can all be explained by applying symmetry. However, this “shifts the question of why a certain force exists to why nature respects its associated symmetry principle” (374). Symmetry seems a consequence rather than a cause of various aspects; therefore, future theorists will need to determine whether some other, currently unknown principle “inexorably leads” to symmetry.
The second question is what space and time really are. Einstein describes space and time and a single multidimensional “fabric” called spacetime. However, what that really means deserves scrutiny. Initially, Newton pronounced that space and time are “eternal and immutable ingredients in the makeup of the cosmos, pristine structures lying beyond the bounds of question or explanation” (377). However, physicists later argued that “space and time are merely bookkeeping devices for conveniently summarizing relationships between objects and events” (377). The question remains whether space and time are social constructs used as shorthand or humans can view themselves as “truly being embedded in something” like spacetime fabric. Greene argues that string theorists must not make assumptions based on previous concepts of spacetime but rather “allow string theory to create its own spacetime” (379). Attempting to find the “correct mathematical apparatus” to do so without dictating “pre-existing notions of space and time” (379) is one of the biggest challenges string theorists currently face.
Third, theorists ask whether string theory will require an overhaul of quantum mechanics. Throughout modern physics, the process of formulation has required starting with the classical framework of physics before modifying various aspects with a quantum-mechanical formula as necessary. However, the symmetries within string/M-theory are “inherently quantum-mechanical symmetries” (382), which implies that a complete formulation of string/M-theory will require an overhaul of quantum mechanics without first relying on classical physics.
The fourth question is whether string theory can be experimentally tested. As of now, the answer is no. This is one of its biggest obstacles. Any proposed theory should be testable before it can be fully accepted. However, current technology and measuring instruments are not powerful enough test string theory with the necessary precision. Theorists continue to work toward this goal. Greene points to the eventual construction of the Large Hadron Collider as string theory’s biggest hope for experimental proof.
Fifth is a question that plagues not only string theorists but scientists in general: whether explanation has limits. As discussed throughout the book, the desire to understand and explain every aspect of the universe is the biggest goal and biggest challenge for all scientists. Theorists believe that string theory offers the best hope of meeting this challenge. However, current failures of any theory to do so may indicate more than technological limitations. Possibly, there “is no explanation for these observed properties of reality” (385). While science has been successful so far, “a limit to comprehensibility” (385) may exist, and humanity may reach an “absolute limit of scientific explanation” (385). However, this possibility must not prevent scientists from trying. As Greene states, “The search for the fundamental laws of the universe is a distinctly human drama, one that has stretched the mind and enriched the spirit” (387). He adds, “We are all, each in our own way, seekers of the truth and we each long for an answer to why we are here” (387). The fear of never reaching an answer should not discourage humans from continuing the search.
Part 4, Chapter 13-Part 5 Analysis
The last chapters of Part 4 and the single chapter of Part 5 focus on the largest implications of string theory for the biggest questions of the universe (black holes and the big bang) and culminate with Greene’s hopes for how string theory will progress. These three chapters are highly speculative, relying on large assumptions (based on the perturbation theory discussed in Chapter 12), big concepts, and hypothetical ideas of how future technological developments may (or may not) confirm those concepts. This speculative nature might reflect the most inherent quality of theoretical physicists and humanity in general. As Greene emphasizes throughout the book, humanity is driven by questions of “why” and “what if,” and none more than theoretical physicists. In this way, the last three chapters most clearly represent The Human Need to Understand as a theme.
Greene’s discussion of how string theory approaches black holes and the big bang, while highly intriguing, also encapsulates many of the book’s interrelated concepts. The building blocks that Greene establishes chapter by chapter reach their final form in revisiting the concept of “realms that are tiny and yet incredibly massive” (4) to examine the point at which the mathematics of general relativity and quantum mechanics break down, as Greene mentioned in the first chapter. In this way, Greene brings the book full circle, back to the topics with which he began. As in Chapter 11, Greene returns to narrative structure to share his role in developing a string theory response to the mysteries of black holes. Once again partnering with his colleague Morrison, and another physicist named Strominger, Greene helped determine the consequences of possible fabric-tearing black hole scenarios, and the book uses a storytelling approach to unravel the mysteries bit by bit. Greene’s personal involvement in, and reflections on, several of the developments of string theory is one of the book’s major highlights. His insider perspective is therefore illuminating.
In Chapter 14, Greene again relies on analogies, returning to previous visuals like throwing a ball, among others. However, even with analogies, the scales involved in the big bang theory are so extreme (both extremely large and extremely small) that conceptualizing it becomes nearly impossible, even to most physicists. To help readers, Greene illustrates the time scales via a diagram in Figure 14.1 (356). However, even this proves largely incomprehensible, and possibly for this reason Greene concludes Chapter 14 by reiterating the anthropic principle, pointing to the limits of knowledge and the human capacity to understand. Few things make those limitations more apparent to the reader than the vast stretches of time and distance involved in the big bang theory, thematically foregrounding The Limitations of Intuition.
In Chapter 15, Greene concludes by acknowledging the current limitations of string theory and sharing his hopes for what will come next. Like many others, this chapter is organized around a list of “five central questions” (374), which he addresses in depth one at a time. These five questions represent the holes in string theory’s capacity to provide a possible unifying theory. As the title of Part 5: “Unification in the Twenty-First Century” suggests, Greene has high hopes that further research, refined mathematical formulas, and advancing technology will eventually lead theorists to the answers they search for. In addition, he remains confident that string theory, or rather its overarching framework, M-theory, will ultimately provide that answer. Many of the ideas Greene concludes with again demonstrate the intertwined nature of two of the book’s major themes: The Human Need to Understand and The Unification of Physics. Specifically, humans’ innate need to understand everything in the universe drives physicists to find a unifying theory of physics because they believe that it must exist.
However, Greene also admits that no concrete proof exists that any such answer does exist; physicists may eventually hit an “absolute limit to scientific explanation” (385) beyond which no explanation will be found. This echoes but is not quite the same as the limitations inherent in the anthropic principle. Most importantly, however, Greene emphasizes that the possibility of never reaching an answer will not discourage scientists from trying to find one. Hope will drive them onward because the human need to understand is central to “the whole human struggle” (387). If nothing else, this need inspires people to stretch their imaginations beyond what intuition might initially allow.
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