by Brian Greene
The Simulated Multiverse, at least in theory, might also be linked to a pared-down version of the Ultimate Multiverse that includes only universes based on computable mathematical structures. Unlike the full-blown version of the Ultimate Multiverse, this more limited incarnation has a genesis story that lifts it beyond mere assertion. The users, real and simulated, who are behind the Simulated Multiverse will, by definition, be simulating computable mathematical structures and thus will have the capacity to generate this part of the Ultimate Multiverse.
Gaining experimental or observational insight into the validity of any of the multiverse proposals is surely a long shot. But it’s not an impossibility. And with the immensity of the potential payoff, if the exploration of multiverses is where the natural course of theoretical research takes us, we must follow the trail to see where it leads.
How Does a Multiverse Affect the Nature of Scientific Explanation?
Sometimes science focuses on details. It tells us why planets travel in elliptical orbits, why the sky is blue, why water is transparent, why my desk is solid. However familiar these facts may be, it is wondrous that we’ve been able to explain them. Sometimes science takes a larger view. It reveals that we live within a galaxy containing a few hundred billion stars, it establishes that ours is but one of hundreds of billions of galaxies, and it provides evidence for unseen dark energy permeating every nook and cranny of this vast arena. Looking back just a hundred years, to a time when the universe was thought to be static and populated solely by the Milky Way galaxy, we can rightly celebrate the magnificent picture science has since painted.
Sometimes science does something else. Sometimes it challenges us to reexamine our views of science itself. The usual centuries-old scientific framework envisions that when describing a physical system, a physicist needs to specify three things. We’ve seen all three in various contexts, but it’s useful to gather them together here. First are the mathematical equations describing the relevant physical laws (for example, these might be Newton’s laws of motion, Maxwell’s equations of electricity and magnetism, or Schrödinger’s equation of quantum mechanics). Second are the numerical values of all constants of nature that appear in the mathematical equations (for example, the constants determining the intrinsic strength of gravity and the electromagnetic forces or those determining the masses of the fundamental particles). Third, the physicist must specify the system’s “initial conditions” (such as a baseball being hit from home plate at a particular speed in a particular direction, or an electron starting out with a 50 percent probability of being found at Grant’s Tomb and an equal probability of being found at Strawberry Fields). The equations then determine what things will be like at any subsequent time. Both classical and quantum physics subscribe to this framework; they differ only in that classical physics purports to tell us how things will definitely be at a given moment, while quantum physics provides the probability that things will be one way or another.
When it comes to predicting where a batted ball will land, or how an electron will move through a computer chip (or a model Manhattan), this three-step process is demonstrably powerful. Yet, when it comes to describing the totality of reality, the three steps invite us to ask deeper questions: Can we explain the initial conditions—how things were at some purportedly earliest moment? Can we explain the values of the constants—the particle masses, force strengths, and so on—on which those laws depend? Can we explain why a particular set of mathematical equations describes one or another aspect of the physical universe?
The various multiverse proposals we’ve discussed have the potential to profoundly shift our thinking on these questions. In the Quilted Multiverse, the physical laws across the constituent universes are the same, but the particle arrangements differ; different particle arrangements now reflect different initial conditions in the past. In this multiverse, therefore, our perspective on the question of why the initial conditions in our universe were one way or another shifts. Initial conditions can and generally will vary from universe to universe, so there is no fundamental explanation for any particular arrangement. Asking for such an explanation is asking the wrong kind of question; it’s invoking single-universe mentality in a multiverse setting. Instead, the question we should ask is whether somewhere in the multiverse is a universe whose particle arrangement, and hence initial conditions, agrees with what we see here. Better still, can we show that such universes abound? If so, the deep question of initial conditions would be explained with a shrug of the shoulders; in such a multiverse, the initial conditions of our universe would be in no more need of an explanation than the fact that somewhere in New York is a shoe store that carries your size.
In the inflationary multiverse, the “constants” of nature can and generally will vary from bubble universe to bubble universe. Recall from Chapter 3 that environmental differences—the different Higgs field values permeating each bubble—give rise to different particle masses and force properties. The same holds true in the Brane Multiverse, the Cyclic Multiverse, and the Landscape Multiverse, where the form of string theory’s extra dimensions, together with various differences in fields and fluxes, result in universes with different features—from the electron’s mass to whether there even is an electron to the strength of electromagnetism to whether there is an electromagnetic force to the value of the cosmological constant, and so on. In the context of these multiverses, asking for an explanation of the particle and force properties we measure is once again asking the wrong kind of question; it’s a question borne of single-universe thinking. Instead, we should ask whether in any of these multiverses there’s a universe with the physical properties we measure. Better would be to show that universes with our physical features are abundant, or at least are abundant among all those universes that support life as we know it. But as much as it’s meaningless to ask for the word with which Shakespeare wrote Macbeth, so it’s meaningless to ask the equations to pick out the values of the particular physical features we see here.
The Simulated and Ultimate Multiverses are horses of a different color; they don’t emerge from particular physical theories. Yet, they too have the potential to shift the nature of our questions. In these multiverses, the mathematical laws governing the individual universes vary. Thus, much as with varying initial conditions and constants of nature, varying laws suggest that it’s as misguided to ask for an explanation of the particular laws in operation here. Different universes have different laws; we experience the ones we do because these are among the laws compatible with our existence.
Collectively, we see that the multiverse proposals summarized in Table 11.1 render prosaic three primary aspects of the standard scientific framework that in a single-universe setting are deeply mysterious. In various multiverses, the initial conditions, the constants of nature, and even the mathematical laws are no longer in need of explanation.
Should We Believe Mathematics?
Nobel laureate Steven Weinberg once wrote, “Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world.”1 Weinberg was referring to the pioneering results of Ralph Alpher, Robert Herman, and George Gamow on the cosmic microwave background radiation, which I described in Chapter 3. Although the predicted radiation is a direct consequence of general relativity combined with basic cosmological physics, it rose to prominence only after being discovered theoretically twice, a dozen years apart, and then being observed through a benevolent act of serendipity.
To be sure, Weinberg’s remark has to be applied with care. Although his desk has played host to an inordinate amount of mathematics that has proved relevant to the real world, far from every equation with which we theorists tinker rises to that level. In the absence of compelling experimental or observational results, deciding which mathematics should be taken seriously is as much art as it is science.
Indeed, this issue is central to all we’ve discussed in this book; it has also informed the book’s title. The breadth of multiverse proposals in Table 11.1 might suggest a panorama of hidden realities. But I’ve titled this book in the singular to reflect the unique and uniquely powerful theme that underlies them all: the capacity of mathematics to reveal secreted truths about the workings of the world. Centuries of discovery have made this abundantly evident; monumental upheavals in physics have emerged time and again from vigorously following mathematics’ lead. Einstein’s own complex dance with mathematics provides a revealing case study.
In the late 1800s when James Clerk Maxwell realized that light was an electromagnetic wave, his equations showed that light’s speed should be about 300,000 kilometers per second—close to the value experimenters had measured. A nagging loose end was that his equations left unanswered the question: 300,000 kilometers per second relative to what? Scientists pursued the makeshift resolution that an invisible substance permeating space, the “aether,” provided the unseen standard of rest. But in the early twentieth century, Einstein argued that scientists needed to take Maxwell’s equations more seriously. If Maxwell’s equations didn’t refer to a standard of rest, then there was no need for a standard of rest; light’s speed, Einstein forcefully declared, is 300,000 kilometers per second relative to anything. Although the details are of historical interest, I’m describing this episode for the larger point: everyone had access to Maxwell’s mathematics, but it took the genius of Einstein to embrace the mathematics fully. And with that move, Einstein broke through to the special theory of relativity, overturning centuries of thought regarding space, time, matter, and energy.
During the next decade, in the course of developing the general theory of relativity, Einstein became intimately familiar with vast areas of mathematics that most physicists of his day knew little or nothing about. As he groped toward general relativity’s final equations, Einstein displayed a master’s skill in molding these mathematical constructs with the firm hand of physical intuition. A few years later, when he received the good news that observations of the 1919 solar eclipse confirmed general relativity’s prediction that star light should travel along curved trajectories, Einstein confidently noted that had the results been different, “he would have been sorry for the dear Lord, since the theory is correct.” I’m sure that convincing data contravening general relativity would have changed Einstein’s tune, but the remark captures well how a set of mathematical equations, through their sleek internal logic, their intrinsic beauty, and their potential for wide-ranging applicability, can seemingly radiate reality.
Nevertheless, there was a limit to how far Einstein was willing to follow his own mathematics. Einstein did not take the general theory of relativity “seriously enough” to believe its prediction of black holes, or its prediction that the universe was expanding. As we’ve seen, others, including Friedmann, Lemaître, and Schwarzschild, embraced Einstein’s equations more fully than he, and their achievements have set the course of cosmological understanding for nearly a century. By contrast, during the last twenty or so years of his life, Einstein threw himself into mathematical investigations, passionately striving for the prized achievement of a unified theory of physics. In assessing this work based on what we know now, one can’t help but conclude that during those years Einstein was too heavily guided—some might say blinded—by the thicket of equations with which he was constantly surrounded. And so, even Einstein, at various times in his life, made the wrong decision regarding which equations to take seriously and which to not.
The third revolution in modern theoretical physics, quantum mechanics, provides another case study, one of direct relevance to the story I’ve told in this book. Schrödinger wrote down his equation for how quantum waves evolve in 1926. For decades, the equation was viewed as relevant only to the domain of small things: molecules, atoms, and particles. But in 1957, Hugh Everett echoed Einstein’s Maxwellian charge of a half century earlier: take the math seriously. Everett argued that Schrödinger’s equation should apply to everything because all things material, regardless of size, are made from molecules, atoms, and subatomic particles. And as we’ve seen, this led Everett to the Many Worlds approach to quantum mechanics and to the Quantum Multiverse. More than fifty years later, we still don’t know if Everett’s approach is right. But by taking the mathematics underlying quantum theory seriously—fully seriously—he may have discovered one of the most profound revelations of scientific exploration.
The other multiverse proposals similarly rely on a belief that mathematics is tightly stitched into the fabric of reality. The Ultimate Multiverse takes this perspective to its furthermost incarnation; mathematics, according to the Ultimate Multiverse, is reality. But even with their less panoptic view on the connection between mathematics and reality, the other multiverse theories in Table 11.1 owe their genesis to numbers and equations played with by theorists sitting at desks—and scribbling in notebooks, and writing on chalkboards, and programming computers. Whether invoking general relativity, quantum mechanics, string theory, or mathematical insight more broadly, the entries in Table 11.1 arise only because we assume that mathematical theorizing can guide us toward hidden truths. Only time will tell if this assumption takes the underlying mathematical theories too seriously, or perhaps not seriously enough.
If some or all of the mathematics that’s compelled us to think about parallel worlds proves relevant to reality, Einstein’s famous query, asking whether the universe has the properties it does simply because no other universe is possible, would have a definitive answer: no. Our universe is not the only one possible. Its properties could have been different. And in many of the multiverse proposals, the properties of the other member universes would be different. In turn, seeking a fundamental explanation for why certain things are the way they are would be pointless. Instead, statistical likelihood or plain happenstance would be firmly inserted in our understanding of a cosmos that would be profoundly vast.
I don’t know if this is how things will turn out. No one does. But it’s only through fearless engagement that we can learn our own limits. It’s only through the rational pursuit of theories, even those that whisk us into strange and unfamiliar domains, that we stand a chance of revealing the expanse of reality.
*Note, as in Chapter 7, that an airtight observational refutation of inflation would require the theory’s commitment to a procedure for comparing infinite classes of universes—something it has not yet achieved. However, most practitioners would agree that if, say, the microwave background data had looked different from Figure 3.4, their confidence in inflation would have plummeted, even though, according to the theory, there’s a bubble universe in the Inflationary Multiverse in which those data would hold.
Notes
Chapter 1: The Bounds of Reality
1. The possibility that our universe is a slab floating in a higher dimensional realm goes back to a paper by two renowned Russian physicists—“Do We Live Inside a Domain Wall?,” V. A. Rubakov and M. E. Shaposhnikov, Physics Letters B 125 (May 26, 1983): 136—and does not involve string theory. The version I’ll focus on in Chapter 5 emerges from advances in string theory in the mid-1990s.
Chapter 2: Endless Doppelgängers
1. The quote comes from the March 1933 issue of The Literary Digest. It is worth noting that the precision of this quote has recently been questioned by the Danish historian of science Helge Kragh (see his Cosmology and Controversy, Princeton: Princeton University Press, 1999), who suggests it may be a reinterpretation of a Newsweek report from earlier that year in which Einstein was referring to the origin of cosmic rays. What is certain, however, is that by this year Einstein had given up his belief that the universe was static and accepted the dynamic cosmology that emerged from his original equations of general relativity.
2. This law tells us the force of gravitational attraction, F, between two objects, given the masses, m1 and m2, of each, and the distance, r, betwe
en them. Mathematically, the law reads: F = Gm1m2/r2, where G stands for Newton’s constant—an experimentally measured number that specifies the intrinsic strength of the gravitational force.
3. For the mathematically inclined reader, Einstein’s equations are Ruv– ½ guvR = 8πGTuv where guv is the metric on spacetime, Ruv is the Ricci curvature tensor, R is the scalar curvature, G is Newton’s constant, and Tuv is the energy-momentum tensor.
4. In the decades since this famous confirmation of general relativity, questions have been raised regarding the reliability of the results. For distant starlight grazing the sun to be visible, the observations had to be carried out during a solar eclipse; unfortunately, bad weather made it a challenge to take clear photographs of the solar eclipse of 1919. The question is whether Eddington and his collaborators might have been biased by foreknowledge of the result they were seeking, and so when they culled photographs deemed unreliable because of weather interference, they eliminated a disproportionate number containing data that appeared not to fit Einstein’s theory. A recent and thorough study by Daniel Kennefick (see www.arxiv.org, paper arXiv:0709.0685, which, among other considerations, takes account of a modern reevaluation of the photograph plates taken in 1919) convincingly argues that the 1919 confirmation of general relativity is, indeed, reliable.