The God Equation

by Michio Kaku

  The Chinese have also expressed interest in building the Circular Electron Positron Collider. Work would begin around 2022, and it might be finished around 2030, at a cost of $5 to $6 billion. It would reach an energy of 240 billion electron volts and would be 100 kilometers around.

  Not to be outdone, the physicists at CERN are planning the successor to the LHC, called the Future Circular Collider (FCC). It would eventually reach an astounding 100 trillion electron volts. It would also be about 100 kilometers around.

  It is not clear if these accelerators will ever be built, but it does mean there is hope for finding dark matter in the next generation of accelerators beyond the LHC. If we discover particles of dark matter, they can then be compared against the predictions of string theory.

  Another prediction of string theory that might be verified by these accelerators is the presence of mini black holes. Since string theory is a theory of everything, it includes gravity as well as subatomic particles, so physicists expect to find tiny black holes in the accelerator. (These mini black holes, unlike stellar black holes, are harmless and have the energy of tiny subatomic particles, not the energy of dying stars. In fact, the Earth is bombarded by cosmic rays much more powerful than any that can be produced by these accelerators, without any harmful effects.)

  Big Bang as Atom Smasher

  There is also the hope that we can take advantage of the greatest atom smasher of all, the Big Bang itself. Radiation from the Big Bang may give us a clue to dark matter and dark energy. First of all, the echo, or afterglow, of the Big Bang is easy to detect. Our satellites have been able to detect this radiation to enormous accuracy.

  Photographs of this microwave background radiation show that it is remarkably smooth, with tiny ripples appearing on its surface. These ripples, in turn, represent tiny quantum fluctuations that existed at the instant of the Big Bang that were then magnified by the explosion.

  What is controversial, however, is that there appear to be irregularities, or blotches, in the background radiation that we cannot explain. There is some speculation that these strange blotches are the remnants of collisions with other universes. In particular, the CMB (cosmic microwave background) cold spot is an unusually cool mark on the otherwise uniform background radiation that some physicists have speculated might be the remnants of some type of connection or collision between our universe and a parallel universe at the beginning of time. If these strange markings represent our universe interacting with parallel universes, then the multiverse theory might become more plausible to skeptics.

  Already, there are plans to put detectors in space that can refine all these calculations, using space-based gravity wave detectors.

  LISA

  Back in 1916, Einstein showed that gravity could travel in waves. Like throwing a stone in a pond and witnessing the concentric, expanding rings it creates, Einstein predicted that swells of gravity would travel at the speed of light. Unfortunately, these would be so faint that he did not think we would find them anytime soon.

  He was right. It took until 2016, one hundred years after his original prediction, before gravity waves were observed. Signals from two black holes that collided in space about a billion years ago were captured by huge detectors. These detectors, built in Louisiana and Washington State, each occupy several square miles of real estate. They resemble a large L, with laser beams traveling down each leg of the L. When the two beams meet at the center, they create an interference pattern that is so sensitive to vibrations that they could detect this collision.

  For their pioneering work, three physicists, Rainer Weiss, Kip S. Thorne, and Barry C. Barish, won the Nobel Prize in 2017.

  For even greater sensitivity, there are plans to send gravity wave detectors into outer space. The project, known as the laser interferometry space antenna (LISA), might be able to pick up vibrations from the instant of the Big Bang itself. One version of the LISA consists of three separate satellites in space, each connected to the others by a network of laser beams. The triangle is about a million miles on each side. When a gravity wave from the Big Bang hits the detector, it causes the laser beams to jiggle a bit, which can then be measured by sensitive instruments.

  The ultimate goal is to record the shock waves from the Big Bang, and then run the videotape backward to get the best guess for the radiation before the Big Bang. These pre–Big Bang waves would then be compared to what’s predicted in several versions of string theory. In this way, one might be able to get numerical data about the multiverse before the Big Bang.

  Using devices more advanced than LISA, one might be able to get baby pictures of the universe. And perhaps even find evidence of the umbilical cord connecting our infant universe to a parent universe.

  Testing the Inverse Square Law

  Another frequent objection to string theory is that it postulates that we actually live in ten or eleven dimensions, for which there is no experimental evidence.

  But this aspect might actually be testable with off-the-shelf instruments. If our universe is three-dimensional, then the force of gravity diminishes as the square of the distance of separation. This famous law of Newton is what guides our space probes millions of miles in space with breathtaking precision, so we can shoot space probes right through the rings of Saturn if we felt like it. But Newton’s famous inverse square law has been tested only over astronomical distances, rarely in the laboratory. If the strength of gravity over small distances does not obey the inverse square law, it would signal the presence of a higher dimension. For example, if the universe had four spatial dimensions, then gravity should diminish as the cube of the distance of separation. (If the universe had N spatial dimensions, then gravity should diminish with the (N −1) power of the distance of separation.)

  But rarely has the force of gravity been measured between two objects in the laboratory. These experiments are difficult to do, since gravitational forces are quite small in the laboratory, but the first measurements have been done in Colorado, and the results were negative—that is, Newton’s inverse square law still holds. (But this means only that there are no added dimensions in Colorado.)

  Landscape Problem

  To a theoretician, all these criticisms are troublesome but not fatal. But what does cause problems for a theoretician is that the model seems to predict a multiverse of parallel universes, many of which are crazier than those in the imagination of a Hollywood scriptwriter. String theory has an infinite number of solutions, each describing a perfectly well-behaved finite theory of gravity, which do not resemble our universe at all. In many of these parallel universes, the proton is not stable, so it would decay into a vast cloud of electrons and neutrinos. In these universes, complex matter as we know it (atoms and molecules) cannot exist. They only consist of a gas of subatomic particles. (Some might argue that these alternate universes are only mathematical possibilities and are not real. But the problem is that the theory lacks predictive power, since it cannot tell you which of these alternate universes is the real one.)

  This problem is actually not unique to string theory. For example, how many solutions are there to Newton’s or Maxwell’s equations? There are an infinite number, depending on what you are studying. If you start with a light bulb or a laser and you solve Maxwell’s equations, you find a unique solution for each instrument. So Maxwell’s or Newton’s theories also have an infinite number of solutions, depending on the initial conditions—that is, the situation you start with.

  This problem is likely to exist for any theory of everything. Any theory of everything will have an infinite number of solutions depending on the initial conditions. But how do you determine the initial conditions of the entire universe? This means you have to input the conditions of the Big Bang from the outside, by hand.

  To many physicists this seems like cheating. Ideally, you want the theory itself to tell you the conditions that gave rise to the Big
Bang. You want the theory to tell you everything, including the temperature, density, and composition of the original Big Bang. A theory of everything should somehow contain its own initial conditions, all by itself.

  In other words, you want a unique prediction for the beginning of the universe. So string theory has an embarrassment of riches. Can it predict our universe? Yes. That is a sensational claim, the goal of physicists for almost a century. But can it predict just one universe? Probably not. This is called the landscape problem.

  There are several possible solutions to this problem, none of them widely accepted. The first is the anthropic principle, which says that our universe is special because we, as conscious beings, are here to discuss this question in the first place. In other words, there might be an infinite number of universes, but our universe is the one that has the conditions that make intelligent life possible. The initial conditions of the Big Bang are fixed at the beginning of time so that intelligent life can exist today. The other universes might have no conscious life in them.

  I clearly remember my first introduction to this concept when I was in the second grade. I remember my teacher said that God so loved the Earth that he put the Earth “just right” from the sun. Not too close, or the oceans would boil. Not too far, or the oceans would freeze. Even as a child, I was stunned by this argument, because it used pure logic to determine the nature of the universe. But today, satellites have revealed four thousand planets orbiting other stars. Sadly, most of them are too close or too far from their star to support life. So there are two ways one can analyze my second-grade teacher’s argument. Perhaps there is a loving God after all, or perhaps there are thousands of dead planets that are too close or too far, and we are on a planet that is just right for sustaining intelligent life that hence can debate this question. Similarly, we may coexist in an ocean of dead universes, and our universe is special only because we are here to discuss this question.

  The anthropic principle actually allows one to explain a curious experimental fact about our universe: that the basic constants of nature seem to be fine-tuned to allow for life. As physicist Freeman Dyson has written, it seems as if the universe knew that we were coming. For example, if the nuclear force were a bit weaker, the sun would never have ignited, and the solar system would be dark. If the strong nuclear force were a bit stronger, then the sun would have burned out billions of years ago. So the nuclear force is tuned just right.

  Similarly, if gravity were a bit weaker, perhaps the Big Bang would have ended in a Big Freeze, with a dead, cold expanding universe. If gravity were a bit stronger, we might have ended in a Big Crunch, and all life would have been burned to death. Yet our gravity is just right to allow for stars and planets to form and last long enough for life to spring up.

  One can list a number of these accidents that make life possible, and each time we are in the middle of the Goldilocks zone. So the universe is one gigantic crapshoot, and we won the roll. But according to the multiverse theory, it means we coexist with a vast number of dead universes.

  So perhaps the anthropic principle can pick our universe from the millions of universes in the landscape, because we have conscious life in our universe.

  My Own Point of View on String Theory

  I have been working on string theory since 1968, so I have my own definite viewpoint. However you look at it, the final form of the theory has yet to be revealed. So it is premature to compare string theory to the present universe.

  One feature of string theory is that it is evolving backward, revealing new mathematics and concepts along the way. Every decade or so, there is a new revelation in string theory that changes our point of view concerning its nature. I have witnessed about three such astonishing revolutions, yet we have yet to express string theory in its complete form. We do not yet know its final fundamental principles. Only then can we compare it with experiment.

  Revealing a Pyramid

  I like to compare it to searching for treasure in the Egyptian desert. Let’s say one day you stumble on a tiny rock sticking up in the desert. After brushing away the sand, you begin to realize that this pebble is actually the top of a gigantic pyramid. After years of excavation, you find all sorts of strange chambers and artwork. In each floor, you find new surprises. Finally, after excavating many floors, you reach the final door, and are about to open it to find out who made the pyramid.

  Personally, I believe we are still not at the bottom floor, since we keep discovering new mathematical layers every time we analyze the theory. There are still more layers to reveal before we find string theory in its final form. In other words, the theory is smarter than we are.

  It is possible to express all of string theory in terms of string field theory in an equation about one inch long. But we need five such equations in ten dimensions.

  Although we can express string theory in field theory form, this is still not possible for M-theory. The hope is that one day physicists may find a single equation that summarizes all of M-theory. Unfortunately, it is notoriously difficult to express a membrane (which can vibrate in so many ways) in field theory form. As a consequence, M-theory consists of scores of disjointed equations that miraculously describe the same theory. If we can write M-theory in field theory form, then the entire theory should emerge from a single equation.

  No one can predict if or when this will happen. But after witnessing the hype around string theory, the public has grown impatient.

  But even among string theorists, there is a certain amount of pessimism about the future prospects of the theory. As Nobel laureate David Gross has mentioned, string theory is like the top of a mountain. As climbers scale the mountain, the top is clearly visible, but it seems to recede the closer you come to it. The goal is tantalizingly close, but seems to be always just out of reach.

  Personally, I think this is understandable, since no one knows when, if ever, we will find supersymmetry in the laboratory, but this has to be put into proper perspective. The correctness or incorrectness of a theory should rest on concrete results, not the subjective desires of physicists. We all hope that our pet theories are confirmed within our lifetime. That is a deeply human desire. But sometimes nature has her own timetable.

  The atomic theory, for example, took two thousand years before it was finally vindicated, and only recently have scientists been able to take vivid images of individual atoms. Even Newton’s and Einstein’s great theories took decades for many of their predictions to be fully tested and verified. Black holes were first predicted in 1783 by John Michell, but only in 2019 did astronomers produce the first conclusive pictures of their event horizon.

  Personally, I think the pessimism of many scientists might be misguided, because the evidence for the theory might be found not in some gigantic particle accelerator but when someone finds the final mathematical formulation of the theory.

  The point here is that perhaps we do not need an experimental proof of string theory at all. A theory of everything is also a theory of ordinary things. If we can derive the mass of the quarks and other known subatomic particles from first principles, that might be convincing evidence that this is the final theory.

  The problem is not experimental at all. The Standard Model has twenty or so free parameters that are put in by hand (such as the mass of the quarks and the strength of their interactions). We have plenty of experimental data concerning the masses and couplings of subatomic particles. If string theory can precisely calculate these fundamental constants from first principles, without any assumptions, then this would, in my opinion, prove its correctness. It would be a truly historic event if the known parameters of the universe could emerge from a single equation.

  But once we have this one-inch-long equation, what do we do with it? How can we escape the landscape problem?

  One possibility is that many of these universes are unstable and decay to our familiar
universe. We recall that the vacuum, instead of being a boring, featureless thing, is actually teeming with bubble universes popping in and out of existence, like in a bubble bath. Hawking called this the space-time foam. Most of these tiny bubble universes are unstable, jumping out of the vacuum and then jumping back in.

  In the same way, once the final formulation of the theory is found, one might be able to show that most of these alternate universes are unstable and decay down to our universe. For example, the natural time scale for these bubble universes is the Planck time, which is 10−43 seconds, an incredibly short amount of time. Most universes only live for this brief instant. Yet the age of our universe, by comparison, is 13.8 billion years, which is astronomically longer than the lifespan of most universes in this formulation. In other words, perhaps our universe is special among the infinity of universes in the landscape. Ours has outlasted them all, and that is why we are here today to discuss this question.

  But what do we do if the final equation turns out to be so complex that it cannot be solved by hand? Then it seems impossible to show that our universe is special among the universes in the landscape. At that point I think we should put it in a computer. This is the path taken for the quark theory. We recall that the Yang-Mills particle acts like a glue to bind quarks into a proton. But after fifty years, no one has been able to rigorously prove this mathematically. In fact, many physicists have pretty much given up hope of ever accomplishing it. Instead, the Yang-Mills equations are solved on a computer.