PDF Summary:The God Equation, by Michio Kaku

The science of physics studies many topics, from the gravity holding planets together to electromagnetism and the microscopic forces inside the heart of the atom. Each of these forces works by different rules, some of which contradict each other. For centuries, scientists have sought a single theory that would unify these disparate fields and provide one equation from which we can derive a complete understanding of the universe. In The God Equation, theoretical physicist Michio Kaku suggests that an approach called “string theory” may be the path to uniting all the different laws of nature.

In this guide, we’ll trace the history of discoveries that unify areas of knowledge and reveal the underlying symmetries of nature. We’ll delve into string theory and define terms and concepts to give readers an understanding of the topic. We’ll discuss research to support string theory’s claims, and we’ll touch on the philosophical implications that a theory of the universe entails.

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Kaku writes that Planck found a solution that was just as shocking to the norms of science as Einstein’s theory of relativity. In essence, Planck determined that the energy released by matter as it cools doesn’t diminish on a linear scale but as tiny bundles called “quanta.” It’s as if when an atom cools, its energy level doesn’t slide down a ramp but instead walks down a flight of stairs where each step is a specific amount of energy. This energy emerges as packets of light—photons—that exactly match the energy of light given off by heated objects.

The Photoelectric Effect

Thanks to symmetry, this process that Kaku describes also works in reverse. Just as electrons emit photons of light as they cool to lower energy levels, if you bombard a substance with light, it imbues electrons with extra energy, breaking some loose in what’s known as the photoelectric effect. This effect doesn’t depend on the intensity of the light, but rather on the frequency of the light waves. This doesn’t make sense under the Newtonian model of physics, which dictates that it’s the amount of energy that matters.

However, in 1905, while still working on his theory of relativity, Einstein used Planck’s equations to explain the photoelectric effect. He showed that when the energy of photons at specific light frequencies matched electrons’ exact quantum energy levels, that alone was sufficient to knock the electrons into higher quantum energy states and even break them free from their atoms. It was for his explanation of the photoelectric effect that Einstein won the Nobel Prize in Physics, not for his work on relativity.

Particles and Waves

The problem with photons being particles of light is that Maxwell’s equations show that light is an electromagnetic wave. It turns out that light has properties of both waves and particles, depending on how it interacts with matter. But Kaku writes that the mystery grew deeper because researchers found that the same was true of electrons. Instead of being envisioned as a particle in orbit around the nucleus, an electron is better described as a wave that wraps the nucleus in a shell. That wave can only exist at specific energy levels that match the quantum orbits in which electrons are found.

(Shortform note: This wavelike twist on our model of the atom is referred to as the electron cloud. While the earlier “solar system” model of the atom presumed that electrons remained at fixed distances from the nucleus, the electron cloud model suggests that electrons can be anywhere around the nucleus—it’s simply much more likely that they’ll be located within specific quantum shells.)

So when we say that electrons are waves, it simply means that you can’t pin one down—you can only determine the probability of its location at any given point, and that probability flows like a wave. This goes against all our prior conceptions of motion. Imagine if you can’t tell where the moon is in the sky—you can only compute where it’s likely to be. This makes no sense in our macroscopic world governed by laws of relativity, but it’s exactly how electrons and photons behave, as well as other particles that we’ll cover later. Clearly, says Kaku, if we’re to uncover one universal theory, it will have to reconcile quantum mechanics with Einstein’s special relativity.

(Shortform note: While Kaku’s focus is on the subatomic world, there’s a large-scale example of a phenomenon in which we can only make probabilistic predictions—the weather. Meteorology relies on chaos theory, a branch of math that calculates probabilities in systems that have so many uncertain variables that completely accurate predictions are impossible. In the world of subatomic particles, it appears that a degree of uncertainty is baked into the very fabric of the universe—for any phenomenon that behaves like a wave, it’s impossible to measure its exact speed and direction at any given moment in time.)

The Heart of the Atom

The quest to unify relativity and quantum theory is ongoing. In the meantime, scientists pushed quantum mechanics further so they could unravel the secrets of the atom. To enter the subatomic realm, we’ll explain researchers’ methods to pry open the atom’s nucleus, the discovery of even smaller subatomic particles, and how combining the mathematical models of the forces which interact inside the atom leads to a theory of how all matter functions.

The only way to look inside atoms is to break them apart. Using accelerators, researchers smash atomic nuclei together and study the pieces found in the wreckage. When this was first done, physicists were surprised to find that protons and neutrons are composed of even smaller particles known as quarks. Studies revealed different types of quarks, and that protons, neutrons, and even stranger particles are made from arrangements of groups of three quarks each. Kaku says the equations governing quarks are symmetrical because you can move around any of the three quark components and the particle they create stays the same.

(Shortform note: In addition to the symmetry that Kaku calls attention to, quarks come in six different types, grouped into three symmetrical pairs—“up” and “down” quarks, “top” and “bottom” quarks, “charm” and “strange” quarks—though only up and down quarks make up protons and neutrons. The other quarks are building blocks of strange, exotic particles that can only exist under special conditions, such as the short-lived J/psi meson that proved the existence of charm quarks. Another example is the Lambda baryon—a neutron-like particle containing strange quarks that may be prevalent in superdense neutron stars.)

What holds quarks together into protons and neutrons, and holds neutrons and protons together in the atom, is the most powerful force known in nature—the strong nuclear force. This force is an extremely short-range field of attraction generated by a particle called a gluon (because it glues subatomic particles together). This field can be described mathematically using a modified version of Maxwell’s equations that covers fields of attraction and repulsion beyond that of electromagnetism. Kaku writes that equations such as these that explain different forces are key steps along the path to a unified theory of physics.

(Shortform note: Though Kaku doesn’t pose it this way, at the subatomic level of quantum mechanics, the forces of nature are actually the result of energy being transferred from one particle to another via smaller particles that act as couriers. The field of the strong force binding the nucleus is the result of gluons zipping back and forth between protons and neutrons, exchanging energy, just as the electromagnetic force binding electrons to their atomic shells is transmitted by the interchange of photons. In this sense, the forces are better described as “interactions” between particles.)

The Weak Interaction

In addition to electromagnetism, which keeps electrons in orbit around the nucleus, and the strong nuclear force, which holds the nucleus together, there is also the “weak interaction,” also known as the weak nuclear force. Whereas the strong force holds atoms together, the weak nuclear force breaks particles apart and can even change them from one form to another.

(Shortform note: Though stronger than gravity, the weak force is less powerful than either the strong force or electromagnetism, and it only operates on the tiniest scale—inside protons and neutrons. The action of the weak force is to transform quarks from one type to another—for example, from top to bottom, or vice versa. This change in state for one quark in a neutron can change the neutron itself into a proton, altering the atomic nucleus of which it is a part. The example of the weak interaction Kaku points to is a form of radiation known as beta decay.)

In beta decay, the weak force breaks a neutron apart into a proton, an electron, and the electron’s phantom partner, the neutrino. Neutrinos have no electric charge and virtually no mass. As such, a neutrino can pass through stars and planets without stopping. Nevertheless, Kaku says the neutrino fills a key gap in the atomic model that was predicted by symmetry—if the proton has an electrically neutral counterpart (the neutron) shouldn’t the electron have one too? Their existence was purely theoretical until they were observed in 1956.

(Shortform note: Kaku doesn’t explain how scientists were able to capture neutrinos if they’re so elusive. In the 1950s, researchers devised an experiment by placing light-emitting material inside a giant water tank positioned near a nuclear reactor. The nuclear plant would theoretically produce trillions of neutrinos per second, an infinitesimally small fraction of which would interact with the matter in the tank, making it glow. In order to detect the light from neutrinos, the tank had to be shielded from every other form of radiation, including cosmic rays from outer space. Even so, it took more than five months of observations to confirm that neutrinos actually exist.)

Once neutrinos were confirmed to exist, they gave insights into the weak nuclear force that let physicists unite it with the electromagnetic force. Today it’s understood that electromagnetism and the weak nuclear force are one and the same, now referred to as the electroweak force. Kaku affirms that this is the path that the search for a unifying theory has taken—merging separate theories about phenomena until there’s only one left.

(Shortform note: Merging the weak and electromagnetic forces isn’t something that’s easily achieved. While Kaku hints at how it’s done mathematically, in practice the two forces only act the same at temperatures that haven't existed since the first picosecond after the Big Bang. The first experimental evidence for the existence of the electroweak force was discovered in 1973 at the European Council for Nuclear Research (CERN) neutrino detector in Switzerland, followed by the discovery of the particles that transmit the electroweak force in 1983 using CERN’s Super Proton Synchrotron.)

The Standard Model

Though scientists have yet to figure out the unifying theory—Kaku’s “God Equation”—physicists have made a great deal of progress using an intermediate step toward that theory referred to as the Standard Model. In this section, we’ll go over the basis of the Standard Model as well as its theoretical problems, including cumbersome mathematics, a mismatch between the theory and the observable universe, and its current inability to account for the force of gravity.

Beginning in the 1970s, physicists stitched together the equations for the strong nuclear force, the weak nuclear force, and electromagnetism to produce what’s now called the Standard Model. This mathematical expression of those fundamental forces presumes that in the extreme conditions that existed moments after the Big Bang, all of the forces acted as one and only diverged as the universe cooled.

(Shortform note: In Astrophysics for People in a Hurry, Neil deGrasse Tyson goes into more detail about the universe’s first moments and their importance to a unified theory of physics. Tyson explains that at the instant of the universe’s inception, the domains of relativity and quantum mechanics overlapped. If scientists can determine how matter, energy, time, and space behaved during the first nanoseconds of existence, they could potentially resolve the discrepancies between relativity and the Standard Model. This research is being done at the Large Hadron Collider in Switzerland, which smashes particles together at extremely high energies in an attempt to replicate the conditions that existed in the universe’s first moments.)

However, the Standard Model is woefully incomplete. One issue that Kaku is quick to point out is that the Standard Model is an unsightly mess—a chimeric creation of theories stitched together in a way that isn’t completely plausible and contains mathematical terms that aren’t greatly understood. On the plus side, the equations for the Standard Model are symmetrical, but Kaku says that in this case, that’s a problem. The universe around us isn’t symmetrical—matter is clumped into stars and galaxies with vast expanses of empty space between them. If the Standard Model were true, the universe would instead be a perfectly uniform sea of particles. Obviously, something is amiss.

For the Standard Model to produce the asymmetrical universe of today, something must have shattered the initial symmetry shortly after the Big Bang occurred. Researchers proposed an undiscovered “Higgs boson” consistent with the Standard Model that unbalanced the cooling universe’s symmetry and led to the variety of particles known today. Sometimes called the “God particle,” the Higgs boson created regions of space within which other particles began to have mass. Kaku writes that because of this imbalance, the original ruling force divided into the strong and weak forces, electromagnetism, and gravity. Confirming the existence of the Higgs boson was one of the missions of the Large Hadron Collider, which it accomplished in 2012.

The Higgs Boson

One point that Kaku doesn’t state explicitly is that the Standard Model’s equations are only symmetrical if you presume that subatomic particles have no mass. Once mass is introduced into the equations, their inherent symmetry falls apart and they become enormously complex and contradictory. In 1964, Peter Higgs proposed a solution—an invisible field permeating space that pushes back on particles as they accelerate, imbuing them with what we perceive as mass. Higgs’s suggestion was initially rejected, though other physicists eventually warmed to the idea. If mass is the product of an external “Higgs field,” then it doesn’t have to be accounted for in the Standard Model’s particle equations.

However, since the Standard Model explains fields as being produced by carrier particles, such as the photon and the gluon, the Higgs field must have a particle too—the Higgs boson. The Large Hadron Collider (LHC) detected the Higgs boson indirectly by smashing other particles together and interpreting the data from their collisions. As subatomic particles break apart, their components decay into a variety of components, and out of every billion LHC collisions, the collider produced a particle that matched predictions of the Higgs boson’s properties.

The Gravity Problem

Despite the work done to confirm the Standard Model, it only applies in the realm of quantum mechanics. To achieve a theory unifying all of physics, gravity will have to be folded into the equations for all the other forces. Here, we’ll explain the struggles involved with incorporating gravity into the Standard Model and several discoveries that have been made along the way.

The trouble with gravity is that it’s by far the weakest of the universal forces, so it’s virtually impossible to observe on the quantum scale. While Einstein defines gravity as a curvature in space, there’s no way to express this in the mathematics of quantum mechanics. Instead, physicists have tried another tack—to model gravity as a force created by a particle. Just as gluons transmit the strong nuclear force, scientists have proposed the existence of gravitons that govern gravitational fields. While plausible, Kaku writes that so far no one’s been able to create a mathematical model of gravitons that produces meaningful results.

(Shortform note: Despite Kaku’s admission that gravitons lack a firm mathematical model, researchers are already thinking about ways that they might be detected experimentally. One approach takes advantage of the idea that force-bearing particles like gravitons and photons can, on rare occasions, convert back and forth. By observing a space where graviton concentration would be highest —such as between colliding black holes—astronomers might witness flashes of light as handfuls of gravitons transform into photons. Another approach takes advantage of the Casimir effect, in which metallic plates are attracted to each other in a vacuum if they’re placed near enough to experience the effect of quantum forces.)

Despite the problems of incorporating gravity into quantum mechanics, that hasn’t stopped people from trying. Famed physicist Stephen Hawking has done so by considering what happens in the subatomic world in the one place where gravity can’t be escaped—the event horizon of a black hole, or the dividing line around a black hole’s zone of no return. Black holes are objects so massive and dense that their gravity stops even light from escaping. Kaku says that in this realm of gravitational extremes, Hawking discovered that quantum forces slowly drain energy and mass from a black hole's gravitational field.

This takes place because, in the subatomic world, empty space isn’t really empty. Kaku points out that in quantum mechanics, empty space is actually a bubbling sea of symmetrical positive and negative particles that pop into being and cancel each other. Normally, these particles vanish before they affect their surroundings, but near a black hole, half of the pair may be sucked in while the other flies free. Since energy can’t be created or destroyed, the energy to create this particle must be subtracted from the black hole, reducing its gravitational pull. In other words, near a black hole, gravity and the quantum world intersect.

Simulating Black Holes in the Lab

Dubbed “Hawking radiation,” the quantum energy drain from a black hole that Kaku describes might be nearly impossible to detect. Hawking’s black hole equations show that in addition to being unbelievably massive, black holes are incredibly cold and grow more so as they increase in mass. Black holes “hot” enough to give off perceptible Hawking radiation would have to be smaller than any known to exist.

However, scientists have found another way. In 2021, researchers used a Bose-Einstein condensate—a state of matter that only exists when particles are cooled almost to absolute zero—as a way to simulate conditions near a black hole. They detected Hawking radiation from the quantum condensate, which they were able to compare to various models of how Hawking radiation should emerge from black holes.

Back to the Beginning

Another point in space and time where quantum mechanics and gravity overlap is in the moment of the Big Bang itself. Kaku writes that because of this, physicists are determined to study and model the first fractions of a second of the universe’s existence. In the first instants of time, all the fundamental forces of nature interacted with each other before branching off into their own domains. We study the conditions in those first moments by recreating them in supercolliders and by looking into the farthest depths of space, which—because light takes time to travel—also entails looking backward in time.

Indeed, Kaku affirms that there are several observations about the Big Bang that can only be explained using quantum mechanics. By treating the Big Bang as a source of quantum radiation, researchers were able to compute the temperature of the Big Bang’s leftover heat, which was confirmed when the Cosmic Microwave Background (CMB) was discovered in 1964. Another issue is the universe’s rate of expansion, as measured by the stretched-out light waves detected from distant galaxies, which is much faster than Einstein’s equations predict. Kaku argues that quantum theory accounts for this discrepancy by showing that the universe expanded more rapidly in its early stages than was previously thought.

Physics of the Young Universe

The early, rapid expansion that Kaku mentions is referred to as the Inflationary Universe Theory. This theory posits that when the fundamental forces began to diverge in the first few seconds after the Big Bang, they unleashed a surge of energy that made the universe expand even more quickly than before. It’s unclear if this is related to the so-called dark energy that’s currently accelerating the universe’s expansion. Regardless, this early expansion took place 400,000 years before the earliest image of the universe we have—the aforementioned Cosmic Microwave Background.

In Astrophysics for People in a Hurry, Neil deGrasse Tyson explains the CMB as a “snapshot” of the early universe at a time when it had cooled enough for photons to pass unimpeded through space without colliding with hot gas particles. Tyson calls attention to the fact that the map of the CMB is lumpy—an uneven mix of hot and cool zones that backs up Kaku’s statement that something unbalanced any symmetry the early universe had.

The Promise of String Theory

Though physicists have yet to reconcile relativity, gravity, and quantum mechanics, Kaku argues we may already know what form the eventual solution will take. The answer comes from the field of string theory, which Kaku himself has been working to advance for all of his professional career. In this section, we’ll explain what string theory is, how it suggests higher orders of symmetry in the universe, and why it appears that the universe we see is merely a projection of something much greater.

The basic assertion of string theory is that all particles can be described as vibrations on subatomic strings that constitute the basis of energy and matter. Kaku writes that each different type of particle is produced by a different vibration. One of string theory’s strong appeals is that its equations produce gravitons as being the lowest string vibration, removing the conflict between Einstein and quantum mechanics. String theory also supports physicists’ long-held belief that in the beginning, there was one unifying force in the cosmos by showing that the particles transmitting those forces are symmetrically interchangeable since they’re all just different frequencies of the same foundational strings.

(Shortform note: A basic layman’s question that Kaku doesn’t address is that if everything is made of vibrating strings, then what are those strings composed of? The answer physicists have to fall back on is that they’re not made of anything smaller—strings are the foundational building blocks of everything. In the old Standard Model, the universe’s fundamental units were a bestiary of zero-dimensional point particles (that also act as waves) interacting in fields of quantum uncertainty. String theory simplifies the Standard Model’s complexity by replacing those points with one-dimensional strings and explaining each different particle interaction as nothing more than different types of deformations on a string.)

Another argument in favor of string theory is that it introduces a new symmetry not present in previous theories. Kaku says that in the Standard Model, there are two families of subatomic particles—bosons representing the fundamental forces, such as gluons, photons, and gravitons; and fermions comprising the building blocks of matter, such as protons, neutrons, and electrons. In the math for string theory, bosons and fermions are interchangeable, just like time and space are in relativity, as well as electricity and magnetism in Maxwell’s equations. This feature of string theory, called supersymmetry, unites every observable property of space, time, energy, and matter into the interactions of vibrating strings.

(Shortform note: The concept of supersymmetry was first introduced in 1974 by Mohammad Abdus Salam and John Strathdee in their work to define the electroweak force. They also introduced the concept of superspace, in which the x-y-z coordinate system to determine the location of any particle includes extra spatial dimensions where the coordinates are complex Grassman numbers. These are derived from a form of algebra invented by Hermann Grassmann, a 19th-century Prussian schoolteacher who had no idea that his work would be important to the realm of quantum physics 100 years later.)

String theory equations have bizarre implications, the most fundamental of which, according to Kaku, is that string theory only works if the universe has ten dimensions instead of the four that we’re aware of (three dimensions of space and one dimension of time). One hypothesis scientists use to explain this is to suggest that the other six dimensions we can’t see are so small that they’re imperceptible even on a subatomic scale. Another idea that Kaku puts forward is that our four-dimensional universe is a holographic image of the actual 10-D universe around us, just as an optical hologram is a 3-D image imprinted on a 2-D surface.

(Shortform note: Despite the impossibility of directly conceptualizing these extra, compacted dimensions Kaku describes, researchers are busy conceiving of ways their presence might be detected indirectly. One avenue involves supposing that the other dimensions aren’t as small as theorized. If so, then the three non-gravitational forces might be combined at much lower energies than previously imagined. Other possibilities are to observe particles disappearing into these dimensions or to watch for the effects of particles that can only exist in those dimensions.)

The trouble with these other six dimensions is that we don’t know what they represent, but their existence is required by string theory’s equations. For this reason among others we’ll discuss, many scientists reject string theory as a piece of mathematical sleight-of-hand.

(Shortform note: Even before the time of Kaku’s writing, the supersymmetry component of string theory was meeting difficulties on the experimental front. Supersymmetry predicts a wide array of undiscovered subatomic particles in addition to those already known, and yet the Large Hadron Collider (LHC) hasn’t produced any trace of these expected new discoveries. While many theorists argue that the LHC may simply not be powerful enough to produce these particles, or that our detection methods are flawed, many others view the LHC’s lack of results in this area as evidence that string theory, while compelling, may nonetheless be wrong.)

The String Theory Debate

Despite the encouraging direction string theory provides to theoretical physics, it hasn’t met with universal acceptance within the scientific community. The objections to string theory revolve around the fact that a scientific hypothesis has to be tested for it to be considered a viable theory, and so far string theory has failed to suggest even a method to test it. We’ll conclude by addressing these concerns and offering several potential avenues by which experimental data might confirm string theory’s claims.

Kaku says the most direct test of string theory would be to produce and observe a graviton to see if it conforms to string theory’s predictions. However, to observe a graviton with our current methods would require a particle accelerator the size of our whole solar system. Another facet of string theory that gives physicists pause is that string theory’s equations don’t just allow for a single universe, but a potentially infinite multiverse. To this, Kaku replies that we don’t even have a final version of string theory’s equations as of yet. Perhaps there really is a multiverse out there, or maybe whatever the true God Equation is, there is only one viable solution, with the others being too unstable to exist.

(Shortform note: The idea of a multiverse has become a staple of popular culture, in particular the Many Worlds Interpretation introduced by Hugh Everett in 1957. This approach to the quantum uncertainty problem suggests that every decision or random occurrence with more than one possible outcome produces one or more branching universes in which those outcomes took place. A different interpretation of the multiverse, the one that Kaku leans toward, suggests that other universes may exist in parallel, and completely separate from, our own. These other universes may have different laws of nature and particle interactions, such that most might be inimical to the development of life forms such as us.)

To the charge that we can’t directly test string theory, Kaku points out that science often tests its theories indirectly. One avenue might be through astrophysicists’ search for the unseen dark matter that permeates the universe. String theory predicts several options for what dark matter might actually be, such as microscopic black holes or particles called photinos. If any of these are eventually found, it would strengthen string theory’s overall standing. Another option comes from the recent discovery of gravity waves, as predicted by Einstein. Kaku writes that if we can detect gravity waves generated by the Big Bang itself, they may confirm or deny string theory’s predictions about the nature of the universe’s origin.

(Shortform note: Earlier in this guide, we discussed the progress already being made to detect gravity waves. Kaku’s other two options may prove more difficult. Attempts to detect photinos in the 1980s and ’90s produced tantalizing hints but no actual results. A search for microscopic black holes in 2010 using the Large Hadron Collider likewise provided no evidence for their existence. A new avenue of approach that’s opened up is to use string theory to make predictions about the universe’s early inflation. If string theory suggests a solution to the question of what took place on the quantum level during that early epoch, then it may make predictions that can be confirmed via astronomy rather than looking at the subatomic scale.)

If we do find that string theory is correct, will it make a difference in our everyday lives? Kaku admits that it probably won’t—the energies involved in making use of string theory are orders of magnitude beyond what we possess. Its biggest impact might be felt in the circles of religion and philosophy. If one equation explains the entirety of the cosmos, it may change how we conceive of God and our place in the universe. Though Kaku describes himself as agnostic, he finds it amazing that all the laws of physics can be summarized succinctly and display such symmetry. For him, the mathematical beauty of the laws of physics opens the door to the possibility that there might actually be a grand design.

(Shortform note: While Kaku takes a cautious approach to linking string theory with the existence of God, others have openly embraced the idea. In Life After Death, Dinesh D’Souza argues that the multiverse interpretation of string theory opens the door not only to God but to the existence of heaven and hell beyond the bounds of our own universe. However, in The Trouble With Physics, Lee Smolin argues that without experimental evidence, string theory has risen to the level of dogma, with many conjectures that must be accepted on faith. In other words, string theory may have stepped beyond science and become a quasi-religion unto itself.)