Our Mathematical Universe: My Quest for the Ultimate Nature of Reality

by Max Tegmark

Physicists have known for a century that solid steel is really mostly empty space, because the atomic nuclei that make up 99.95% of the mass are tiny balls that fill up merely 0.0000000000001% of the volume, and that this near-vacuum only feels solid because the electrical forces that hold these nuclei in place are very strong.

Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic orbits of flying rocks

Darwin’s theory thus makes the testable prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down.

“If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words?”

physics is the ultimate intellectual adventure, the quest to understand the deepest mysteries of our Universe.

There’s no better guarantee of failure than convincing yourself that success is impossible, and therefore never even trying.

Amusingly, Christopher Columbus totally bungled this by relying on subsequent less-accurate calculations and confusing Arabic miles with Italian miles, concluding that he needed to sail only 3,700 km to reach the Orient when the true value was 19,600 km. He clearly wouldn’t have gotten his trip funded if he’d done his math right, and he clearly wouldn’t have survived if America hadn’t existed, so sometimes being lucky is more important than being right.

As the gas cloud gradually contracts, any slight rotation of the cloud gets amplified, just as a figure skater spins faster when she pulls her arms closer to her body. The centrifugal forces from this ever-faster rotation prevents gravity from crushing the gas cloud down to a point—instead, it’s crushed into a pizza shape, just as when the pizza baker near my old elementary school spun his dough to flatten it out. The main ingredients of all such cosmic pizzas are hydrogen and helium gas, but if the ingredient list also contains heavier atoms such as carbon, oxygen, and silicon, then while the center of this gas pizza forms a star, the outer parts may clump into other colder objects, planets, which become revealed once the newborn star blows away the rest of the pizza dough.

Wouldn't This mean that depending on the distance from the sun the planets should made off of heavier elements primarily? As if the pizza was a giant centrifuge, splitting the components as it spins. However, that is not the case

The most common isotope of uranium atoms spontaneously decays into thorium and other lighter atoms at such a rate that half of the atoms have fallen apart after 4.47 billion years. Such radioactive decays generate enough heat to keep Earth’s core nice and toasty for billions of years, explaining why Earth is so warm even if it’s way older than 20 million years. Moreover, by measuring what fraction of the uranium atoms in a rock have decayed, you can determine the age of the rock, and in this way, some rocks from the Jack Hills of western Australia have been found to be over 4.404 billion years old. The record age for meteorites is 4.56 billion years, suggesting that both our planet and the rest of our Solar System formed in the ballpark of 4.5 billion years ago—in good agreement with those much cruder estimates from tides.

Basically, the same battle between gravity and pressure that formed our pizza-shaped Solar System repeats itself on a vastly larger scale, compressing a much larger region of gas into a pizza shape millions to trillions of times heavier than the Sun. This collapse turns out to be quite unstable, so it doesn’t lead to a solar system on steroids with a single mega-star surrounded by mega-planets. Instead, it fragments into countless smaller gas clouds that form separate solar systems: thus, a galaxy has been born. Our Solar System is one of hundreds of billions in one of these pizza-shaped galaxies, the Milky Way, and we orbit around it once every couple of hundred million years,

But this woud imply a lot of interstellar gas in between where stars formed ? This explanation is missing something

To me, Friedmann is one of the great unsung heroes of cosmology. While writing this, I couldn’t resist reading his original 1922 paper,

Five years later, history repeated itself: an MIT graduate student, the Belgian priest and astrophysicist Georges Lemaître, again published Friedmann’s Big Bang solution, which he had been unaware of and had rediscovered. And once again, it was largely ignored by the scientific community. What finally made people take note of the Big Bang idea wasn’t new theoretical work, but new measurements.

General relativity liberalizes the speed limit: whereas special relativity says that no two objects can move faster than light relative to one another under any circumstances, general relativity merely insists that they can’t move faster than light relative to one another when they’re in the same place—in contrast, the galaxies speeding away from us superluminally are all very far from us. If we think of space as expanding, then we can rephrase this by saying that nothing is allowed to move faster than light through space, but space itself is free to stretch however fast it wants to.

If our Universe is only 14 billion years old, how can we see objects that are 30 billion light-years away? How did their light have time to reach us? Moreover, we just figured out that they’re receding from us faster than the speed of light, which makes the notion that we can see them sound even weirder. Here the answer is that we’re not seeing these distant galaxies where they are now, but where they were when they emitted the light that reaches us now.

To me, a key lesson from both Newton and Friedmann is this simple mantra: “Dare to extrapolate!” Specifically, take your current understanding of the laws of physics, apply them in a new uncharted situation, and ask whether they predict something interesting that we can observe. Newton took the laws of motion that Galileo had established on Earth and extrapolated them to the Moon and beyond. Friedmann took the laws of motion and gravity that Einstein had established in our Solar System and extrapolated them to our entire Universe.

No matter how emphatically we scientists claim to be rational seekers of truth, we’re as prone as anyone to human foibles such as prejudice, peer pressure and herd mentality.

In other words, Gamow predicted that our Universe began with a hot Big Bang, and that plasma once filled all of space. What’s exceptionally interesting about this is that the prediction is testable: whereas cold hydrogen gas is transparent and invisible, hot hydrogen plasma is opaque and glows brightly, like the surface of the Sun. This means that when we gaze ever farther into space as in Figure 3.3, we should encounter old galaxies nearby, then young galaxies beyond them, then transparent hydrogen gas, then a wall of glowing hydrogen plasma. We can’t see beyond this wall, because it’s opaque

the WMAP project leader, Chuck Bennett, almost killed himself keeping it on schedule: David Spergel, another key contributor to the project, told me that Chuck collapsed and had to be hospitalized for three weeks after launch. Moreover, they made all their data publicly available online, so that cosmologists around the world could take a crack at reanalyzing it themselves. Cosmologists like me.

But gravity can only amplify small fluctuations into larger fluctuations—it can’t create fluctuations out of nowhere. If something is perfectly smooth and uniform, gravity will keep it that way forever, unable to create any dense clumps, let alone galaxies. This means that, early on, there must have been small seed fluctuations for gravity to amplify, acting like a form of cosmic blueprints that determined where galaxies would form.

Perhaps the need for dark matter and dark energy could be eliminated just as the epicycles were, by discovering a still more accurate law of gravity?

one of the most beautiful ideas in Einstein’s gravity theory is that geometry isn’t just mathematics: it’s also physics. Specifically, Einstein’s equations show that the more matter space contains, the more curved space gets. This curvature of space causes things to move not in straight lines, but in a motion that curves toward massive objects—thus explaining gravity as a manifestation of geometry.

after spending its first 7 billion years slowing down, the cosmic expansion started speeding up again and has accelerated ever since!

Basically, measuring redshifts and velocities is easy in astronomy while measuring distances is hard, so Hubble’s law can save you work:

Dark matter clusters, dark energy doesn’t. Dark matter dilutes as it expands, dark energy doesn’t. Dark matter attracts, dark energy repels. Dark matter helps galaxies form, dark energy sabotages.

Our physical space doesn’t come with centimeter marks built in the way a ruler does, nor does our Universe come with a bunch of clocks pre-installed. Instead, any observer needs to define her own measurement rods and clocks, which in turn define her notion of space and time.

The only circumstance when we can definitely say that an event on Mars happened before an event on Earth is if we can send a message from Mars after the Mars event that reaches Earth before the Earth event.

how can the laws of physics allow different laws of physics? As we’ll now see, the key idea is that fundamental laws of physics, which by definition hold anywhere and anytime, can give rise to a complicated physical state of affairs where the effective laws of physics inferred by self-aware observers vary from place to place.

If we’re living in a random habitable universe, the numbers should still look random, but with a probability distribution that favors habitability.

The funny unit MeV is the amount of motion energy an electron picks up if you use a million volts to accelerate it.

old Cold War joke about how, in the West, everything that wasn’t forbidden was allowed, while in the East, everything that wasn’t allowed was forbidden.

At a more technical level, some particle physicists like to glibly answer the question “What’s a particle?” by saying, “It’s an element of an irreducible representation of the symmetry group of the Lagrangian.” That’s quite a mouthful, and enough to stop most budding conversations dead in their tracks, but it’s a completely mathematical thing, just a bit more general than the concept of a set of numbers. And yes, sure, string theory or a competitor may deepen our understanding of what particles really are, but all the leading theories out there simply replace one mathematical entity with another.

For example, if the quantum numbers from Table 7.1 turn out to correspond to different types of superstring vibrations, then you shouldn’t think of these strings as fuzzy little objects with intrinsic properties like being made out of braided golden-brown cat hairs, but rather as purely mathematical constructs that physicists have dubbed “strings” simply to emphasize their one-dimensional nature and to make an analogy with something that feels less mathematical and more familiar.

I just downloaded a list of over 20,000 spectral lines from http://physics.nist.gov/cgi-bin/ASD/lines1.pl that have had their frequency painstakingly measured in laboratories around the world, and by capturing the patterns and regularities in these numbers, the Schrödinger equation can data-compress them down to just three numbers: the so-called fine-structure constant α ≈ 1/137.036, which gives the strength of electromagnetism, the number 1836.15, which is how many times heavier the proton is than the electron, and the orbital frequency of hydrogen.

Reading Everett’s book taught me a lesson not only in physics but also in sociology: I learned the importance of going back and checking the source material for yourself rather than relying on secondhand information. It’s not only in politics that people get misquoted, misinterpreted and misrepresented, and Everett’s Ph.D. thesis is a great example of something that, to first approximation, everyone in physics has an opinion about and almost nobody has read.

quantum observation isn’t about consciousness, but simply about the transfer of information.

Criterion 3 places a limit on how long you can run your quantum-suicide experiment in practice before fluke events save your life.

This is ironically a concsciousness centric view of quantum mechanics.

Spacetime certainly isn’t made of the crude voxels we use to simulate tomorrow’s weather, which is one of the reasons why weather forecasts are often inaccurate. Yet this idea that there’s a bunch of numbers at each point in spacetime is quite deep, and I think it’s telling us something not merely about our description of reality, but about reality itself. One of the most fundamental concepts in modern physics is that of a field, which is just this: something represented by numbers at each point in spacetime. For example, there’s a temperature field corresponding to the air around you: there’s a well-defined temperature at each point, totally independent of any human-invented voxels, and you can measure the temperature number by holding a thermometer there—or your finger, if you don’t need great accuracy.

As a first example, let’s look at the magnetic field. It’s represented by not one (like temperature) but three numbers at each point in spacetime, encoding both a strength and a direction.

A second example is the electric field, which is also represented by a triplet of numbers encoding strength and direction.

These electric and magnetic fields can be elegantly unified into what’s known as the electromagnetic field, represented by six numbers at each point in spacetime.

if our physical world is a mathematical structure, then all the light in our Universe (which feels quite physical) corresponds to six numbers at each point in spacetime (which feels quite mathematical). These numbers obey the mathematical relations that we know as Maxwell’s equations,

what I’ve just described was our understanding of electricity, magnetism and light in classical physics. Quantum mechanics complicates this picture, but without making it any less mathematical, replacing classical electromagnetism with quantum field theory, the bedrock of modern particle physics. In quantum field theory, the wavefunction specifies the degree to which each possible configuration of the electric and magnetic fields is real. This wavefunction is itself a mathematical object, an abstract point in Hilbert space.

quantum field theory says that light is made of particles called photons, and, crudely speaking, the numbers constituting the electric and magnetic fields can be thought of as specifying how many photons there are at each time and place. Just as the strength of the electromagnetic field corresponds to the number of photons at each time and place, there are other fields corresponding to all the other elementary particles known. For example, the strengths of the electron field and the quark field relate to the numbers of electrons and quarks at each time and place.

In quantum field theory, the wavefunction specifies the degree to which each possible configuration of each of these fields is real.

Some people find it emotionally displeasing to think of themselves as a collection of particles.

This suggests that the continuity business is a red herring, and that there simply is no new physical process to be discovered that somehow makes certain observer moments feel connected, thereby explaining our familiar feeling that time flows.

My guess is that we’ll one day understand consciousness as yet another phase of matter. I’d expect there to be many types of consciousness just as there are many types of liquids, but in both cases, they share certain characteristic traits that we can aim to understand.

Very speculative

For any two mathematical structures, you can define a new one by combining all their elements and relations. Many structures on the master list are of this composite type, and when studying the Level IV multiverse, it makes sense to ignore them. This is because there are no relations connecting the two parts, which means that a self-aware observer in one of its parts will be forever unaware of and unaffected by the existence of the other part, so she can just as well act as though the other part didn’t exist—or wasn’t part of her mathematical structure. The only way in which composite structures may perhaps matter is if they enter into the solution of the measure problem, altering the likelihood you’d assign to living in different mathematical structures.