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

by Max Tegmark

trees are made of air, primarily. When they are burned, they go back to air, and in the flaming heat is released the flaming heat of the Sun which was bound in to convert the air into tree. And in the ash is the small remnant of the part which did not come from air, that came from the solid earth, instead.

Wow

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.

Whatever a politician wanted to do, he or she could find an economist as advisor who had argued for doing precisely that. Franklin D. Roosevelt wanted to increase government spending, so he listened to John Maynard Keynes, whereas Ronald Reagan wanted to decrease government spending, so he listened to Milton Friedman.

How could an infinite space get created in a finite time?

wow

In conclusion, the only information we have about stars is in their faint light that reaches us, but through clever detective work, we can decode this light into information about their distance, size, mass, composition, temperature, pressure, magnetism and any solar system they may host.

In summary, the space we live in might go on forever and it might not—both possibilities are perfectly reasonable according the best theory we have for the nature of space, Einstein’s general relativity.

Topology. Torus

we’ve discovered that this is all part of a much grander drama, where generation upon generation of galaxies are born, interact and eventually die in a cosmic ecosystem of sorts. So could there be a third level in this dramaturgy, whereby even universes are created and die? In particular, is there any indication that our Universe itself had some sort of beginning? If so, what happened, and when?

Red light has the lowest frequency of all the colors in the rainbow, so if a galaxy is moving away from us, the colors of all its spectral lines will be redshifted, shifted toward redder colors, and the higher its speed, the greater its redshift. If the galaxy is moving toward us, its colors will instead be blueshifted toward higher frequencies.

Don’t galaxies receding faster than the speed of light violate relativity theory? Hubble’s law v = Hd implies that galaxies will move away from us faster than the speed of light c if their distance from us is greater than c/H ≈ 14 billion light-years, and we have no reason to doubt that such galaxies exist,

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.

When we gaze out into our Universe with our best telescopes, we see something similar: nearby are lots of large and mature galaxies like our own, but very far away, we see mostly small baby galaxies that don’t yet look fully developed. Beyond them we see no galaxies at all, merely darkness. Since it takes light longer to reach us from farther away, gazing into the distance is equivalent to observing the past.

Gazing into the distance is equivalent to observing the past

“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.

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.

If it gets hot enough, the bumps get so violent that the atoms break apart and the electrons and protons go their separate ways—a hydrogen plasma is simply such a soup of free electrons and protons.

Plasma

this is what we should see in all directions, since wherever we look, we’re also looking back in time. It therefore looks to us like we’re surrounded by a gigantic plasma sphere.

As Fred Hoyle was the first to realize, this coincidence enabled stars in the late stages of their lives to turn helium into carbon, oxygen and most of the other atoms that you and I are made of.

As our Universe expanded and cooled, quarks assembled into protons (hydrogen nuclei) and neutrons, which in turn fused into helium nuclei. Then these nuclei formed atoms by capturing electrons, and gravity clumped these atoms into the galaxies, stars and planets that we observe today.

Main

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.

Main

We know of four fundamental forces of nature, and three of them have taken turns driving this clustering process: first the strong nuclear force pulled the nuclei together, then the electromagnetic force made the atoms and the molecules, and finally gravity built the grand structures that adorn our night sky.

after our Big Bang, it became clear that gravity wouldn’t have had time to amplify this faint clustering into today’s cosmic large-scale structure unless some invisible form of matter contributed extra gravitational pull. This mysterious stuff is known as dark matter,

white dwarfs today, created by stars past. Many of them are continually gaining weight by gobbling up gas from dying companion stars that they’re orbiting. Once they become officially overweight (which happens when they reach 1.4 times the mass of the Sun), they suffer the stellar equivalent of a heart attack: they become unstable and detonate in a gigantic thermonuclear explosion—a Type Ia supernova.

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.

Q: Did our Big Bang happen at a single point? A: No.

?

Q: Where in space did our Big Bang explosion happen? A: It happened everywhere, at an infinite number of points, all at once.

Q: How could an infinite space get created in a finite time? A: There’s no explanation—the equations simply assume that as soon as there was any space at all, it was infinite in size.

it couldn’t just have been a crazy fluke coincidence that infinitely many separate regions of space underwent Big Bang explosions all at once—some physical mechanism must have caused both the exploding and the synchronizing. One unexplained Big Bang is bad enough; an infinite number of unexplained Big Bangs in perfect synchronization strains credulity.

According to Einstein’s theory of gravity, a substance whose density is undilutable can “inflate,” doubling its size at regular intervals, growing from a subatomic scale to a size vastly larger than our observable Universe in a split second and effectively putting the bang into our Big Bang.

Big bang

The inflation theory says that our baby Universe grew much like a human baby: an accelerating growth phase where the size doubled at regular intervals was followed by a more leisurely decelerating growth phase.

A cell in your nose has the same DNA as a cell in your toe because they have a common progenitor: they’re both produced by successive doublings of your very first cell.

Lifespan

Surely, mass can’t just be created from nothing? Interestingly, Einstein offered us a loophole through his special relativity theory, which says that energy E and mass m are related according to the famous formula E = mc2. Here c = 299,792,458 meters per second is the speed of light, and because it’s such a large number, a tiny amount of mass corresponds to a huge amount of energy: less than a kilogram of mass released the energy of the Hiroshima nuclear blast. This means that you can increase the mass of something by adding energy to it.

you can make a rubber band very slightly heavier by stretching it: you need to apply energy to stretch it, and this energy goes into the rubber band and increases its mass.

Energy adds weight

A rubber band has negative pressure because you need to work to expand it. For a substance with positive pressure, like air, it’s the other way around: you need to do work to compress it. In summary, the inflating substance has to have negative pressure in order to obey the laws of physics, and this negative pressure has to be so huge that the energy required to expand it to twice its volume is exactly enough to double its mass.

gravity is caused not only by mass, but also by pressure. Since mass can’t be negative, the gravity from mass is always attractive. But positive pressure also causes attractive gravity, which means that negative pressure causes repulsive gravity!

The gravitational field, which is responsible for all gravitational forces, has negative energy. And it gets more negative energy every time gravity accelerates something.

Doesn’t creation of the matter around us from almost nothing by inflation violate energy conservation? We’ve seen that the answer is no: all the required energy was borrowed from the gravitational field.

This so-called snowflake fractal, invented by the Swedish mathematician Helge von Koch, has the remarkable property that it’s identical to a magnified piece of itself. Inflation predicts that our baby Universe was similarly indistinguishable from a magnified piece of itself, at least in an approximate statistical sense.

We know that inflation ends in at least some places, since 14 billion years ago, it ended in the part of space that we now inhabit. This means that there must be some physical process which can get rid of the inflating substance, causing it to decay into ordinary non-inflating matter, which then keeps expanding, clustering, and ultimately forming galaxies, stars and planets as we described in the last chapter.

main

For each volume of inflating substance (symbolized by a cube) that decays into a non-inflating Big Bang universe like ours, two other inflating volumes don’t decay, instead tripling their volume. The result is a never-ending process where the number of Big Bang universes increases as 1, 2, 4, etc., doubling at each step. So what we call our Big Bang (one of the flashes) isn’t the beginning of everything, but the end of inflation in our part of space.

what we’ve called our Big Bang wasn’t the ultimate beginning, but rather the end—of inflation in our part of space.

Interesting

In almost all parts of space, inflation will eventually end in a Big Bang like ours.

There will nonetheless be some points in space where inflation never ends.

How could an infinite space get created in a finite time? It sounds impossible. But as I mentioned, inflation is like a magic show where seemingly impossible tricks happen through creative use of the laws of physics. Indeed, inflation can do something even better, which I think is its most amazing trick of all: it can create an infinite volume inside a finite volume! Specifically, it can start with something smaller than an atom and create an infinite space inside of it, containing infinitely many galaxies, without affecting the exterior space.

Once inflation ends in a given region, the traditional Big Bang story from the last two chapters starts unfolding there, with a hot cosmic fusion reactor eventually cooling to form atoms, galaxies, and perhaps observers like us.

main

according to Einstein’s theory of general relativity, an observer living in one of these galaxies will perceive space and time differently than I’ve defined them with my axes in the drawing.

in water

What caused our Big Bang? A: The repeated doubling in size of an explosive subatomic speck of inflating material.

Q: Where in space did our Big Bang explosion happen? A: In that tiny region—but inflation stretched it out to about the size of a grapefruit growing so fast that the subsequent expansion made it larger than all the space that we see today.

a space that isn’t just huge but truly infinite, with infinite numbers of galaxies, stars and planets.

Our Universe The part of physical reality we can in principle observe

inflation predicts that there is. Your doppelgänger’s universe (page 126), if it exists, is a sphere of the same size centered over there, none of which we can see or have any contact with yet, because light or other information from there hasn’t had time to reach us. This is the simplest (but far from the only) example of parallel universes. I like to call this kind, a distant region of space the size of our Universe, a Level I parallel universe. All the Level I parallel universes together form the Level I multiverse.

level I universe

The most sensitive test of all is to use the most distant thing we can see, the cosmic microwave background, and look for matching patterns in opposite directions as in Figure 6.1—many research teams, Angélica and I included, tried this and found nothing.

See the back of your head