The Biggest Ideas in the Universe 1, page 1
Praise for Sean Carroll
‘As a ten-year-old physics enthusiast, I would have loved The Biggest Ideas in the Universe. With this book, Sean Carroll rejects traditional elitism in physics and welcomes in anyone who knows only a little algebra but wants to understand the whole universe.’
Chanda Prescod-Weinstein, author of The Disordered Cosmos: A Journey into Dark Matter, Spacetime, and Dreams Deferred
‘Sean Carroll is a wizard of empathy. In this short book, the first of three on The Biggest Ideas in the Universe, he anticipates what’s always confused you about physics and then gently guides you to enlightenment… and ultimately, to newfound wonder.’
Steven Strogatz, author of The Joy of X and Infinite Powers
‘No-nonsense, not-dumbed-down explanations of basic laws of the universe that reward close attention.’
Kirkus
‘Sean Carroll’s greatest gift isn’t that he’s an expert on the fundamentals of physics, which he is, but that he never speaks down to his reader. He assumes that anyone, even the uninitiated, can learn to understand the formulae that underlie complicated concepts like space and time. It is a pleasure to read his work, a greater pleasure still to get a world-class education from such a witty, thoughtful teacher.’
Annalee Newitz, author of The Future of Another Timeline
‘Carroll has the remarkable ability of putting the reader utterly at ease with his lucid and addictive prose.’
Jim Al-Khalili
‘Sean Carroll is a fantastically erudite and entertaining writer.’
Elizabeth Kolbert
ALSO BY SEAN CARROLL
Something Deeply Hidden
The Big Picture
The Particle at the End of the Universe
From Eternity to Here
To Jennifer
CONTENTS
Introduction
ONE Conservation
TWO Change
THREE Dynamics
FOUR Space
FIVE Time
SIX Spacetime
SEVEN Geometry
EIGHT Gravity
NINE Black Holes
Appendices
Appendix A: Functions, Derivatives, and Integrals
Appendix B: Connections and Curvature
Index
INTRODUCTION
My dream is to live in a world where most people have informed views and passionate opinions about modern physics. Where you knock off after a hard day at work, head down to the pub with friends, and argue over your favorite dark-matter candidate, or competing interpretations of quantum mechanics. A world where, as kids are running around at a birthday party, one parent says, “I don’t see why anyone thinks there should be new particles near the electroweak scale,” and another immediately replies, “Then how in the world are you going to address the hierarchy problem?” People have opinions, after all, about supply-side economics or critical race theory. Why not inflationary cosmology and superstring theory?
That’s not quite the world in which we live. Even more than most other academic disciplines, physics is a field by and for specialists. Practitioners talk to one another in a highly specialized jargon, one that is dominated by mathematical concepts most people have never heard of, much less mastered. There are sensible reasons why this is the case, but it doesn’t have to be this way. The situation is due in large part to the ways in which physicists tend to share their knowledge with the rest of the world.
If you are a non-expert interested in learning about modern physics, you have basically two options. One is to remain at a popular level of explanation, where you can learn about some of the relevant concepts without digging into the technical or mathematical details. You can read books, go to lectures, watch videos, listen to podcasts. The good news is that we do have a vibrant ecosystem of such resources, and it’s possible to learn quite a bit, albeit in a somewhat haphazard way. But at the end, you know you’re not getting the real stuff. What you get are images and metaphors, rough translations of the underlying mathematical essence into ordinary language. You can go an impressive distance down this route, but something vital will always be missing.
The other route is to become a physics student. That could be literally at a university, or by assembling the right textbooks and online resources. Along the way you will need to become proficient at quite a bit of mathematics: calculus and differential equations most importantly, but also aspects of vector analysis, complex numbers, linear algebra, and more. The journey will be rewarding, but frustratingly slow. It typically takes at least a year of introductory courses before a student ever hears about relativity or quantum mechanics. And most physics students can get an undergraduate degree—or even go all the way to obtain a PhD—without learning about particle physics, black holes, or cosmology. Those goodies are reserved only for specialists in particular subfields.
The gap between learning physics as an interested amateur, relying on metaphors and murky translations, and becoming a credentialed expert, comfortable with pushing around equations of intimidating complexity, is wide but not unpassable. Just because I don’t want to be a professional race-car driver doesn’t mean I shouldn’t be allowed to drive at all. Surely there is a way to engage with some of the authentic essence of modern physics—even if that means looking at a few equations—without slogging through years of a standard curriculum.
You’ve come to the right place!
The Biggest Ideas in the Universe is dedicated to the idea that it is possible to learn about modern physics for real, equations and all, even if you are more amateur than professional and have every intention of staying that way. It is meant for people who have no more mathematical experience than high school algebra, but are willing to look at an equation and think about what it means. If you’re willing to do that bit of thinking, a new world opens up.
Here is the thing about equations: They are not that scary. They are just a way to compactly summarize a relationship between different quantities. It’s one thing to be told that, according to Einstein’s theory of general relativity, “mass and energy cause spacetime to curve.” It is quite another to be given Einstein’s equation:
The English-language sentence gives you a kind of feeling for what general relativity is about, but the equation tells you what is really going on, in precise and unambiguous terms. You can read all of the words you like, but until you understand this equation, you won’t really understand Einstein’s theory.
The problem is that this equation is utterly opaque if you don’t know what the symbols mean. It’s gibberish. To wrap your head around it, you need to understand the individual roles of all the numbers and letters, including the subscripts μ and ν, which are from the Greek alphabet, for goodness’ sake. There are good reasons why typical physics students take years to make it that far.
But you will get that far by reading this book. By the time you reach Chapter 8, you will understand what all the symbols in Einstein’s equation mean, how they fit together, and what they are telling us about spacetime and gravity. The equation might involve Greek letters, but coming to understand it is enormously easier than, say, learning to speak and write actual Greek.
Most popular books assume that you don’t want to make the effort to follow the equations. Textbooks, on the other hand, assume that you don’t want to just understand the equations, you want to solve them. And solving these equations is enormously more work and requires enormously more practice and learning than “merely” understanding them does.
Let me elaborate on this solving/understanding distinction, because it will be the key to the remarkably fast progress we’ll be able to make. Einstein’s equation doesn’t just relate some specific collection of mass and energy to the curvature of some specific spacetime. It is a completely general relationship, of the form “you give me some distribution of mass and energy, and I will tell you how spacetime curves in response to that.” Carrying out this promise is what we mean by “solving the equation.”
Sometimes solving an equation is easy: If the equation is x = y2, and we’re told that y = 2, the solution is x = 4. Not so hard. But real-world physics equations are more complicated than that, involving ideas from calculus (the mathematics of continuous change) and other advanced concepts. Solving such equations can become a full-time occupation for working physicists. Therefore, sensibly enough, their education consists in large part in learning to solve equations. Any physics student will tell you that the difficult part of their years in school isn’t going to the lectures, it’s doing the problem sets that professors keep handing out as if the students have nothing else to do that weekend.
Here in The Biggest Ideas in the Universe, we’re not going to teach you how to solve the equations. But you will learn to understand the equations, even ones that are considered relatively advanced by physics-textbook standards. That turns out to be enormously easier. These books are dedicated to the belief that the ideas of modern physics—the real ones, not watered-down metaphorical versions—can be accessible to anyone willing to do just a little bit of thinking about the equations and what they mean.
Okay, but what are these ideas that we’re talking about? There are many of them, as you might imagine. Enough that we’ve divided the material into a three-part series: Space, Time and Motion; Quanta and Fields; and Complexity and Emergence. The trilogy format has proven successful for The Lord of the Rings and other popular franchises.
The book you hold i
n your hands, Space, Time and Motion, focuses on the framework of classical physics pioneered by Isaac Newton, which held sway until the quantum revolution of the twentieth century. But do not fear, we will not be spending too much time on pulleys and inclined planes, as important as those are. The ambit of classical physics includes deep questions about the nature of space, time, and change, and we won’t be afraid to sprinkle some philosophical considerations in among our equations. It also includes the theory of relativity, all the way up to Einstein’s ideas about curved spacetime, and consequences such as black holes. So this book starts with ideas that are centuries old but will take us right up to modern research-level concepts.
In Quanta and Fields we will discuss sexy quantum ideas like entanglement and Schrödinger’s cat, but mostly we will take the opportunity to learn about quantum field theory and particle physics, the best modern take on the fundamental laws of nature. The final installment, Complexity and Emergence, is where we admit that the world isn’t made of just two or three particles. Interesting things happen when systems consist of a large number of moving parts.
That’s a lot of concepts. And yet, almost all of them are within the realm of physics and related areas. This is not to disparage the equally big and important ideas from other areas of science (or the arts and humanities, for that matter), but one has to draw the line somewhere.
Another place we’ve drawn the line is between “ideas we have good reason to believe are true” and “promising speculations.” While physics textbooks tend to stick to ideas that have established their usefulness, popular-level treatments will cheerfully dive into concepts that are still entirely hypothetical. That’s a perfectly sensible thing to do; researchers spend most of their time at the frontier, thinking about possibilities that haven’t become part of settled lore. Our goal here is to stick to ideas that we have excellent reason to think will still be part of the working physicist’s tool kit a hundred years from now.
It is a pleasure to acknowledge the enormous help I have received along the way. Scott Aaronson, Justin Clarke-Doane, and Matt Strassler provided invaluable feedback and saved me from more than a few infelicities of expression. Jason Torchinsky made the beautiful illustrations. My editor, Stephen Morrow, was supportive and insightful as always, and my agent, Katinka Matson, helped shape the form of a complicated project. Alice Dalrymple, Tiffany Estreicher, Dora Mak, Nakeesha Warner, and Melanie Muto were crucial in the production process. The book grew out of a series of videos I made during the COVID-19 pandemic, inspired by online playwriting classes given by my friend Lauren Gunderson. And of course I cannot properly express my gratitude to Jennifer Ouellette for writing advice, moral support, and more.
To see the videos, as well as find other supplementary materials, visit:
https://preposterousuniverse.com/biggestideas/
ONE
CONSERVATION
Look around. If you’re like most people, you have a body. It’s located somewhere. Chances are that you are surrounded by a variety of other objects, located other places. Tables, chairs, a floor, ceiling, walls, maybe trees or a body of water if you’re outside. All of these objects exist, with certain locations and properties, and those locations and properties can change with time. You can scoot your chair nearer to a wall, or farther away. You drink a glass of water, absorbing its substance into your body. If instead you put the glass on a table and leave it there, the water will eventually evaporate into the air.
That’s how we think about the world from an immediate, human-scale perspective. There is stuff, which is located in space. (By “space” we don’t mean “outer space,” just the three-dimensional realm through which things move.) This stuff might change, or it might remain constant over time. Physics is the study of all that stuff, and its behavior, at the most basic level we can think of. What is all that stuff, really? How do different objects relate to one another? How do they change with time? What is “time,” and for that matter what is “space,” when you get right down to it?
One of the most enjoyable features of physics is how quickly we go from mundane observations—look at that stuff, behaving in that way!—to profound questions about the nature of reality. The key is that things don’t just happen—all of the happenings fit into certain patterns. It’s those patterns that we call the laws of physics, and our job is to uncover them.
The simplest pattern of all is the fact that certain things remain constant even as time passes. Contemplating that basic feature of reality is a great jumping-off point for our investigations, which will get pretty wild soon enough.
PREDICTABILITY
We take for granted that the world around us is at least a little bit predictable. If there is a table in a room, and we turn to face away from it for just a second, we expect the table to still be there when we turn back. If we place an apple on the table, we expect the table to support it, rather than the apple falling right through to the floor. As much as we might lament how difficult it is to predict the weather or future election outcomes, we should be impressed by how much reliable predictability there is.
Physics is made possible by this predictability. It may not be absolute, but we can somewhat anticipate what’s going to come next in the world if we know what’s going on right now. The most basic kind of predictability is conservation, the fact that some things don’t change at all.
Conservation is just how physicists refer to “staying constant over time.” You may have heard that energy is conserved, for example. Energy isn’t a kind of substance, like water or dirt. It’s a property that things have, depending on what they are and what kind of situation they’re in. There is no “energy fluid” that flows from place to place. There are simply objects that have positions and velocities and other properties, and we can associate a certain amount of energy with them because of those facts.
An object can have energy because it is moving, because it’s located at a high elevation, because it’s hot, because it’s massive, because it’s electrically charged, or for other reasons. Under the right circumstances, those forms of energy can be converted back and forth between each other. The energy that a wineglass has just from being located on a table can, if the glass is knocked off the edge, rapidly be converted into energy of motion as it falls, and then into heat and noise and other forms of dissipated energy as it breaks on the floor. Conservation of energy is simply the idea that the total energy, given by adding up all the individual forms, remains constant throughout the whole process.
(Wait—is this circular reasoning? Are we merely inventing a bunch of quantities that add up to a constant number by definition, calling that “energy,” and congratulating ourselves for discovering a law of physics? No. There is an independent way to define energy and then show that it’s conserved, based on the fact that the laws of physics don’t themselves change over time. But you’re asking the right kind of question.)
As simple an idea as we can imagine—there is a quantity that doesn’t change, it stays the same as time passes. But conservation of energy and other quantities isn’t just a gentle, unintimidating place from which to launch a survey of all of physics. It’s logically the right place, since an understanding of conservation was the first step in the transition from pre-modern to modern science.
FROM NATURES TO PATTERNS
Put yourself in the mindset of humans trying to understand the world before physics in its modern form came along. The Greek philosopher Aristotle is usually chosen as an exemplar, though other ancient thinkers would have thought similarly. To greatly simplify a complex and subtle set of ideas, Aristotle separated the way things move into “natural” and “unnatural” (or “violent”) motions. He thought of the world as fundamentally teleological—oriented toward a future goal. Objects have natural places to be or conditions to be in, and they tend to move to those places. A rock will fall to the ground and sit there; fire will rise to the heavens.
