"A porthole into another world."—Scientific American
"Brings science dissemination to a new level."—Science
The most trusted explainer of the most mind-boggling concepts pulls back the veil of mystery that has too long cloaked the most valuable building blocks of modern science. Sean Carroll, with his genius for making complex notions entertaining, presents in his uniquely lucid voice the fundamental ideas informing the modern physics of reality.
Physics offers deep insights into the workings of the universe but those insights come in the form of equations that often look like gobbledygook. Sean Carroll shows that they are really like meaningful poems that can help us fly over sierras to discover a miraculous multidimensional landscape alive with radiant giants, warped space-time, and bewilderingly powerful forces. High school calculus is itself a centuries-old marvel as worthy of our gaze as the Mona Lisa. And it may come as a surprise the extent to which all our most cutting-edge ideas about black holes are built on the math calculus enables.
No one else could so smoothly guide readers toward grasping the very equation Einstein used to describe his theory of general relativity. In the tradition of the legendary Richard Feynman lectures presented sixty years ago, this book is an inspiring, dazzling introduction to a way of seeing that will resonate across cultural and generational boundaries for many years to come.
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About the Author
Read an Excerpt
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.
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 what physicists call "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 ground. 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.
Here on Earth, in Aristotle's view, if everything were in its natural state, things would be motionless. It requires some external influence to get things moving, and even then the motion will only be temporary. You can pick up a rock and throw it; that's an unnatural or violent motion. But eventually the rock will come back down, maybe bounce around a bit, and return to its natural state at rest on the ground.
He's not wrong, at least in a wide variety of circumstances. If you're sitting with a coffee cup on the table in front of you, by itself the cup will just sit there. You can make it move by pushing it, but when you stop pushing, it will come to rest again. We can extrapolate this, Aristotle imagines, to a basic feature of the universe. Objects are naturally at rest, and motion only occurs when something pushes them away from this natural state.
This picture fits less well with other cases that were known even in Aristotle's day. Ancient Greeks were well acquainted with arrows flying through the air. The initial force may be applied by the bowstring, but it's clear that the arrow keeps going long after it has left the bow. Why doesn't the arrow just fall to the ground? What keeps it from expeditiously returning to its natural state?
This was a question that great minds puzzled over for hundreds of years. It took a while, but the answer ultimately led to a wholesale overthrow of Aristotle's teleological view of the universe. It was replaced with a picture in which objects don't evolve toward ultimate goals; instead, they obey laws that predict what will happen the very next moment based on what's happening now.
Conservation of Momentum
An important step was taken by John Philoponus, an Alexandrian thinker in the sixth century. He suggested that the bowstring imparted a certain quantity, later dubbed "impetus," to the arrow, which kept it moving for a while before eventually dissipating away. A simple suggestion, perhaps, but an important move away from thinking in terms of forward-looking purposes and replacing them with properties that exist in the moment.
Philoponus's idea was developed further by Ibn Sn (Avicenna), an eleventh-century Persian polymath. It was Ibn Sn who took the crucial step of arguing that impetus is not just temporary. Every object has a certain amount of impetus (equal to zero for a stationary object, some larger number if the object is moving), and that amount remains constant unless a force somehow pushes on it.
In this new picture, the reason why rocks and coffee cups come to rest is not because that's their natural state; it's because forces-friction, air resistance-gradually degrade the impetus from the body. In the vacuum of empty space, Ibn Sn suggested, there would be no air resistance, and a moving body would keep moving at a constant velocity in perpetuity. This was a wildly speculative thought experiment one thousand years ago, but today we regularly build spacecraft that move between the planets at basically a constant velocity (apart from the gentle tug of gravity). In the fourteenth century, French philosopher Jean Buridan proposed a mathematical formula for the impetus, equating it to the weight of an object times its speed.
What we have here is the birth of a law of physics: conservation of momentum. The rough idea of some "quantity of motion" being conserved came along before anyone could pinpoint precisely what that quantity was. This is a standard story of progress in theoretical physics: We put forward a new concept, work to characterize it in quantitative terms, then take that quantitative expression-an equation-and ask how it comports with other phenomena we observe in the world. Today we know that momentum is mass times velocity (at least until relativity comes along and complicates things a bit).
One problem with Buridan's definition of impetus as "weight times speed" is that "weight" isn't an intrinsic property of an object, because it depends on the amount of gravity pulling on it-your weight would be lower on the moon than it is on Earth, and you would be weightless if you were in a spaceship coasting between the planets. Mass, on the other hand, is an intrinsic property; roughly speaking, mass is the resistance that an object has to being accelerated. It takes a lot of force to accelerate a high-mass object to a certain speed, and only a little force to accelerate a low-mass object to that speed.
Similarly, speed and velocity are subtly different. Speed is a number, a certain number of meters per second. Whereas velocity is a vector-a quantity with both a magnitude and a direction. In fact, the magnitude of the velocity vector is precisely what we call "speed," but the velocity also points in some specified direction. So you have the same speed if you're driving north at 90 km/hour as you do if you're driving south at 90 km/hour, but your velocity is different.
We denote vectors by drawing little arrows over an appropriate symbol, so the velocity of an object is typically written . We very often care about the size, or magnitude, of a vector, which is written with the same symbol but without the arrow: The magnitude of a vector is simply v.
The arrow notation makes sense because we often represent a vectorial quantity by literally drawing an arrow that points in the direction of the vector, and whose length is proportional to the magnitude of the vector. Alternatively, we can represent a vector in terms of its components-the contributions it gets from different directions. If you are traveling exactly northward, the component of your velocity in the east/west direction is zero.
It is easy to add vectors together. Just imagine placing the beginning of the second vector at the end of the first, so we define a third resulting vector by traveling down the first and then the second. If the two vectors we're adding together point (almost) along the same direction, the total vector will be (almost) as long as the sum of their magnitudes, but if they point in (almost) opposite directions, the resulting vector can be much shorter.
Buridan and his predecessors didn't think in terms of vectors, which were gradually developed by a number of thinkers in the nineteenth century, including German mathematician August Ferdinand Mšbius (of "Mšbius strip" fame); Irish mathematician William Rowan Hamilton; German polymath Hermann Grassmann; and English mathematician Oliver Heaviside. So it's no surprise that it took a while to get the right definition of momentum.
These days the momentum vector is usually denoted . (The letter m is reserved for mass, so we take the symbol from the Latin word for momentum, petere.) With all that in mind, the expression for momentum is the simplest thing in the world:
Our first official equation. The momentum vector points in the same direction as the velocity vector, and their magnitudes are proportional. Proportionality will be a crucial concept for us: It means that a multiplicative change in one quantity implies a multiplicative change in the other. If you double the velocity, you double the momentum. The factor relating the two is called the "constant of proportionality," although in some equations it might not actually be constant. In this case it is: It's just the mass of the object.
The power of even a basic equation like this should be evident. We're not saying that the momentum of some particular object just happens to be equal to its mass times its velocity; we're saying that there is a universal relationship between momentum, mass, and velocity, which always takes precisely this form for every object. (When relativity comes along, some of the explicit forms of the equations we'll see here are going to have to be tweaked, but the basic principles are largely the same.)
An equation like this has no "causality" built into it; it's a rigid relationship between the quantities involved, and it reads equally well left-to-right or right-to-left. We can manipulate the equation in any way that does the same operation to both sides, such as dividing by m to get . We can therefore say, "If I know the velocity of an object, I multiply by its mass to get the momentum," or equally well, "If I know the momentum of an object, I divide by its mass to get the velocity."
Table of Contents
1 Conservation 7
2 Change 29
3 Dynamics 55
4 Space 89
5 Time 117
6 Spacetime 139
7 Geometry 173
8 Gravity 211
9 Black Holes 239
Appendix A Functions, Derivatives, and Integrals 269
Appendix B Connections and Curvature 281