Extra-dimensional models address such particle physics problems, but they also use ideas from string theory. After discussing the basics of particle physics, we’ll introduce the fundamental motivation and concepts in string theory. We won’t derive models directly from string
theory, but string theory contains some of the elements that are used when developing extra-dimensional models.
This review covers a lot of ground because research on extra dimensions ties together many theoretical advances in the two major strands of particle physics—model building and string theory. Some familiarity with many of the most interesting recent developments in these fields will help you to better understand the motivations and the methods underlying the development of extra-dimensional models.
However, in case you want to jump around, I will end each of the review chapters with a bulleted list of vital concepts that we will refer to later on when we return to extra-dimensional model building. The bullets will serve as a short cut, a summary, in case you want to skip a chapter or if you want to focus on the material we’ll turn to later on. I might occasionally refer to points that aren’t in the bullets, but the bullets will review the key ideas that are essential to the major results in the later part of the book.
In Chapter 17 we’ll start to explore extra-dimensional braneworlds—theories that propose that the matter of which our universe is composed is confined to a brane. Braneworld ideas have provided new insights into general relativity, particle physics, and string theory. The different braneworlds I’ll present make different assumptions and explain different phenomena. I’ll summarize the distinctive features of each model with bullets at the end of these chapters as well. We don’t yet know which, if any, of these ideas correctly describe nature. But it’s entirely conceivable that we’ll ultimately discover that branes are a part of the cosmos, and that we—along with other, parallel universes—are confined to them.
One thing I have learned from this research is that the universe often has more imagination than we do. Sometimes its properties are so unexpected that we stumble across them only by accident. Discovering such surprises can be amazing. Our known physical laws turn out to have startling consequences.
Let’s now begin our exploration of what those laws are.
Relativity: The Evolution of Einstein’s Gravity
The laws of gravity are very, very strict.
And you’re just bending them for your own benefit.
Billy Bragg
Icarus (Ike) Rushmore III couldn’t wait to show Dieter his new Porsche. But as proud as he was of his car, he was even more excited about his Global Positioning System (GPS) that he had recently designed and installed himself.
Ike wanted to impress Dieter, so he convinced his friend to drive with him to the local track. They got in the car, Ike programmed in their destination, and the two of them set off. But to Ike’s chagrin, they ended up in the wrong place—the GPS system didn’t work nearly as well as he had thought it would. Dieter’s first thought was that Ike must have made some ridiculous error, like confusing meters and feet. But Ike didn’t believe he could have made such a stupid mistake, and he bet Dieter that wasn’t the problem.
The next day, Ike and Dieter did some troubleshooting. But to their dismay, when they went for a drive the GPS was even worse than before. Ike and Dieter searched again for the problem and finally, after a frustrating week, Dieter had an epiphany. He did a quick calculation and made the startling discovery that without accounting for general relativity, Ike’s GPS system would build up errors at the rate of more than 10 km each day. Ike didn’t think his Porsche was fast enough to warrant relativistic calculations, but Dieter explained that the GPS signals—not the car—travel at the speed of light. Dieter modified the software to account for the changing gravitational field the GPS signals had to pass through. Ike’s system then worked as well as the readily available commercial variety. Relieved, Ike and Dieter began to plan a road trip.
At the beginning of the last century the British physicist Lord Kelvin said, “There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.”
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Lord Kelvin was famously incorrect: very soon after he uttered those words, relativity and quantum mechanics revolutionized physics and blossomed into the different areas of physics that people work on today. Lord Kelvin’s more profound statement, that “scientific wealth tends to accumulate according to the law of compound interest,”
†
is certainly true, however, and is especially appropriate to these revolutionary developments.
This chapter explores the science of gravity, and how it evolved from the impressive achievement of Newton’s laws to the revolutionary advances of Einstein’s theory of relativity. Newton’s laws of motion are the classical physics laws that scientists used for centuries to compute mechanical motion, including motion caused by gravity. Newton’s laws are magnificent, and they allow us to make predictions of motion that work spectacularly well—well enough to send men to the Moon and satellites into orbit, well enough to keep the superfast trains in Europe on the tracks when rounding corners, well enough to prompt the search for the eighth planet, Neptune, based on peculiarities in Uranus’s orbit. But alas, not well enough for an accurate GPS system.
Incredibly, the GPS system now in use requires Einstein’s theory of general relativity to achieve its one-meter accuracy. Determinations of the variation in snow depth on Mars using laser ranging data from orbiting spacecraft also incorporate general relativity, and yield values with an unbelievable precision of 10 cm. Certainly, at the time it was developed, no one—not even Einstein—anticipated such practical applications of a theory as abstract as general relativity.
This chapter will explore Einstein’s theory of gravity, a spectacularly accurate theory that applies to a wide range of systems. We’ll begin by briefly reviewing Newton’s gravitational theory, which works fine for the energies and speeds we encounter in daily life. We’ll then move on to the extreme limits in which it fails: namely, very high speed (close to the speed of light) and very large mass or energy. In these limits, Newtonian gravity is superseded by Einstein’s theory of relativity. With Einstein’s general relativity, space (and spacetime) evolved from a static stage to a dynamical entity that can move and curve and have a rich life of its own. We’ll consider this theory, the clues that led to its development, and some of the experimental tests that convince physicists that it’s right.
Newtonian Gravity
Gravity is the force that keeps your feet on the ground and is the source of the acceleration that returns a tossed ball to Earth. In the late sixteenth century, Galileo showed that this acceleration is the same for all objects on the surface of the Earth, no matter what their mass.
However, this acceleration does depend on how far the object is from the Earth’s center. More generally, the strength of gravity depends on the distance between the two masses—gravity’s pull is weaker when objects are farther apart. And when what creates the gravitational attraction is not the Earth, but some other object, gravity’s strength will depend on the mass of that object.
Isaac Newton developed the gravitational force law that summarizes how gravity depends on mass and distance. Newton’s law says that the force of gravity between two masses is proportional to the mass of each of them. They could be anything: the Earth and a ball, the Sun and Jupiter, a basketball and a soccer ball, or any two objects you please. The more massive the objects, the greater the gravitational attraction.
Newton’s gravitational force law also says how the gravitational force depends on the distance between the two objects. As discussed in Chapter 2, the law says that the force between two objects is
proportional to the inverse square of their separation. This inverse square law was where the famous apple entered in.
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Newton could deduce the acceleration due to the Earth’s gravitational pull on an apple located near the Earth’s surface and compare it with the acceleration induced on the Moon, which is located sixty times further away than the Earth’s surface is from its center. The acceleration of the Moon due to the earth’s gravity is 3,600 times smaller (3,600 is the square of 60) than the acceleration of the apple. This is in accordance with the gravitational force decreasing as the square of the distance from the Earth’s center.
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However, even when we know the dependence of the gravitational attraction on mass and distance, we still need another piece of information before we can determine the overall strength of gravitational attraction. The missing piece is a number, called
Newton’s gravitational constant
, that factors into the calculation of any classical gravitational force. Gravity is very weak, and this is reflected in the tiny size of Newton’s constant, to which all gravitational effects are proportional.
The Earth’s gravitational pull or the gravitational attraction between the Sun and the planets might seem pretty big. But that’s only because the Earth, the Sun, and the planets are so massive. Newton’s constant is very small, and the gravitational attraction between elementary particles is an extremely weak force. This feebleness of gravity is itself a big puzzle that we will return to later on.
Although his theory was correct, Newton delayed its publication for twenty years, until 1687, while he tried to justify a critical assumption of his theory: that the Earth’s gravitational pull was exerted as if its mass were all concentrated at the center. While Newton was hard at work developing calculus to solve this problem, Edmund Halley, Christopher Wren, Robert Hooke, and Newton himself made tremendous progress in determining the gravitational force law by analyzing the motion of the planets, whose orbits Johannes Kepler had measured and found to be elliptical.
These men all made major contributions to the problem of planetary motion, but it is Newton who gets credited with the inverse square
law. That is because Newton ultimately showed that elliptical orbits would arise as a result of a central force (that of the Sun) only if the inverse square law was true, and he showed with calculus that the mass of a spherical body did in fact act as if it were concentrated at the center. Newton did, however, acknowledge the significance of others’ contributions in his words, “If I have seen further, it is because I have stood upon the shoulders of giants.”
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(However, rumor has it that he said this only because of his intense dislike for Hooke, who was very short.)
In high school physics, we learned Newton’s laws and calculated the behavior of interesting (if somewhat contrived) systems. I remember my outrage when our teacher, Mr Baumel, informed us that the gravitational theory we had just learned was wrong. Why teach us a theory that we know to be incorrect? In my high school view of the world, the whole merit of science was that it could be true and reliable, and could make accurate and factual predictions.
But Mr Baumel was simplifying, perhaps for dramatic effect. Newton’s theory was not wrong: it was merely an approximation, one that works incredibly well in most circumstances. For a large range of parameters (speed, distance, mass, and so on), it predicts gravitational forces quite accurately. The more precise underlying theory is relativity, and you only make measurably different predictions with relativity when you are dealing with extremely high speeds or large amounts of mass or energy. Newton’s law predicts the motion of a ball admirably well, since neither of the above criteria apply. To use relativity to predict the motion of a ball would be pure silliness.
In fact, Einstein himself initially thought of special relativity merely as an improvement on Newtonian physics—not as a radical paradigm shift. This, of course, grossly underplays the ultimate significance of his work.
Special Relativity
A very reasonable thing to expect from physical laws is that they should be the same for everyone. No one could blame us for questioning their validity and utility if people in different countries or sitting on moving trains or flying on an airplane experienced different physical laws. Physical laws should be fundamental and hold true for any observer. Any differences in calculations should be due to differences in environment, not the physical laws. It would be very strange indeed to have universal physical laws that required a particular vantage point. The particular quantities you might measure could depend on your reference frame, but the laws that govern these quantities should not. Einstein’s formulation of special relativity ensures that this is the case.
In fact, it’s somewhat ironic that Einstein’s work on gravity is referred to as “the theory of relativity.” The essential point that drove both special and general relativity was that physical laws should apply for everyone, independent of their reference frame. In fact, Einstein would have preferred the term
Invariantentheorie
.
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In a letter Einstein wrote in 1921 in reply to a correspondent who had suggested he reconsider the name, he admitted that the term “relativity” was unfortunate.
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But by that time, the term was too well entrenched for him to attempt to change it.
Einstein’s first insight about reference frames and relativity came from thinking about electromagnetism. The well-known theory of electromagnetism from the nineteenth century was based on Maxwell’s laws, which describe the behavior of electromagnetism and electromagnetic waves. The laws gave correct results, but everyone initially falsely interpreted the predictions in terms of the motion of an
aether
, a hypothesized invisible substance whose vibrations were supposed to be electromagnetic waves. Einstein realized that if there were an aether, there would also be a preferred observational vantage
point, or frame of reference: the one in which the aether is at rest. He reasoned that the same physical laws should apply to people who are moving at constant velocity
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with respect to each other and with respect to someone at rest—that is, in frames of reference that physicists refer to as inertial frames. By requiring that
all
physical laws, including those of electromagnetism, should hold for observers in all inertial reference frames, Einstein was led to abandon the idea of the aether and, ultimately, to formulate special relativity.