Deriving the Lorentz transformations

Relativity theory is provocative. It incites an urge to show that the theory is inconsistent or that one can beat the speed of light. In this chapter we examine both issues. As a prelude, we need to study how Alice and Bob assign coordinates to an event, that is, how they specify the location and time of an event. Then we need to learn how those assignments are related. Once we have those relations, we can compute how differently Alice and Bob perceive the velocity of some object. And that is a major step toward trying to beat the speed of light.

Alice and Bob specify the location of an event – a firecracker going off, say – by measuring distances from the origins of their respective reference frames. Figure 13.1 displays this. The distance Bob assigns is denoted x; the distance Alice assigns is called x. Alice specifies the time by using a clock that is present at the event, is at rest in her frame, and is synchronized with the master clock at the origin of her frame. Bob does the same, using a clock at rest in his frame. This procedure – using clocks at the event itself – is a matter of convenience and is something we discussed earlier, in section 9.5. The times are denoted t for Bob's value and t' for Alice's.

Hertz revisited

In chapters 5 and 6, we noted that Heinrich Hertz discovered the photoelectric effect while investigating electromagnetic waves. A strong spark produces not only visible light but also ultraviolet light. When ultraviolet light from a spark in Hertz's transmitter circuitry struck the metal of his detector, the ultraviolet ejected some electrons. Those electrons, now present in the gap across which a spark was to jump, made it easier for a spark to jump. After all, a spark is an electric current, and the presence of the ejected electrons gave the spark a head start (for air is ordinarily a poor conductor of electricity, having few mobile charged particles). What Hertz perceived directly was that, when ultraviolet light shone on the detector, the spark was easier to start and could be made to jump a larger gap.

This was fascinating, and Hertz did spend six months establishing conclusively the essential role of ultraviolet light, but then he returned to electromagnetic waves.

First observations

How should we begin? Surely with Isaac Newton. He published his first scientific paper in 1672, and it was on light. In response to an objection raised against his paper, he wrote

• For the best and safest method of philosophizing seems to be, first diligently to investigate the properties of things and establish them by experiment, and then to seek hypotheses to explain them. For hypotheses ought to be fitted merely to explain the properties of things and not attempt to predetermine them, except in so far as they can be an aid to experiments.

So let us get a light source and begin investigating. I will suppose that you have access to some apparatus – at least in the sense of seeing lecture demonstrations – and so I will describe things as though we were doing the experiments together.

We take a strong flashlight or a 35-millimeter slide projector and shine the light horizontally. Clapping a pair of well-used erasers produces a cloud of chalk dust, and we see the light beam piercing the cloud as a bright shaft of light. This gives us our first observation:

Observation 1. Light goes in a straight line from a luminous source.

Reflection and refraction

Next we tip our source – the projector, say – so that the light enters the water in an aquarium tank. Figure 1.1 depicts what happens.

In chapter 11, I noted that the equation E=mc2 has given rise to a host of misconceptions. There are other issues associated with the equation, too, among them an issue of definition, and this appendix addresses some of those issues. It is written primarily for an instructor, but a student who has read appendix B will be able to understand most of it and will gain a greater perspective.

Mass in history

The word “mass” has a long history and a correspondingly long string of meanings and nuances. Max Jammer provides a fine discourse on the topic in his Concepts of Mass in'Classical and Modern Physics, cited in the “Additional resources” for chapter 11; I rely on Professor Jammer for the early history. In Roman times, the Latin word “massa” denoted a lump of dough. This sense of “lump of stuff” is what we have in mind when we “hang a mass from a spring.” By Newton's time, the word “massa” (still in Latin, of course) had come to denote “quantity of matter,” a valuable concept but one that defied attempts to make it both operationally precise and independent of any prior notion of density.

Newton recognized clearly the physical notion of inertia and hence what we would today call “inertial mass.” It was Leonhard Euler, however, who first defined inertial mass operationally as the ratio of force to acceleration (more precisely, as the ratio of their magnitudes).

Preview

The last three chapters developed a wave theory of light. Indeed, they suggested that light is an electromagnetic wave, and then they marshaled experimental support for that view. The wave theory of light is correct – but it is not the whole story. Some other idea complements that theory. In this chapter, new experiments compel us to develop the alternative idea, and in chapter 7 we learn how the wave theory and the new idea complement each other.

The photoelectric effect

In this day and age, we are accustomed to the idea that sunlight is a source of energy. Let us investigate the connection between light (in general) and energy.

If you are not thoroughly familiar with how the word “energy” is used in physics, now is the right time to read appendix A, Energy. Knowing what physicists mean with their words will help you to understand the line of reasoning.

Electron escape

When an electron absorbs light, it acquires additional energy. If the electron is in a piece of metal, it may be able to escape from the metal. The positive charges in the metal tend to hold the electrons back and prevent their escape. (To be sure, the electrons can move about quite freely inside a good electrical conductor.)

The electric field

Let us recall the plan we adopted at the close of section 3.1:

1. (a) Learn about waves.

2. (b) Find that light does indeed have a wave-like character.

3. (c) Learn what the “something” is out of which light waves are formed.

In subsequent sections, we found that waves do have the properties of reflection and refraction that light possesses. Then we shifted perspective and found that light has the property of interference that waves possess. In short, chapters 3 and 4 taught us that light does indeed have a wave-like character.

Before we can address part (c) of the plan, we need to know certain aspects of electricity and magnetism. So we turn now to those twin subjects.

Sometimes, when you pull a sweater off over your head on a day when the air is particularly dry (perhaps indoors in winter), you hear a crackling sound and may even feel an electric spark. The frictional rubbing of the sweater on your shirt or blouse separates some electric charges, one from another; the crackling and the spark are a miniature form of thunder and lightning.

More deliberate rubbing of two different objects separates electric charges more effectively.

Chapter 9 began with the assertion, “Space and time are different from what you thought they were like.” By now, probably you agree. This brief chapter draws together verbally the remarkable properties of the space and time in which we live.

The interconnection of space and time

Space and time are intrinsically different from each other. We determine locations in space by laying off a wooden meter stick or by stretching a metal tape, but we determine “locations in time” by reading a clock, be it a pendulum clock or a digital watch or a cesium beam atomic clock. Such an intrinsic difference was appreciated before Einstein's 1905 relativity paper, and it persists to this day. Nonetheless, relativity theory showed space and time to be more closely linked than people had thought. Let us review the connections.

The linkage of space and time shows up most clearly when Alice and Bob – in motion relative to each other – compare the locations in space and time that they assign to the same events. Our investigation of the relativity of simultaneity provides an excellent context. Look again at figures 9.6 and 9.7. Focus first on the event of light's reaching marker (a), fixed in Alice's frame. For Bob, the location in space of that event depends on both the distance of marker (a) from the origin of Alice's frame and on how long she has been moving since the origins of the two frames passed each other.

The college catalog said

• Physics 104. Newton to Einstein: The Trail of Light

• The course will follow the trail of light from Newton's corpuscles to Einstein's relativity. The major theoretical landmarks will be the wave-particle duality and the special theory of relativity.

It was the spring term of 1987, and the course was brand-new. Out of my previous experience with courses for non-scientists, I expected a class of 20 to 30 students. If enrollment became a fad, then the class might grow to 50 or 60. I was not prepared for what happened. When pre-registration was over, 141 students had signed up.

As an aside, let me remark that I was also scheduled to teach the calculusbased introductory course, ordinarily our largest course. That spring I taught more students than the rest of the department, all put together.

My intention was to teach the course in alternate years (to maximize the number of times I would teach it before burning out). When I taught the course the second time, in the fall of 1988, over 170 students sought admission. Reluctantly, I held the line at 150, my saturation point when reading essays and correcting exams.

During the last week of the semester, I ask my students for advice about improving the course, and I also ask them, “What, for you, is the most valuable thing you learned or ‘got’ from this course?” In 1988, my younger son, then a senior in high school and ever skeptical of his father's teaching methods, happened to read the responses first.

Theory building

By now we have a substantial number of observations; we know quite a bit about how light behaves. Let us try our hand at building a theory of light: what light is, and why it behaves as it does.

Our first observation – that light goes in straight lines from a luminous source – suggests that we need the notion of something that moves through space. The simplest thing is a particle, a “little baseball.” Perhaps the luminous source emits little light particles, a stream of them. Figure 2.1 illustrates this notion.

Isaac Newton developed the same idea. He worked on light – experimentally and theoretically – most of his long life. At the start of chapter 1, we noted that Newton's first paper (published in 1672) was on optics, and his interest in the subject goes back at least to his student days at the University of Cambridge, England, perhaps to the year 1663.

Newton could not abide criticism, and so he went to great lengths to avoid even the possibility of it. Because Robert Hooke and he readily came to acrimonious disagreement, Newton withheld the publication of his book on optics until after Hooke's death. Finally, in 1704, Newton published his treatise Opticks.

Energy, mass, and momentum reviewed

Easily the most famous result of relativity theory is the equation E=mc2. In this chapter, we learn where the equation comes from, what it means, and what it does not mean. To do so, we need to be clear about what certain terms in physics mean, and so this section reviews the essential concepts. If you have not yet read appendix A, “Energy,” or have not done so recently, please read it before proceeding any farther in this chapter. Then this section will indeed be a review.

Because this section is a review, the topics are presented in telegraphic style.

As figure 11.1 shows, a moving object can be used to lift a weight. Thus there is energy associated with motion, called “kinetic” energy, from the Greek verb “kinein,” meaning “to move.”

The higher the object starts, the more kinetic energy it has when it hits. Thus the greater the height, the more potential for kinetic energy at the bottom. Physics introduces the idea of a “gravitational potential energy,” an energy associated with position in the earth's gravitational field.