This concludes Part I of this book. I hope you have been able to grasp an outline of what relativity is all about. Let us summarize what we have learned:

• The “Special Theory of Relativity” was constructed by Einstein to resolve the mystery of the speed of light. Einstein's solution was that the concept of simultaneity depended on the frame of reference. And the rule that relates the observations from different frames was given by the Lorentz transformation.

The predictions of Special Relativity such as time dilation and Lorentz contraction are as infamous as they are famous. The reason for the notoriety is due to the apparent paradoxical nature of the prediction: say we have two frames, A and B, moving relative to each other. According to Special Relativity, the observer in frame A will observe the clock in frame B to run slower than the clock in frame A, and the ruler in frame B to be shorter than the ruler in frame A. The observer in frame B will observe the exact opposite. Now how can both points of view be true at the same time?

Of course, the two points of view are NOT true at the same time. They are both true because they are NOT at the same time.

Before and after

As we have seen, the only way to reconcile the experimentally observed fact that the speed of light does not depend on the inertial frame with our belief in an objective reality, was to abandon the notion that “at the same time” meant the same thing for all observers. Two events that are “at the same time” in one frame may not be “at the same time” in another frame, and vice versa.

Now, some of you may have already realized that this could lead to a problem with the notion of causality, namely, the notion of cause and effect. If an event A is the cause of another event B, then A must happen before B, and B must happen after A. But according to Einstein, the chronological order in which two events happen can depend on the frame of reference!

To make our discussion concrete, consider again the tree planted firmly in the ground, the car moving to the right with speed ½c in the tree-frame, and the ball moving to the right with speed ⅘c in the tree-frame (which corresponds to ½c in the car-frame).

The mystery of the speed of light

Now the surprising thing about the Galilei–Newton theory that we have been discussing so far is that it is wrong. It is not wrong in the sense that it is completely wrong, but wrong in the sense that there is a limit to its applicability and in certain cases it does not work. And that case involves the speed of light.

The speed of light in a vacuum is very very fast. It is 299 792 458 meters per second, or roughly 3 × 108 meters per second. Since we do not want to end up writing this big number repeatedly, we will just represent it with the letter c. To give you an idea just how fast this is, it is fast enough to circumnavigate the Earth seven and a half times per second. The time it takes for light to travel 30 centimeters (about a foot) is only 1 nano-second, which is 0.000 000 001 seconds.

Because c is so large, it was very difficult to measure what it was for a long time. Galileo himself tried it but did not succeed. But by the end of the nineteenth century, the technology had advanced to the point that very accurate measurements of c were possible.

Albert Einstein's Special Theory of Relativity, or Special Relativity for short, came into being in 1905 in a paper with the unassuming title of “On the electrodynamics of moving bodies.” As the title suggests, Special Relativity is a theory of “moving bodies,” that is: motion. In particular, it is a theory of how motion is perceived differently by different observers. Since motion is the process in which an object's location in space changes with time, any theory of motion is also a theory of space and time. Therefore, Special Relativity can be said to be a theory of how space and time are perceived differently by different observers. The “electrodynamics” part of the paper title refers to the fact that the theory has something to do with light, which is an electromagnetic wave. As we will learn in this book, the speed of light in vacuum, which we will call c, plays a very special role in the theory of relativity.

Einstein (1879-1955) was not the first to construct a successful theory of motion. Building upon pioneering work by Galileo Galilei (1564-1642), Sir Isaac Newton (1642-1727) had constructed theories of motion and gravity which were spelled out in his famous book Philosophiae Naturalis Principia Mathematica, which is so famous that when people say the Principia, it is understood that they are referring to Newton's book.

This concludes Part III of this book. I hope you have been able to grasp the basic logic of where the equation E = mc2 comes from. To summarize the important points:

• In Einstein's relativistic dynamics, the state of motion of an object is represented by an arrow called the energy–momentum vector. The vector's time-component (vertical component) is the energy, and the space-component (horizontal component) is the momentum, and they represent what might be called the “tenacity of the motion” or the “tendency of the motion to continue as is” in their respective directions in spacetime.

• The energy–momentum vector depends on the frame from which the observation is being made. However, the area of the diamond with the energy–momentum vector as one of its sides and the diagonals at 45° from the horizontal is invariant and equal to (mc)2 where m is the object's mass.

• Changes in the motions of objects are represented by changes in their energy–momentum vectors. In a system of interacting objects, the energy–momentum vector of each individual object will change via interactions, but the total energy–momentum vector of the system will be conserved.

• […]

Addition of velocities

Solve the following problems pictorially using spacetime diagrams. (Do not resort to the equation provided in the endnotes of Chapter 4.)

1. A tree is at rest on the ground, and a car is traveling to the right at speed ½c. If a ball is traveling to the left at speed ½c in the tree-frame, what is its speed in the car-frame?

2. A tree is at rest on the ground, and a car is traveling to the right at speed ½c. If a ball is traveling to the left at speed ⅓c in the tree-frame, what is its speed in the car-frame?

3. A tree is at rest on the ground, and a car is traveling to the right at speed ½c. If a ball is traveling to the right at speed ¼c in the tree-frame, what is its speed in the car-frame?

4. A tree is at rest on the ground, and a car is traveling to the right at speed ½c. If a ball is traveling to the right at speed ¼c in the car-frame, what is its speed in the tree-frame?

5. A tree is at rest on the ground, and a car is traveling to the right at speed ⅓c. If a ball is traveling to the right at speed ⅓c in the car-frame, what is its speed in the tree-frame?

Basic questions

Let us recall the basic questions we asked about the motion of an object in section 2.1, namely:

As we have seen, in order to answer these questions we must first choose a frame, and the answers depended on our frame choice.

Let us actually try this out. Consider the car moving along a straight horizontal road as shown in the figure. In the tree-frame, at every instant in time the clock fixed to the tree (though it's not drawn on the tree, assume that it is) will give some reading t (in seconds) and at the same instant the car will be somewhere along the road at some position x (in meters). The figure shows the sequence of this position from time t = 0 seconds to t = 4 seconds in 1-second intervals. As you can see, we can tell from the figure that:

The energy-momentum vector

Next, let us consider the case when the velocities of objects are close to the speed of light c, and the Lorentz transformation must be used for relating observations from different inertial frames.

Assume that a baseball which was initially at rest is accelerated to half the speed of light by an impact with a bat at point A on the spacetime diagram. The questions we wanted answers for in Chapter 11 were:

As in the Newtonian case, let's represent the “state of motion” of an object with a vector on the spacetime diagram. First, to represent the “state in which the object is at rest,” we follow the Newtonian case and use a vector pointing vertically up from the spacetime origin with length proportional to the object's mass. (See the figure on the opposite page, top-left.) Next, the “state in which the object is moving with velocity v” is the same as “the state in which the object is at rest” but observed from an inertial frame moving at velocity –υ relative to the first.

Questions about motion

When studying the motion of an object, the most basic questions we would like to ask are things like:

In order to answer these questions, we need to know the object's location in space at each instant in time, so that we can keep track of how it is changing as time progresses. If we find that the object's location in space is not changing with time, that is, if it stays at the same place, then we can say that it is “at rest,” while otherwise we can say that it is “moving.” If the object is “moving,” we can specify its direction of motion by saying things like “it is moving to the left” or “it is moving to the right,” and we can figure out its speed by determining by how much its location in space is changing per unit time.

Once we have these basic questions under control, we can then start to ask more advanced questions like: