The dark and the light

The Universe, as we gaze at it at night, is a vast, predominantly dark and for the most part unknown expanse, interspersed with myriads of pinpricks of light. When we consider that these light spots are at enormously large distances, we realize that they must be incredibly bright in order to be visible at all from so far away. Occasionally, some of these specks of light get much brighter, and some of them which were not even seen with the naked eye before become in a few days the brightest spot in the entire night sky, their brightness having increased a billion-fold or more against the immutable-looking dark background. Thus, we have come to realize that the Universe is characterized by what Renaissance artists called chiaroscuro, referring to the contrast between light and dark, which is both stark and subtle at the same time. In the case of the Universe, the contrasts can be enormous and surprisingly violent, as well as of a subtlety which beggars the imagination. In this book we will focus on these contrasts between the vast, unknown properties of the dark Universe and its most violent outpourings of energy, light and particles.

According to current observations and our best theoretical understanding, the Universe is made up of different forms of mass, or rather of mass-energies, since as we know from special relativity, to every mass there corresponds an energy E = mc2 and vice versa, where E is energy, m is mass and c is the speed of light.

Einstein's celebrated Theory of Relativity is one of those scientific theories whose name is so famous that most people have heard of it, but very few people actually know what the theory says, or even what the theory is about. You, too, have probably heard the name, perhaps referred to in a science fiction novel or movie, even if you do not know much about it. And you may have received the impression that it is a very esoteric and difficult theory that could only be understood and appreciated by a select few.

The aim of this book is to show you that that impression is wrong. The Theory of Relativity comes in two flavors, the Special and the General, and if we limit our attention to the Special Theory of Relativity (SR), which is a theory of motion, it is not a particularly difficult theory at all and can be understood by anyone, perhaps “even a child.” By “be understood” here, I do not mean that anyone can develop a vague idea of what the theory is saying, but that anyone can understand it in its full glory beginning from its basic tenets to all of its logical consequences.

All the problems in this chapter are qualitative and you will be able to solve them if you can read spacetime diagrams. Try them out to test your understanding of Special Relativity.

Reading the spacetime diagram

Street lamps

Five street lamps, numbered 1 through 5, are located on a straight line along the x-axis equal distance apart as shown in the figure. They turn on at points A, B, C, D, and E, respectively, on the spacetime diagram.

1. In what order do the lamps turn on in the ground-frame?

2. In what order does the light from the lamps reach the observer at the origin x = 0?

3. A car is moving to the right at constant speed relative to the ground. At t′ = t = 0, it is at x′ = x = 0. The space- and time-axes in the moving frame of the car are tilted with respect to those of the rest frame as shown in the spacetime diagram. In what order do the lamps turn on in the car-frame?

4. In what order does the light from the lamps reach the observer riding the car?

5. Where is the car when the light from street lamp 4 reaches it?

The mass-momentum vector

Situations that can be addressed within Newtonian dynamics are cases in which the velocities of objects are much slower than the speed of light c, and the Galilei transformation suffices as the transformation from one inertial frame to another.

Now, we would like to answer the questions posed in the previous section using diagrams. But for that we must be able to represent pictorially what we mean by the term “same impact.” In the current case, the “impact” we are talking about is that which accelerates the baseball from “a state in which it is at rest” to “a state in which it is traveling at +1 meters per second.” But this in turn means that we must first be able to represent pictorially the “state of the baseball moving at velocity υ” for generic velocities υ.

So how can we do this? Of course, the motion of any object is described by its worldline on the spacetime diagram, and the object's velocity is encoded in the slope of the worldline. But there are two reasons why the worldline is not an appropriate representation of the “state of motion” of an object:

1. Depending on whether the object is a baseball, a ping-pong ball, or a bowling ball, the amount of “stuff” that is moving is different, but the worldline does not give you that information.

2. […]

This brings us to the end of my exposition of Einstein's Special Theory of Relativity (SR). I have attempted to explain everything that is usually explained using equations using drawings only so that you can literally see what I am talking about. I hope you have found this approach more tractable, eye-opening, and fun.

Partly because of its name, Einstein's “Theory of Relativity” is often misunderstood to have discarded Newton's notions of space and time that were both “objective” and “absolute,” and to have pronounced that both space and time were “relative,” and even “subjective” concepts. In truth, Einstein was a firm believer in objective reality, and SR assumes the existence of an “objective” and “absolute” spacetime. All SR is claiming is that when the motion of objects in spacetime is observed from different inertial frames, things like velocity and length will be frame-dependent. And this dependence comes about because the way the time- and space-axes are introduced into the “absolute” spacetime differs from inertial frame to inertial frame. The frame-dependence of the time-axis already existed in Newton's theory, and as a consequence velocity, not surprisingly, was frame-dependent. In SR, however, in addition to a frame-dependent time-axis, the concept of simultaneity depends on the frame and results in a frame-dependent space-axis also.

This book explains Einstein's Special Theory of Relativity (SR) using diagrams only. Readers who are used to thinking of physics as a vast labyrinth of equations may feel somewhat uneasy about this unconventional approach and fear that it risks losing important information about SR that can only be conveyed via equations. However, this fear is not only unfounded but actually reversed: it is the equations that fail to convey the essence of SR that diagrams can easily display right in front of your eyes. After all, SR, and also the General Theory of Relativity (GR), are about the geometry of the spacetime that we inhabit, and what can best describe geometry if not diagrams? Equations are simply inadequate, to wit, one diagram is worth a thousand equations.

So if you are a reader for whom equations are anathema, rest assured that you will get much more out of this book than any physics student will get out of a textbook full of equations. If you are a physics student, this book will provide you with a deep enough understanding of SR that will enable you to reproduce any equation you may need from scratch, if such a need ever arises, and also prepare you for GR as well.

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).

Recall the questions about motion that I listed as the “more advanced” ones in section 2.1:

Let us ask these questions of the motion of a baseball whose worldline is shown here on the spacetime diagram. The worldline has a kink at point A, the time-coordinate of which is t = 3 seconds, so we can tell that the velocity of the baseball changed at t = 3 seconds. The baseball has been hit by a bat at this point. The worldline before A is vertical, so the baseball was at rest before being hit. From the slope of the worldline after A, we can tell that the velocity of the baseball after being hit was +1 meters per second. So the answers to the above questions in this case are:

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.

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