Biographies & Memoirs

Chapter 11

The Equation

In This Chapter

bullet Understanding the relativity of mass

bullet Putting E = mc2 on paper

bullet Waiting for a reaction

bullet Becoming a professor

I n September of 1905, only three months after he published his paper on special relativity, Einstein sent off the last paper of his miracle year. This paper, only three pages long, was published in November.

In his beautiful paper, which Einstein titled “Does the inertia of a body depend on its energy content?”, Einstein says that the results of his previous investigation “led to a very interesting conclusion, which will be derived here.” What this technical expert third class at the Bern patent office deduced was that mass and energy are equivalent. Einstein condensed his deduction into one simple and powerful equation, E = mc2.

In this chapter, I show you how Einstein used ideas that were commonly known at the time to get to his uncommon conclusion. I show you how this paper (along with the other contributions Einstein made during his miracle year) led to him finally getting a professorship that allowed him to have more time to work on his amazing ideas, thought experiments, and theories.

Bringing Mass into the Equation

When physicists use the word energy (the big E in the equation), they mean the ability to do work. The c stands for the speed of light, which is constant throughout the universe, regardless of how you move or where you are. (I discuss this fact in detail in Chapters 9 and 10.) What about the m in the equation? We know it stands for mass, but what exactly does that mean? Like the word energy, mass entered our everyday language and lost some of its initial meaning. We tend to use the word interchangeably with weight, which is not entirely accurate.

Measuring our laziness

Mass measures the resistance that you feel when you try to change the motion of an object. This resistance is called inertia.

We use the word inertia in our everyday language to indicate sluggishness or inaction. To overcome it, you need to get up and get going. As I explain in Chapter 4, Isaac Newton’s first law of motion is that objects have a tendency to stay where they are, either resting or in uniform motion. His second law of motion states that to overcome inertia — to get objects to change their motion — you need to apply a force.


Mass is a measure of inertia, a measure of an object’s laziness, if you like — its desire to maintain the way it is moving (or staying still).

But keep in mind that there is a second way to measure mass — according to weight, the gravitational attraction of an object to the Earth. Physicists distinguish between the two measurements of mass, calling the first one inertial mass and the second one gravitational mass. In Chapter 12, I tell you more about this distinction and show you that these two masses are actually the same. (Einstein used this fact as the basis for his general theory of relativity.)

The name inertial mass is very descriptive. A massive object is one with a large inertial mass, presenting a large resistance to any attempts to change its motion. It’s more difficult to move than an object with small inertial mass, and, if it’s already moving, it’s more difficult to stop than a light object. We’re all familiar with this idea. However, sometimes the rough surfaces objects move on or the rubbing of their internal components complicates their motions. It’s easier to see the differences between large and small inertial mass in space, where objects have no drag or friction.

For example, during the Hubble Space Telescope servicing mission in 2002, astronauts had to install the Advanced Camera for Surveys, a large new camera with a mass about the same as that of a golf cart. During the mission, the astronauts had a difficult time moving it and positioning it correctly because of the camera’s large inertial mass. Pushing it was about as difficult as pushing a golf cart on smooth level ground on Earth if the wheels are well-lubricated.


Gravity doesn’t affect an object’s inertial mass. The camera’s inertial mass is a property of the camera, which stays the same regardless of where it is. You’d have a hard time moving a cart carrying a massive satellite that’s being readied for launch from Earth. And an astronaut in the cargo bay of the Space Shuttle would have the same difficulty moving it in space (see Figure 11-1). The satellite’s large inertial mass is the same here on the ground or up in space.

Figure 11-1: An astronaut in the cargo bay of the Space Shuttle has the same difficulty maneuvering a massive object as we do here on the ground.

Figure 11-1: An astronaut in the cargo bay of the Space Shuttle has the same difficulty maneuvering a massive object as we do here on the ground.

Courtesy NASA

The title of Einstein’s E = mc2 paper — “Does the inertia of a body depend on its energy content?” — makes more sense when you know the meaning of inertial mass. He is asking the reader if the inertial mass of an object changes with the amount of energy in the body. He shows in his paper that it does.

Realizing that mass is relative

Einstein discovered that the mass of an object is relative. In Chapter 10, I discuss how space and time are interconnected. In a nutshell, when you and I are standing still, we are moving together through time. If you start moving relative to me, I see that you are converting part of your motion through time into motion through space. If you keep moving faster and faster, you’ll be converting more of your motion through time into motion through space.


If you could convert all your motion through time into motion through space, you’ll be moving at the speed of light, in essence stopping time. But you can’t completely stop time. Which means that you can’t move at the speed of light. Nothing, except light, can. For light, there is no motion through time; all its motion is through space, at the speed of light (c), of course.

Suppose that you are traveling in a very advanced interstellar spaceship that has already reached 0.99c. What happens when your skipper wants to get even closer to c? To accelerate, his engine has to supply the force that overcomes the ship’s inertia and increases the speed. That’s how the skipper reached his current speed. But now the ship doesn’t respond as easily. It has more inertia, and he needs to supply a large force to achieve a small increase.

The ship’s inertial mass, its resistance to being accelerated, increases as the speed increases, making it more and more difficult to accelerate the ship to a speed closer to the speed of light. If the ship could somehow reach the speed of light, its inertial mass would be infinite.


According to Einstein’s special relativity, then,

The inertial mass of an object in motion relative to us increases with the speed, becoming infinitely large at the speed of light.

Because an infinite mass would require an infinite force to accelerate, which means that you would need an infinite amount of energy to supply this force, we conclude that

No object can travel at the speed of light.

Choosing c2

Einstein’s famous equation, E = mc2, involves the square of the speed of light, c. Why did he choose c-squared and not c-cubed? Einstein’s equation is an energy equation, and it must have the general properties of energy.

The properties of energy began to be discovered during Newton’s lifetime, starting when his contemporary, Christian Huygens, was preparing a paper to present before the Royal Society of London in January of 1669.


Christian Huygens

The Dutch scientist Christian Huygens was one of the most famous scientists in Europe and certainly the pride of his homeland. He came from good stock: His father was Constantin Huygens, a towering figure in Dutch literature.

The young Huygens was influenced by the ideas on space of his friend, the French philosopher and mathematician René Descartes. Huygens published several important mathematical papers that made him well known before he was 30 years old.

While helping his brother build a telescope, Huygens became interested in optics and invented new methods to grind lenses. He also became interested in astronomy and discovered the Orion Nebula, a large cloud of interstellar dust that still fascinates amateur astronomers all over the world. (Today, we know that this cloud is one of many birthplaces of stars.) Using a telescope that was 7 meters (23 feet) long, he discovered the rings of Saturn and its largest moon, which he named Titan.


In the paper, Huygens talked about the collisions of objects and tried to clarify confusion that existed at the time about the physics of these collisions. Specifically, physicists were arguing about the motion of two small metal balls that swing and collide repeatedly. (You’ve no doubt seen a modern version of such a device in novelty shops, similar to the one shown in Figure 11-2.) The scientists couldn’t understand why, after one such collision, the incoming ball stops completely while the struck ball swings back to the same height that the first ball had before the collision. Nothing in their existing body of knowledge about energy and speed explained why, for example, the first ball would not bounce back at half the speed as before the collision, while the second one would take off with the other half of the speed.

Figure 11-2: The total energy of the swinging steel balls colliding with each other stays unchanged.

Figure 11-2: The total energy of the swinging steel balls colliding with each other stays unchanged.

Huygens clarified the behavior of the balls in his paper. He explained that if you take the product of the mass of each ball, multiply it by the square of the speed, and add these two quantities, the sum is always the same before and after each collision. The product of the mass times the square of the velocity of each ball gives you the energy of motion of the ball.


The energy of each ball changes when the balls collide, but at each collision, the energy is exchanged between the balls so that the total amount stays the same. This idea became the powerful principle of conservation of energy, which I discuss in Chapter 5. The constant exchange of energy back and forth, while keeping the total amount fixed, explains why the struck ball has to always take off and reach the same height, time after time.

According to Huygens, the mass of an object times the square of the speed at which it is moving gives you the object’s energy of motion. (Actually, it gives you one-half of that value, but that fact isn’t important at this point.) Einstein’s energy equation is also the product of the mass of the object times the square of the speed, but in this case, it’s not the object’s speed but the speed of light. Why the speed of light and not the speed of the object? Read on to find out.

Formulating E = mc2

In the short paper that features Einstein’s most famous equation, he showed that if an object at rest in the laboratory emits light (such as an atom undergoing radioactive decay), its energy changes, because part of the energy is carried away by the emitted light.

From the conservation of energy principle, Einstein knew that the energy the emitted light carried away came from the object itself. The total energy of the object decreased. That much was known. But Einstein took an additional step. He used his special relativity equations to calculate the energy difference of an atom emitting light while at rest in the laboratory and while the body is moving relative to the laboratory. His calculations showed him that the mass of the atom decreased after the emission of light.

At this point Einstein did what he did better than anyone else: He extended his statement. He analyzed the emission of light by an atom, saying that its mass decreases every time it gives off light.


Then he made the big step: the generalization. He was not simply making a statement about an atom emitting light. He said that his discovery applied to all matter. Einstein stated that the mass of a body is a measure of its energy content, and that this fact is always true. According to Einstein,

An object’s mass is a form of energy, and energy carries mass. Mass and energy are two forms of the same thing.

These two sentences are the meaning of E = mc2.

In 1905, when this patent clerk was making his bold statement, scientists were only beginning to accept the existence of atoms. Radioactive decay, the spontaneous emission of light and charged particles from atoms, had been discovered only five years earlier.

Questioning his own conclusions

Einstein ends his short paper with the statement that perhaps it would be possible to test his theory using bodies with an energy content that changes dramatically, like radium salts. “If the theory agrees with the facts,” he wrote in his paper, “then radiation carries mass between emitting and absorbing bodies.”

A few weeks later, he wrote to his friend Conrad Habicht about the paper, expressing some doubts. He told him that his conclusion about light carrying mass with it was a seductive idea, but that for all he knew, the Lord might be playing a trick on him and laughing about it.


The Lord wasn’t playing tricks. What Einstein discovered not only explained the workings of the sun but also made possible technological advances ranging from smoke alarms to PET scanners. It’s hard to think of an area of physics developed over the last 100 years where E = mc2didn’t play an important role. Forty years after Einstein proposed the equation, World War II precipitated a dramatic demonstration of E = mc 2 with the development of the nuclear bomb and its use over two cities in Japan.

But all that technology had to wait until the correct understanding of the atom was in place, which was decades after Einstein wrote his paper. In 1905, scientists had only a very primitive idea of what an atom was. Just a couple years earlier, Einstein himself provided several irrefutable proofs of the existence of the atom for the diehards that weren’t accepting its existence.


Why c 2?

Einstein’s E = mc2 tells us that energy and mass are equivalent. You can change one into the other. And c 2 is the conversion factor telling you how they are linked. When you convert inches to centimeters, both measuring length, the conversion factor is 2.5. When you convert mass into energy or energy into mass, the conversion factor is the square of the speed of light. Because c is so large (about 300,000 kilometers per second) and its square much larger (90,000,000,000 kilometers2 per second2), a tiny amount of mass can be converted into a huge amount of energy.


Running to gain mass

As Einstein said, the mass of an object is a form of energy, and energy is a form of mass. A bird standing on a tree branch has less energy than when it is flying. If you could measure the bird’s mass with extreme precision, you’d find that it’s very slightly larger when flying than when sitting still. But a single feather from the bird’s wing has a mass billions of times larger than the mass increase due to the energy of motion. Similarly, when you run, your mass increases because of your motion. However, you lose more mass by radiating heat and sweat than you gain by running.

Consider this example: Take two small balls and join them with a spring. Pull them apart until the spring breaks. If you had a super-precise balance, you could measure the mass before and after you break the spring (see Figure 11-3). You’d find that the mass of the two balls is smaller when they’re pulled together by the spring than it is after you break the spring and pull them apart. If you’re using marbles with a mass of 100 grams (3.5 ounces), the mass increase is about one-billionth of a gram. A balance with the precision to measure this small increase doesn’t exist.

Figure 11-3: The total mass of the balls and the spring is smaller when the balls are together than when they are apart.

Figure 11-3: The total mass of the balls and the spring is smaller when the balls are together than when they are apart.

Cutting a strong thread

During the 1920s, physicists discovered the correct composition of the atom: a tiny nucleus made up of two types of particles — the positively charged protons and the electrically neutral neutrons — which is surrounded by a cloud of negative electrons. During this decade, they also developed quantum physics, the physics of the atom, which had started with Einstein’s 1905 paper on the photoelectric effect. (For the details on these developments, see Chapters 15 and 16.)

Physicists found that the particles that form the nucleus — the positive protons and the neutral neutrons — are held together by a very strong force, the nuclear force. This nuclear force is the thread that keeps them together. If you could cut this thread, they would fly apart, because the positive protons repel each other.

Much like the two balls held together by a spring, the mass of an atom’s nucleus is smaller than the mass of its neutrons and protons. Einstein’s E = mc2 says that mass is a form of energy. The nucleus, with less mass, has less energy than its component particles when they are apart.

Here’s how this energy difference comes about. The nuclear force provides a very strong thread tying these particles together in the nucleus. You’d need to do a great deal of work (use a large amount of energy) to pull them apart. This situation is similar to the case of two powerful magnets sticking together with their north and south poles facing each other. You have to work hard to separate them (see Figure 11-4).

Figure 11-4: You need to work hard to pull these magnets apart.

Figure 11-4: You need to work hard to pull these magnets apart.

After you manage to separate the two powerful magnets, it’s easy to bring them together again. They actually do the work for you. Not only do they come together on their own, but they pull you along. To do that, they must use their own energy. So when they are together, they have less energy than when they are apart.


Did anyone read it?

Einstein knew that his work on relativity was “revolutionary.” He said so in a letter he wrote to his friend Conrad Habicht in May of 1905, which became one of the most famous letters of all time. At the time, Einstein was working on four papers on the existence of atoms and molecules (which he later sent to the Annalen der Physik), and on the special theory of relativity. The E = mc2 paper wasn’t on his mind. He thought of it only after he’d submitted his relativity paper.

Einstein’s sister, Maja, wrote in her book about her brother’s early years that after the publication of his papers of 1905, Einstein was anxious to see the response of the physics community to his work. He didn’t have to wait long. In May of 1906, Einstein said that his papers were receiving much acknowledgment and generating new investigations, and that he had even received a letter about the relativity papers from the great Max Planck.



Like the magnets sticking together, the protons and neutrons sticking together to form the nucleus have less energy than the separate particles. The “missing” energy is the energy that keeps the nucleus together, which is called the binding energy of the nucleus.

According to Einstein’s E = mc2, then,

Things sticking together have less mass than when you pull these same things apart.


Weighing a speck of soot

The smallest mass ever measured with a scale is that of a speck of soot. The “scale” was made with miniature tubes of carbon (called carbon nanotubes) as springs. The scientists who made the measurement at Georgia Tech placed the speck on one of the tiny tubes and started vibrating the tube by applying an electric charge to a nanotube placed near a probe with the opposite charge. The speck weighed down the vibrating nanotube, and the researchers were able to calculate the mass of the speck. The mass of the speck of soot was 22 femtograms, or 22 millionths of a billionth of a gram. However, measuring a mass difference that small is still not possible.


Introducing Professor Einstein

After 1905, Einstein’s isolated and quiet life as a lone scientist working as a clerk in a patent office came to an end. Max Planck, the well-known German physicist who had introduced the idea of the quantum of energy only five years earlier, became very interested in Einstein’s relativity. In 1906, he became the next scientist to write a paper on relativity.

Soon, and largely because of Planck, other scientists took notice. Well-known physicists and recent PhD’s came to Bern, wanting to meet Einstein and to work with him for a couple of months. Einstein’s sister, Maja, wrote that soon after the papers of 1905, he started receiving letters that were addressed to “Professor Einstein at the University of Bern.”

Einstein was quite proud of his growing recognition. In May of 1906, he wrote to his friend Maurice Solovine telling him how his work was becoming highly regarded and that even Professor Planck had written to him about his theories.

Looking for a job again

Encouraged by his growing fame, Einstein decided to try again for an academic career. Things would surely be different than they were when he first graduated, he thought, with all these important papers generating excitement in the scientific community. He had also just received his PhD with the acceptance of his thesis at the University of Zurich, so he could now be called Dr. Einstein. He found that using his title was sometimes advantageous. He commented in a letter that his title considerably smoothed relationships with people.

The first step toward entering the academic world in Germany and Austria at the time was to become what was called a Privatdozent, an instructor at a university who received no salary but collected fees from the students. To be hired as a Privatdozent, you had to present an original paper and give a demonstration lecture.

Einstein sent an application to the University of Bern, encouraged by his PhD thesis advisor, Professor Alfred Kleiner. If selected, Einstein would still be able to keep his day job at the patent office.

Einstein waited impatiently for the reply — so impatiently that he couldn’t resist writing letters to the dean and to the head of the faculty, promising that if hired, he would develop an exciting course for the students.

Weeks went by with no response. Finally, the letter arrived in the mail. It was a rejection.

Cutting through red tape

Einstein’s application was rejected by the University of Bern because he hadn’t included the required original scientific paper. This mistake was strange because Einstein had more papers than anyone else applying for this type of position. He also was very interested in entering the academic world, and this position was his ticket. And he was clearly already experienced in preparing complex papers for publication in scientific journals. An application for a position would be comparatively simple. Nevertheless, Einstein messed it up.

Einstein must’ve been unhappy with the rejection at the time. Years later, however, he referred to it as an amusing example of academic red tape.

When Professor Kleiner heard about the rejection, he convinced Einstein to resubmit. Einstein sent in a more complete application, and the university officials, now worried that they were rejecting a rising star, reconsidered and hired him. Einstein quickly accepted the offer. He was 29.

The first course that Einstein taught at the University of Bern was “The Theory of Radiation,” offered during the winter term of 1908–1909. Only four students enrolled, and all were his own friends. The class was held at night, after he left his day job at the patent office. During the second semester, he had only one student.

Einstein wasn’t happy with his night job. Teaching so few students wasn’t that interesting, and he still had to prepare the lectures as if he were teaching a full class. On top of that, he now had very little time to do research, which is what he really loved to do. But he was determined to become a professor, so he kept going.

Facing politics in Zurich

In 1909, shortly after Einstein started working as a Privatdozent at the University of Bern, Professor Kleiner set out to hire a theoretical physicist at the University of Zurich. He had two good candidates for the position: his former assistant, Frederich Adler, and his former PhD student, Einstein.

During committee deliberations for the position, it became clear that Einstein was going to be the top candidate. In an attempt to cushion the bad news, Kleiner told Adler that he wasn’t on the list of finalists. Adler wrote to his father that day, telling him that he wasn’t going to get the position but that the man who most likely would get it was his former classmate Einstein. Adler said that, apart from his own disappointment, he would be very pleased if Einstein did get it, because people in Switzerland and even in Germany were feeling that it was a scandal to have a man like that sitting in the patent office.

The final vote of the committee was ten for Einstein and one abstention. The committee forwarded its recommendation to the university administration for final approval.


The other Dr. Einstein

Einstein’s sister, Maja, attended the University of Bern at the time that Einstein was a Privatdozent there and would occasionally drop in for one of his classes. She’d attended the University of Berlin for two years but transferred to Bern.

On December 21, 1908, Maja received her PhD magna cum laude in Romance languages from the university, becoming the second Dr. Einstein in the family.


The administration was opposed to Einstein’s candidacy on political grounds. For the most part, members of the administration were Social Democrats who favored Adler, a member of the same party. Adler felt uncomfortable with this situation. He didn’t want to be appointed for political reasons and knew that the faculty hadn’t chosen him. He decided to remove his name from consideration. In his letter to the administration, Adler said that if the university had the opportunity to hire a man like Einstein, it would be absurd to hire him instead. He added that his abilities were in no way a match for Einstein’s.

A short time later, Einstein received an offer for a position as a professor at the University of Zurich. He declined because he didn’t like the salary, which was about half of what he was making at the patent office. Kleiner went back to the administration, and they agreed to match Einstein’s current salary. Einstein accepted the second offer.

Learning to teach

Einstein resigned from the Bern patent office effective October 15, 1909. His appointment at the University of Zurich was as associate professor of theoretical physics with a salary of 4,500 francs, plus certain lecture and examination fees. He started his university position the same day that he left the patent office.

His duties included teaching six to eight hours of classes and seminars per week, as well as advising students. His first graduate student was Hans Tanner. (But Tanner didn’t get his PhD with Einstein, because by the time he finished his studies, Einstein had moved on to another university.)

Einstein wasn’t a good lecturer during his early years as a professor. He came to class wearing pants that were too short and carried around a piece of paper containing his lecture notes.

With time, his lectures improved, and eventually, his students came to like him. They liked his informal style and his open, easygoing approach. He allowed his students to interrupt the lecture at any time with any questions, and he even went along with the students to the local coffee houses where he sat and explained physics to them.

When Einstein was offered a position at the University of Prague in 1910, the Zurich students sent a petition to the university administration asking them to do whatever they could to keep Einstein there. Right away, the university raised his salary to 5,500 francs and asked the department to look into reducing his teaching load. Einstein turned down the offer from Prague.

Einstein taught at the University of Zurich from October 1909 to March 1911. In these two years, he published eleven papers on theoretical physics, a remarkable output for a new professor. Most of these papers dealt with the problems of radiation and were connected with his idea of the quantum of light that he had proposed in his first paper of the miracle year.

Even while at the patent office, Einstein had been thinking about extending his special theory of relativity to include nonuniform motion. As he found out a couple years later, this task wasn’t going to be easy.


Honorary degree

One day, a short time before Einstein started his new position at the University of Zurich, he received a large envelope at the patent office that contained a stylish sheet of paper with ornate Latin printing on it. Einstein thought that it was some sort of advertisement and threw it in the trash without reading it. Later, Einstein learned that the piece of paper was an invitation to receive an honorary doctorate at the University of Geneva.

Because Einstein didn’t respond to the invitation, the university officials asked one of his former students, Louis Chavan, to convince him to attend the ceremonies in Geneva. Chavan succeeded in convincing Einstein but didn’t tell him what the purpose of the trip was.

When Einstein arrived at the hotel where most of the honorees were staying, he learned that they were all there to receive an honorary degree. Einstein hadn’t brought proper attire and ended up marching in the procession and attending the entire ceremony wearing street clothes and a straw hat, while everyone else wore academic robes.


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