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Physics + MathPhysics & Math

The Theory Behind the Equation

Imagine a police officer chasing after a speeding motorist. If he drives fast enough, the officer knows that he can catch the motorist. Anyone who has ever gotten a ticket for speeding knows that. But if we now replace the speeding motorist with a light beam, and an observer witnesses the whole thing, then the observer concludes that the officer is speeding just behind the light beam, traveling almost as fast as light. We are confident that the officer knows he is traveling neck and neck with the light beam.

ByMichio KakuNova

But later, when we interview him, we hear a strange tale. He claims that instead of riding alongside the light beam as we just witnessed, it sped away from him, leaving him in the dust. He says that no matter how much he gunned his engines, the light beam sped away at precisely the same velocity. In fact, he swears that he could not even make a dent in catching up to the light beam. No matter how fast he traveled, the light beam traveled away from him at the speed of light, as if he were stationary instead of speeding in a police car.

But when you insist that you saw the police officer speeding neck and neck with the light beam, within a hairsbreadth of catching up to it, he says you are crazy; he never even got close. To Einstein, this was the central, nagging mystery: How was it possible for two people to see the same event in such totally different ways? If the speed of light was really a constant of nature, then how could a witness claim that the officer was neck and neck with the light beam, yet the officer swears that he never even got close?

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Newton / Maxwell
Einstein realized that the world described by Isaac Newton (left), in which one could add and subtract velocities, and that described by James Clerk Maxwell, in which the speed of light is constant, could not both be right.
(Newton) Courtesy of the Trustees of the Portsmouth Estate

Newtonian theory was a self-contained system, resting on a few assumptions. If only one of these assumptions were changed, it would unravel the entire theory in the same way that a loose thread can unravel a sweater. That thread would be Einstein's daydream of racing a light beam.

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Special relativity is born

One day around May of 1905, Einstein went to visit his good friend Michele Besso, who also worked at the patent office, and laid out the dimensions of the problem that had puzzled him for a decade. Using Besso as his favorite sounding board for ideas, Einstein presented the issue: Newtonian mechanics and Maxwell's equations, the two pillars of physics, were incompatible. One or the other was wrong. Whichever theory proved to be correct, the final resolution would require a vast reorganization of all of physics.

Einstein in the Bern patent office in 1904, just months away from the brilliant insight that led to his theory of special relativity—and, a few weeks later, to E = mc2
© Einstein Archives, Hebrew University of Jerusalem

Einstein went over and over the paradox of racing a light beam. He would later recall, "The germ of the special relativity theory was already present in that paradox." They talked for hours, discussing every aspect of the problem, including Newton's concept of absolute space and time, which seemed to violate Maxwell's constancy of the speed of light. Eventually, totally exhausted, Einstein announced that he was defeated and would give up the entire quest. It was no use; he had failed.

Although Einstein was depressed, his thoughts were still churning in his mind when he returned home that night. In particular, he remembered riding in a streetcar in Bern and looking back at the famous clock tower that dominated the city. He then imagined what would happen if his streetcar raced away from the clock tower at the speed of light. He quickly realized that the clock would appear stopped, since light could not catch up to the streetcar, but his own clock in the streetcar would beat normally.

Clock Tower
A streetcar trundles below the clock tower in Bern that Einstein made famous with his thought experiment about racing a light beam.
© Underwood & Underwood/CORBIS

Then it suddenly hit him, the key to the entire problem. Einstein recalled, "A storm broke loose in my mind." The answer was simple and elegant: time can beat at different rates throughout the universe, depending on how fast you moved. Imagine clocks scattered at different points in space, each one announcing a different time, each one ticking at a different rate. One second on Earth was not the same length as one second on the moon or one second on Jupiter. In fact, the faster you moved, the more time slowed down. (Einstein once joked that in relativity theory, he placed a clock at every point in the universe, each one running at a different rate, but in real life he didn't have enough money to buy even one.) This meant that events that were simultaneous in one frame were not necessarily simultaneous in another frame, as Newton thought. He had finally tapped into "God's thoughts." He would recall excitedly, "The solution came to me suddenly with the thought that our concepts and laws of space and time can only claim validity insofar as they stand in a clear relation to our experiences.... By a revision of the concept of simultaneity into a more malleable form, I thus arrived at the theory of relativity."

Thank you, I've completely solved the problem.

For example, remember that in the paradox of the speeding motorist, the police officer was traveling neck and neck with the speeding light beam, while the officer himself claimed that the light beam was speeding away from him at precisely the speed of light, no matter how much he gunned his engines. The only way to reconcile these two pictures is to have the brain of the officer slow down. Time slows down for the policeman. If we could have seen the officer's wristwatch from the roadside, we would have seen that it nearly stopped and that his facial expressions were frozen in time. Thus, from our point of view, we saw him speeding neck and neck with the light beam, but his clocks (and his brain) were nearly stopped. When we interviewed the officer later, we found that he perceived the light beam to be speeding away, only because his brain and clocks were running much slower.

The paper that changed everything

The day after this revelation, Einstein went back to Besso's home and, without even saying hello, he blurted out, "Thank you, I've completely solved the problem." He would proudly recall, "An analysis of the concept of time was my solution. Time cannot be absolutely defined, and there is an inseparable relation between time and signal velocity." For the next six weeks, he furiously worked out every mathematical detail of his brilliant insight, leading to a paper that is arguably one of the most important scientific papers of all time. According to his son, he then went straight to bed for two weeks after giving the paper to his wife Mileva to check for any mathematical errors. The final paper, "On the Electrodynamics of Moving Bodies," was scribbled on 31 handwritten pages, but it changed world history.

In the paper, he does not acknowledge any other physicist; he only gives thanks to Michele Besso. It was finally published in Annalen der Physik in September 1905, in volume 17. In fact, Einstein would publish three of his pathbreaking papers in that famous volume 17. His colleague Max Born has written, volume 17 is "one of the most remarkable volumes in the whole scientific literature. It contains three papers by Einstein, each dealing with a different subject and each today acknowledged to be a masterpiece." (Copies of that famous volume sold for $15,000 at an auction in 1994.)

Journal Cover
Volume 17 of the German physics journal Annalen der Physik, in which Einstein published no fewer than three groundbreaking papers at age 26.
Courtesy © 2005 Wiley-VCH, Germany

With almost breathtaking sweep, Einstein began his paper by proclaiming that his theories worked not just for light, but were truths about the universe itself. Remarkably, he derived all his work from two simple postulates applying to inertial frames (i.e., objects that move with constant velocity with respect to each other):

  1. The laws of physics are the same in all inertial frames.
  2. The speed of light is a constant in all inertial frames.

These two deceptively simple principles mark the most profound insights into the nature of the universe since Newton's work. From them, one can derive an entirely new picture of space and time.

Length, like time, is relative

First, in one masterful stroke, Einstein elegantly proved that if the speed of light was indeed a constant of nature, then the most general solution was the Lorentz transformation*. He then showed that Maxwell's equations did indeed respect that principle. Last, he showed that velocities add in a peculiar way. Although Newton, observing the motion of sailing ships, concluded that velocities could add without limit, Einstein concluded that the speed of light was the ultimate velocity in the universe. Imagine, for a moment, that you are in a rocket speeding at 90 percent the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90 percent the speed of light. According to Newtonian physics, the bullet should be going at 180 percent the speed of light, thus exceeding light velocity. But Einstein showed that because meter sticks are shortening and time is slowing down, the sum of these velocities is actually close to 99 percent the speed of light. In fact, Einstein could show that no matter how hard you tried, you could never boost yourself beyond the speed of light. Light velocity was the ultimate speed limit in the universe.

We never see these bizarre distortions in our experience because we never travel near the speed of light. For everyday velocities, Newton's laws are perfectly fine. This is the fundamental reason why it took over 200 years to discover the first correction to Newton's laws. But now imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps one inch in length, for example, although its height would remain the same. Because the passengers in the car are compressed down to one inch, we might expect them to yell and scream as their bones are crushed. In fact, the passengers see nothing wrong, since everything inside the car, including the atoms in their bodies, is squeezed as well.

As the car slows down to a stop, it would slowly expand from one inch to about 10 feet, and the passengers would walk out as if nothing happened. Who is really compressed? You or the car? According to relativity, you cannot tell, since the concept of length has no absolute meaning.

Other scientists came close to discovering relativity before Einstein, including the Dutch physicist Hendrik Lorentz (seated fourth from left) and the French mathematician Henri Poincaré (seated far right, next to Marie Curie). Einstein is standing second from right in this photo from a 1911 conference.
© Hulton-Deutsch Collection/CORBIS

The greatest afterthought in history

Einstein then pushed further and made the next fateful leap. He wrote a small paper, almost a footnote, late in 1905 that would change world history. If meter sticks and clocks became distorted the faster you moved, then everything you can measure with meter sticks and clocks must also change, including matter and energy. In fact, matter and energy could change into each other. For example, Einstein could show that the mass of an object increased the faster it moved. (Its mass would in fact become infinite if you hit the speed of light—which is impossible, which proves the unattainability of the speed of light.) This meant that the energy of motion was somehow being transformed into increasing the mass of the object. Thus, matter and energy are interchangeable. If you calculated precisely how much energy was being converted into mass, in a few simple lines you could show that E = mc2, the most celebrated equation of all time.

the equation
The world's most famous equation, as it appears in modified form in a manuscript on special relativity theory that Einstein wrote in 1912
© Einstein Archives, Hebrew University of Jerusalem

Since the speed of light was a fantastically large number, and its square was even larger, this meant that even a tiny amount of matter could release a fabulous amount of energy. A few teaspoons of matter, for example, has the energy of several hydrogen bombs. In fact, a piece of matter the size of a house might be enough to crack the Earth in half.

Imagine the audacity of such a step ... every speck of dust becoming a prodigious reservoir of untapped energy.

Einstein's formula was not simply an academic exercise, because he believed that it might explain the curious fact discovered by Marie Curie, that just an ounce of radium emitted 4,000 calories of heat per hour indefinitely, seemingly violating the first law of thermodynamics (which states that the total amount of energy is always constant or conserved). He concluded that there should be a slight decrease in its mass as radium radiated away energy (an amount too small to be measured using the equipment of 1905). "The idea is amusing and enticing; but whether the Almighty is laughing at it and is leading me up the garden path—that I cannot know," he wrote. He concluded that a direct verification of his conjecture "for the time being probably lies beyond the realm of possible experience."

Why hadn't this untapped energy been noticed before? He compared this to a fabulously rich man who kept his wealth secret by never spending a cent.

Banesh Hoffman, a former student, wrote, "Imagine the audacity of such a step.... Every clod of earth, every feather, every speck of dust becoming a prodigious reservoir of untapped energy. There was no way of verifying this at the time. Yet in presenting his equation in 1907 Einstein spoke of it as the most important consequence of his theory of relativity. His extraordinary ability to see far ahead is shown by the fact that his equation was not verified ... until some twenty-five years later."

Once again, the relativity principle forced a major revision in classical physics. Before, physicists believed in the conservation of energy, the first law of thermodynamics, which states that the total amount of energy can never be created or destroyed. Now physicists considered the total combined amount of matter and energy as being conserved.

Editor's Notes

*Named for the Dutch physicist Hendrik Lorentz, who calculated them, the Lorentz transformations are the distortions of space and time inherent in the equations for light, i.e., Maxwell's equations. These transformations state that the faster you move, the slower time beats for you and the more compressed you become. (At the speed of light, hypothetically time would stop and distances would shrink to nothing, both of which are impossible.) These transformations are necessary to keep the speed of light a constant in all inertial frames.

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