Was Einstein Smart?
There was a young lady named bright
who could travel faster than light.
She started one day in a relative way
and arrived the preceding night.
” author unknown
Some years ago my then-New Yorker colleague John McPhee was in the process of writing a profile of the physicist Ted Taylor. Taylor had begun his career as a nuclear weapons designer at Los Alamos. One of his creations was the largest purely fission bomb ever detonated ” the so-called Ivy King ” which was exploded in November 1952, in the Pacific.
By 1956 he had become disillusioned with working on nuclear weapons for military purposes and he allowed himself to be recruited to lead a new, visionary, indeed incredible, project ” the Orion ” to design a space ship to be used for planetary exploration that would be powered by a sequence of small nuclear explosions. This enterprise ” which ended unsuccessfully in 1965 ” was located at the then-new General Atomics Company in La Jolla, Calif. Taylor, in turn, began recruiting other scientists to join him in La Jolla. One of his early, and most important, recruits was Freeman Dyson. Dyson, who was a professor at the Institute for Advanced Study in Princeton, was known throughout the physics community as a mathematical genius. This was McPhee’s dilemma.
He had been told by everyone that Dyson was a genius, and he had interviewed him several times, but as a good reporter, McPhee wanted his own evidence that this was true. That is why he approached me. He knew that I had worked with Dyson on the Orion and that we had remained good friends, so he thought I might have an idea of how he could go about this. The problem was that McPhee did not know any mathematics. He was in the position of someone who wanted to write about Bach, but was tone-deaf. Such an individual would have to take the word of others that Bach was a great musician.
I gave the matter some thought and finally came up with a suggestion. I would give McPhee a mathematics problem he could understand, one that I thought was pretty tricky. He could then try to solve it and, after in all likelihood failing, he could go to Dyson and ask for help. He could first find out if Dyson had heard of the problem ” hopefully not ” then he could give it to him and watch what happened.
The problem I gave McPhee was that of the 12 balls. You have 12 balls that appear to be identical. However one of them ” the “guilty” ball ” is either lighter or heavier than the others. You also have a balance scale ” a scale that has platforms on either side of a balance, on which you can put some of the balls. For example, you might try to balance two balls against two balls. If the scale was, say, unbalanced, you would know the guilty ball was among the four. The problem is to devise a method by which, in at most three weighings, you can find which ball is guilty and whether it is heavier or lighter.
I told McPhee my history with this problem. I had heard of it when I was a junior in college. I was up most of the night until I finally solved it. I was very pleased with myself. I had a date the next day to play chess with the most brilliant undergraduate in mathematics and physics. As we were setting up the pieces I gave him the problem. Not only did he solve it before we finished setting up the pieces, but he was generalizing it. Suppose you have “m” balls, how many weighings, “n,” would it take? You can’t do it with two balls and, with three, it takes two weighings, and so on.
There is smart, and there is smart.
As far as I know, McPhee never tried this so I don’t know what would have happened. Historically speaking, it would have been interesting to see what Einstein would have done with such a problem. I am not aware that he had much interest in puzzles like this. I don’t think he played chess and I doubt he played bridge. Besides, there is always the difference between being smart and seeming smart.
For example, Niels Bohr, who was after Einstein the greatest physicist of the 20th century, was certainly smart ” but he did not seem smart. He may even have been dyslexic. He had a ponderous intelligence which he used to crush problems like a bulldozer running over rocks.
Would Einstein have seemed smart? I never met Einstein, so I do not have firsthand knowledge. But I can offer two witnesses. The first is Philipp Frank. He was my first great physics teacher.
Professor Frank, who died in 1966 at the age of 82, was born in Vienna in 1884. He took his Ph.D. in theoretical physics at the University of Vienna in 1907. Even then he was as much interested in the philosophy of science as he was in physics, and in the year of his doctorate he wrote a paper on the meaning of the law of causality. Einstein read the paper and wrote Frank a note to the effect that while he thought the paper’s logic was all right, it did not completely satisfy him. This began a friendship that lasted until Einstein’s death in April 1955. One consequence was that when Einstein left the German University in Prague in 1912, he recommended Professor Frank as his successor. Frank remained there until 1938, when he emigrated to the United States, eventually ending up at Harvard, which is where I met him in 1948.
When I first knew him, Professor Frank had just published a biography of Einstein called “Einstein, His Life and Times.” I spent a good deal of time talking with him about Einstein ” much of it in the Hayes Bickford Coffee shop in Harvard Square, which was the closest Cambridge equivalent to the Viennese coffeehouses of his youth. I once asked him, if I had met Einstein when he did ” both men were still in their 20s ” would Einstein have seemed smart? Professor Frank told me that he would have seemed very smart. He added that Einstein was much given to what Professor Frank called “kreks” ” cracks, jokes ” some of which got him in trouble.
This was not the image of Einstein I had, as a sort of Jewish saint. It was during this sainthood period ” indeed at the end of it ” that my second witness saw Einstein. This was T.D. Lee, who won the Nobel Prize in Physics in 1957. He shared the prize with his collaborator, C.N. Yang, who was a professor at the Institute for Advanced Study, as was Lee. They won it for a discovery they had made involving elementary particles, but this work was almost a distraction from what they were spending most of their time on ” statistical mechanics. This is the discipline, created in the 19th century, that studies the average behavior of systems of particles so numerous that it is a practical impossibility to describe them individually.
Statistical mechanics was a field that Einstein was particularly fond of. His first papers in physics ” the ones that preceded the relativity paper of 1905 ” concerned its foundations. This work was never especially recognized because what followed it overshadowed it, and because others did more or less the same thing. But throughout his life Einstein kept returning to statistical mechanics.
Lee, who told me about it, and Yang decided that they would call on Einstein and tell him what they had done in that field. Lee was not sure what to expect. This was, although they did not know it, close to the time of Einstein’s death. He was in his mid-70s and Lee was not sure if he still had any interest in the subject. Two things surprised him about the visit. The first was Einstein’s hands. To Lee they seemed very large and very powerful. He seemed like a physically strong man. The second thing he did not expect was Einstein’s almost instantaneous grasp of what he and Yang had been doing. The subject had evolved tremendously since Einstein’s work on it, but he nonetheless understood the new developments and even asked some searching questions.
This persuades me that Einstein was smart, but I have the advantage of having studied his physics for decades. I already knew he was smart. To fully convince you, I would have to give you a course in modern physics. I would have to explain to you that Einstein’s footprint is everywhere.
To give a few examples: In 1916, he published a paper on the emission and absorption of radiation, which is the basis of the laser. This was a year after he had published his paper on the general theory of relativity and gravitation which replaced Newton’s theory of gravitation. It was the year before he published a codicil to the theory that introduced the idea of a “cosmological constant,” which may be the basis of the dark energy that is presently accelerating the expansion of the universe. And this was just part of the work of those three years! It does not even touch on the papers he published in 1905, his “miracle year,” in which he laid the foundations of modern physics.
As I said, to understand the scope of this I would need to give you a course in modern physics. I think, however, if I focus on one thing, you may get the idea. I am going to explain to you how Einstein changed our notion of time. I will need only one bit of mathematics ” the Pythagorean theorem of Euclidean geometry.
As you know, the Pythagorean theorem tells us that a2 + b2 = c2. When he was a schoolboy, Einstein found his own proof of it, which is better than solving the problem of the 12 balls. We will use the theorem in due course.
In so far as I understand the creative process that led Einstein to formulate his theory of relativity in 1905, the most difficult step had to do with the nature of time. If we think about it at all, I suspect that our notion of time is about the same as Newton’s. He distinguished between “absolute” time and “common” time. Absolute time, which Newton also called “duration,” flows on without any reference to clocks or observers or anything else. On the other hand, common time is what we measure with clocks. It is subject to the vagaries of the clocks we have at hand.
Newton formulated his physics in terms of absolute time. He published his detailed theory in 1686, in the Principia, and for the next two centuries it went pretty much unchallenged. People simply accepted this notion of an absolute time common to all observers. We now come to 1905. Einstein was 26 and working in the Swiss National Patent Office in Berne. This was a serious job ” examining patents ” which he took seriously. The physics he did on the side.
For about a decade he had been puzzling over the following. In Newton’s physics there is a principle of relativity. It had been emphasized first by Galileo. As far as I know, it was Professor Frank who introduced the term “Galilean relativity” to describe it.
Suppose, Galileo noted, you are on a sailing ship that is at rest with respect to the sea. Furthermore, suppose you drop an object from the mast so that it falls straight down. The object will land at the base of the mast. Now suppose the ship is in motion and that this motion is perfectly uniform ” no acceleration ” and you perform the same experiment, where will the object land? The answer is in exactly the same place.
We might think of this by imagining the ship is stationary and the sea is somehow being moved underneath it with a uniform motion. In this case we are not surprised at the result above.
Galilean relativity, which is built into Newton’s laws, is the proposition that we can never distinguish by any mechanical experiment of this kind between a moving ship and a stationary sea and a stationary ship and a moving sea, providing that these motions do not involve accelerations. But Einstein realized that there is more to physics than Newtonian mechanics. In particular, there is electromagnetism, which includes the propagation of light, which is an electromagnetic wave.
Einstein imagined an experiment with light. For this one, we will use a train. Our train is, in the beginning, at rest with respect to the tracks. I want to shave in my compartment, so I rig up a mirror and a bulb behind my head. From its reflection I can see myself. Now we imagine that the train is set into uniform motion and we perform the same activity. What do we expect to happen?
We know that light moves with a huge but finite speed. If we call this speed “c,” then, experimentally, c = 2.9979… x 1010 centimeters per second, which transforms into about 186,000 miles a second ” a huge speed. This is the speed of light in the vacuum. Light moving in matter is slowed down. In this shaving experiment, light is emitted by the bulb, then travels in a fraction of a second to the mirror and then bounces back to my eye. That is how I see my face. So far so good.
Now let us imagine that just as the light is emitted the train is set into motion and begins moving uniformly on the rails with the speed of light. You get the picture. The light is now trying to catch up with the mirror. Can it do so?
If we are Newtonians the answer is no. The mirror is moving at a speed equal to that of light and the light will never gain on it. You will never see your reflection.
Why did this bother Einstein? It bothered him because it was a violation of the principle of relativity, according to which the laws of physics were the same in a system at rest and a system moving uniformly, no matter what the speed. It seemed as if once the train moved with the speed of light the laws changed, since you could no longer see your face in the mirror.
If this does not trouble you, then you are in good company. None of Einstein’s contemporaries were troubled either. They didn’t mind, if they noticed, that on the one hand you had Newtonian mechanics, in which the relativity principle held and, on the other, you had electromagnetism, where apparently it didn’t ” something that Einstein found profoundly disturbing.
All of this does not seem to have much to do with time, but I am coming to that.
In his relativity theory of 1905 ” the so-called “special” theory, because it dealt only with uniform motions (the general theory of 1915 dealt with all motions) ” Einstein solved the mirror problem by fiat. One of the postulates of the theory is that any observer will measure, in the vacuum, the same speed of light “c,” no matter at what speed the light source is moving with respect to the observer. This is what is called the principle of “constancy” ” the constancy of the speed of light with respect to these observers.
Let us digest this for a moment by comparing it to sound. If I emit a sound wave by, say, banging on a drum, I can catch up to it by moving faster than the speed of sound. That is what supersonic airplanes do. Einstein’s postulate says that you can never have a superluminous airplane. The speed of light “c” is the universal speed limit. Once a light wave is emitted you can never catch up to it.
At the time that Einstein made this assumption there was no direct experimental evidence to support it. Now we have elementary particles produced in accelerators that move almost at the speed of light so we can observe what happens. But there was no direct evidence against it. It was even suggested by the theory of electricity and magnetism that had been invented in the mid-19th century by the Scottish physicist James Clerk Maxwell.
On the other hand, if you believed in Newton’s dynamical theory this speed limit was impossible. According to Newton, if you applied a force to an object long enough you could accelerate it to any speed you liked. Thus Einstein had to choose between Newton and Maxwell, and he chose Maxwell.
Many years ago I had the chance to visit Einstein’s house in Princeton. It was soon enough after his death that his study was about the same as when he used it. He had an etching of Maxwell on the wall. There had been an etching of Newton, but it had come out of its frame and had been replaced by a bit of modern art.
A radical assumption like this must have radical consequences. It does, and I would like to explore what the assumption implies about time. The only kind of time that interests me is what Newton called “common” time ” the kind that is measured by clocks. In truth, I can make no precise sense out of what Newton meant by “absolute” time. Common time is good enough for me ” clock time. A clock is any mechanism that has a periodic behavior. This can be the oscillating of an atom or the beating of a human heart. There are good clocks and bad clocks, depending on how dependable the periodic behavior is. Time is measured in terms of the number of these oscillations that occur between events. I am going to analyze in detail a particularly simple clock ” a so-called “light clock.” We will make use of the figure below.
The Light Clock
We have a light source that emits light that travels in a straight line to a mirror. The light is then reflected back to a detector located at the source. If this detector is a mirror then the light will just bounce back and forth. We will call the time it takes to go from the source to the upper mirror the basic unit of time for this clock ” the equivalent of the second. If we call this time “t0,” and the distance between the two mirrors “d,” then we have d = ct0 ” or equivalently, t0 = d/c. Simple enough.
Now we put this device in motion. We can suppose the clock is transported to the right with a speed “v.” We observe the clock from our stationary onlooking post. The light now, as we observe it, follows a different path. It moves along the hypotenuses of the right triangles in the figure, whose length I will call “h.” The side of the triangle across from the hypotenuse is still “d.”
This requires some commentary, because in relativity there is funny business with lengths. Moving rulers become shorter, as viewed by a resting observer. But that does not happen if the ruler is at right angles to the direction of motion. That is sort of plausible. The ruler would not “know” in which direction to contract. From the figure it is very clear that h is greater than d. Thus the time interval t = h/c is greater than t0 = d/c.
Let us stop here and contemplate this. Putting it graphically, it says that a clock in motion is slower than an identical clock at rest. We see that this is true for the light clock, and Einstein’s theory predicts that it is true of any clock. Furthermore, it does not matter if we are moving past the clock, or the clock is moving past us ” the moving clock is slower. This is important, because you might think that something odd has happened to the clock because it is moving. The same effect is observed if you are moving.
How big is this effect? To answer this, we need to know “h,” and here Pythagoras comes in at last. Look at the third figure. If the moving clock is moving with a speed “v” during the time “t,” then the distance the mirror is displaced is “vt.” Thus we have h2 = v2t2 + d2 = c2t2 or, solving for “t” we have t = t0 / (1-v2/c2) 1/2.
This is one of the most famous formulae in the theory of relativity. You see that it has the remarkable property that when v = c the denominator is zero, which means that the moving clock is now infinitely slower than the resting clock. Time comes to a stop. You also see that when “v” is zero the two times are identical. Because “c” is so huge compared to any “v” we encounter in daily life, this effect escaped observation prior to Einstein’s calling attention to it.
Before concluding, I am going to present two applications of this “time dilation.” The first is a laboratory application and, indeed, was one of the first that measured, more or less directly, the effect of time dilation. Most elementary particles are unstable. After being created they “live” on the average for a certain time, the particle’s “lifetime.” Usually one gives the lifetime, as measured by a clock that is moving with the particle. But this clock appears slow if the particle is in motion and one compares it to a clock at rest. To such an observer the particle lives longer than its resting lifetime. This results in the particle’s leaving a longer track in a detector than you would predict if you ignored time dilation. This effect has been observed countless times.
The second application is a little science-fictional. It is usually referred to as the traveling twins. It is something that was implicit in Einstein’s 1905 paper, but it was only put in these terms a few years later. You have two identical twins, one of whom goes on a round trip in space while the other stays home. The human heart is a kind of clock, so that when the traveling twin returns, the theory predicts that he, or she, is younger than the stationary twin. In 1911 Einstein gave a little talk about this, in which he said:
“If we placed a living organism in a box … one could arrange that the organism, after an arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while correspondingly, organisms which had remained in their original positions had already long since given way to new generations. For the moving organism, the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light.”
When Professor Frank used to describe this he would say that the moral was, travel and you will live longer. However, if you put a few numbers into the formula that describes this effect, then you would learn that if you took a trip of 10,000 miles, moving at a speed of 1,000 mph, you would return only about a 10th of a millionth of a second younger than your twin. For longevity you are better off going to the gym.
I hope I have explained enough so that the next time you hear a physicist say, “He ” or she ” is very smart, but no Einstein,” you will understand what is meant.
Jeremy Bernstein, whose biography of J. Robert Oppenheimer, “Oppenheimer: Portrait of an Enigma,” will be published in April, asks readers not to send him solutions to the 12-ball problem.