Einstein’s Entanglements

Collected here are three short pieces on Albert Einstein’s evolving views toward quantum theory, Einstein and J. Robert Oppenheimer’s contrasting conclusions about the creation of black holes, and Einstein’s longstanding friendship with Michele Besso.

Quantum Misgivings

In 1923, Louis de Broglie suggested as part of a proposed PhD thesis that particles like electrons might also have the character of a wave. His thesis advisor, Paul Langevin, sent a copy of the thesis to Albert Einstein, who replied that he found de Broglie’s ideas interesting and that he had some similar ideas. But what those ideas were exactly, we never learned.

Two years later, Erwin Schrödinger found the equation that describes these waves.¹ Einstein was delighted. That is, until Max Born, among others, showed that the waves were not what Einstein was expecting. Rather than acting like light waves that oscillated in space, as both Einstein and Schrödinger had imagined, these were, in fact, waves of probability. Where they had large amplitudes, a particle was more likely to be found. At this point, Einstein and quantum theory parted company. In 1926, he wrote to Born:

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but it does not really bring us any closer to the secret of the “old one” [Einstein’s affectionate way of referring to God]. I, at any rate, am convinced that He is not playing dice.²

This marked the beginning of a period in which Einstein sought to demonstrate that the theory itself was wrong. This, in turn, led to monumental debates with Niels Bohr. In truth, Einstein did not emerge from these exchanges as the winner. He eventually settled on the view that the theory was not a complete description of reality; he spent the rest of his life trying to find a suitable replacement. But working without any guidance from experiment, there was little chance for success.

Many people tried to induce Einstein back into the mainstream. John Wheeler told me that when his student Richard Feynman produced a new formalism for quantum theory, Wheeler thought that it was so beautiful that his colleague would surely be converted. Einstein, of course, was having none of it.

Einstein was a classical physicist at heart. As Jean Cocteau once said, poets tend to sing from their family trees. Einstein’s family tree was classical physics, the physics he learned as a student. While quantum theory was, in a sense, his child, it was not a child that he was prepared to accept.

During his lifetime, Einstein wrote a vast number of letters. One cannot help but wonder how he had time for anything else. On April 16, 1926, he penned a note to Schrödinger. Although relatively brief, it is one of the most remarkable letters he ever sent. Einstein writes that he had just heard about Schrödinger’s wave equation from Max Planck, and it seemed to him that there was something wrong with it. He writes out the equation as

$div grad\phi +\frac{E^2}{b^2}\frac{\phi }{\left(E-\phi \right)}=0$.

This equation, he notes, is defective. For two independent systems the energies must be additive, which is clearly not the case for this equation. Einstein then proposes another equation to resolve the dilemma. The equation he offers is, in fact, Schrödinger’s original equation. Einstein had misremembered it.

The Fifth Solvay International Conference took place in Brussels in October 1927.³ Einstein arrived at the conference determined to show that quantum theory was wrong. To help make his point, he offered a gedankenexperiment intended to demonstrate that the Heisenberg uncertainty relation between position and momentum, ΔxΔp > ħ/2, was not generally true. Einstein described an apparatus as follows.⁴ A diaphragm is suspended from a pair of springs that allow it to move up and down freely. Any displacements can be measured using a pointer and a scale. When a particle passes through a narrow slit in the diaphragm, its momentum is transferred to the diaphragm. The particle’s momentum is measured by the displacement of the springs, while the slit measures its position. It might then appear that both the position and momentum of the particle could be measured simultaneously to arbitrary accuracy. Bohr pointed out that Einstein had assumed that the position of the pointer could be measured to arbitrary accuracy without disturbing the momentum of the diaphragm, and hence the measurement of the particle’s momentum. When the uncertainty in this measurement is taken into account, the uncertainty between the particle’s position and momentum is restored. Bohr had won.

Undeterred, Einstein arrived at the Sixth Solvay Conference in 1930 having formulated another, far more diabolical Gedanken-machine. This new device was designed to show that the energy–time relation ΔEΔt ≥ ħ/2 was not valid. This uncertainty relation is on a different footing from the others, which involve noncommuting operators—time cannot be represented by an operator, while energy can.

Einstein described another box, this time suspended by a single spring. The box contained radiation and could be weighed, it was assumed, with arbitrary accuracy. At a predetermined time, according to an accurate clock, a shutter would open and a single photon would escape. Since photons carry energy, the mass of the box would change. This box would then be reweighed, from which the change in energy could be determined. The result would be a simultaneous measurement of time and energy, defeating the uncertainty principle. This demonstration gave Bohr a sleepless night, but the next day he was triumphant. Einstein had forgotten his own relativity theory!

In arriving at his solution, Bohr analyzed how this measurement would actually be made. In the equation

$\frac{\Delta T}{T}=g\frac{\Delta q}{c^2}$,

the accuracy with which a determination could be made in terms of the displacement of the box is represented by Δq.⁵ T is the time interval for this to take place in the absence of gravitation, and ΔT is the time interval in the presence of gravitation.

Next, the notion of impulse is needed. This is the force applied over a time interval dt and is equal to the change in momentum during this same time. In this case, it is Tmg, where m is the mass of the box. The mass is only known with an accuracy of Δm. The impulse then becomes TΔmg. For the measurement of the momentum to be meaningful, this impulse has to be greater than the uncertainty in the momentum Δ, hence

TΔmg > Δp.

Substituting the formula for T yields

TΔmc² > ΔpΔq > ħ/2.

Since Δmc² is ΔE, the uncertainty principle is saved. At this point, Einstein stopped claiming that the theory was wrong. But for him, the theory still did not adequately describe reality. In this, he found an unexpected ally in Schrödinger.

In December 1935, Schrödinger published a long article in Naturwissenchaften entitled “Die gegenwärtige Situation in der Quantenmechanik (The Present Status of Quantum Mechanics).”⁶ This is the paper in which he described what has come to be known as the cat paradox.⁷

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance so small that perhaps [emphasis original] in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the [Geiger] counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say the cat still lives if [emphasis original] meanwhile no atom has decayed. The first atomic decay would have poisoned it. The Ψ-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.⁸

Putting aside the issue of producing a wave function for a macroscopic cat, what Schrödinger is really discussing in this excerpt is the phenomenon he termed entanglement. In this situation, the cat is described by a wave function

${\Psi }_{cat}={\Psi }_{dead}+{\Psi }_{alive}$.

If the cat is replaced with spin in two directions, up and down, the real quantum mechanical issue becomes clear. Writing the spin wave function as

${\Psi }_{spin}={\Psi }_{up}+{\Psi }_{down}$

describes a situation in which the spin is neither up nor down. This is the true novelty of quantum mechanics. When observed, the cat will be either dead or alive. The observation projects out one piece of the wave function, while the other is simply detached; the same goes for the spins.

There is much in Schrödinger’s 1935 paper about measurement. Though Einstein hinted at it, he never really came to grips with what is now known as the measurement problem. Put simply, quantum mechanics, as characterized by the Schrödinger equation, cannot describe the kind of projections that Schrödinger is describing. The solutions to the Schrödinger equations are reversible in time, while projections are not. Projections have no inverses. Various attempts have been made to solve the measurement problem, but, in my view, none of them is completely satisfactory.

Schrödinger was inspired by a paper published in May 1935 by Einstein, Boris Podolsky, and Nathan Rosen.⁹ “The appearance of [their] work,” Schrödinger writes, “was the impetus to the present—shall I say paper or general confession?”¹⁰ At the time, Podolsky, who seems to have actually written the paper, was a fellow at the Institute for Advanced Study. Rosen was a much younger assistant to Einstein. Podolsky and Rosen, and subsequently Schrödinger, made their argument in terms of position and momentum; this approach lends their papers a classical air. It is much simpler to frame the argument in terms of spin.

Particles like electrons have a spin of one half in the usual units. With respect to some axis, the spin can only point in one of two directions, up and down. Suppose there are a pair of electrons in a state described by the wave function as

u(1)_upu(2)_down.

Suppose also that there are two observers equipped with magnets. The magnets can interact with the spin of the electrons, causing the electrons with spin up to follow one trajectory and the electrons with spin down to follow another. This will constitute a measurement of the electron’s spin orientations. There is nothing here, it should be noted, that involves entanglement.

Consider another state in which the spins of paired electrons amount to an overall spin of zero. This is known as a singlet state and, normalization aside, it can be described as

u(1)_upu(2)_down – u(1)_downu(2)_up.

In this scenario, there is a 50-50 chance whether, say, electron one is spin up or spin down. This is an entangled state. If nothing is done to destroy the entanglement, there will be a perfect anticorrelation between the two spin directions, no matter how far apart the magnets are. If one observer sees spin up, the other will see spin down, and vice versa. In a letter to Born, Einstein described this sort of thing as “spukhafte Fernwirkung (spooky action at a distance).”¹¹ In this instance, the term “action at a distance” is an apparition from classical physics. There is no action here, spooky or otherwise.

In common with position and momentum, the values of the spins in different directions are connected by an uncertainty principle. For any measurement of the value of the spin in direction z, the value in direction x, as far as that experiment is concerned, is completely indeterminate. A subsequent measurement in direction x will reveal the same anticorrelation. Confronted with this situation, Einstein, Podolsky, and Rosen concluded that the wave function description could not be the whole story. There must be some underlying theory that makes sense of it all. In 1944, Einstein wrote to Born:

We have become Antipodean in our scientific expectations. You believe in the God who plays dice, and I in complete law and order in a world which objectively exists, and which I, in a wildly speculative way, am trying to capture. I firmly believe [emphasis original], but I hope that someone will discover a more realistic way, or rather a more tangible basis than it has been my lot to find. Even the great initial success of the quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility. No doubt the day will come when we will see whose instinctive attitude was the correct one.¹²

Erwin Schrödinger died on January 4, 1961. Along with a small group of colleagues, I had visited him in his apartment in Vienna the previous spring. There was no cat. Schrödinger did not like cats. As we were leaving, he remarked, “There is one thing we have lost since the Greeks.” We paused. “Modesty,” he added, in his lightly accented English. I have no idea what he meant, and I regret not asking him.¹³

Black Holes

On May 10, 1939, a paper by Einstein entitled “On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses” was received by Annals of Mathematics.¹⁴ On July 10, 1939, a paper by J. Robert Oppenheimer and Hartland Snyder, “On Continued Gravitational Contraction,” was received by Physical Review.¹⁵ Oppenheimer and Snyder’s paper was the first to be published, appearing at the beginning of September; Einstein’s paper followed a month later. The two papers, which are both correct within the limits of their assumptions, offered opposing views on the creation of black holes. In his, Einstein showed why black holes could not be created. Oppenheimer and Snyder, on the other hand, demonstrated how to create them. This note explains how they could both be right.

The story begins in 1915. Karl Schwarzschild was stationed on the Russian front. He had volunteered for the German army at the outbreak of war. At the time of enlisting, Schwarzschild was a professor of physics and director of the Astrophysical Observatory in Potsdam. While serving in the army, he still found time to continue his scientific research. Schwarzschild was familiar with the general theory of relativity, the first version of which had been published in early December 1915. Prior to its publication, Einstein had presented the new theory to the Prussian Academy of Sciences during a lecture in Potsdam on November 25.¹⁶ In his spare time at the front, Schwarzschild solved a problem in relativity that Einstein did not think had an exact solution. His letter to Einstein describing the solution was sent on December 22, less than a month after the lecture in Potsdam.¹⁷

Under Newtonian gravitation, using the force law between two masses M and m, where a is acceleration and G the gravitational constant, 6.67408(31) × 10^–11 m³ kg^–1 s^–2, the equation for Newton’s law is

$\frac{GmM}{r^2}=ma$.

Finding the trajectories of gravitating objects involves solving the vector form of this differential equation. In general relativity, however, there is no gravitational force—gravity alters the geometry of spacetime. An exact solution is a closed expression for this alteration, such that

${ds}^2=g_{ij}{dx}^i{dx}^j$,

where ds² represents the square of the infinitesimal distance between two spacetime events, g_ij is the metric tensor, and repeated indices are summed one to four.

Einstein had not sought exact expressions for this metric in contrived situations. Instead, he focused on approximate expressions in more physically realistic situations, such as planetary orbits or the bending of starlight as it passed the sun. He replied to Schwarzschild the following month:

I read your paper with utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation.¹⁸

In devising his solution, Schwarzschild had analyzed the gravitation produced by a spherical mass of constant density. This scenario had long been studied under Newtonian gravitation and is sometimes referred to as the Kepler problem. From the observations of Tycho Brahe, Johannes Kepler determined that the planetary orbits were elliptical rather than circular. Isaac Newton, and others, showed that the orbits produced by such a spherical mass acting on a test mass would be conic sections: an ellipse (the circle was a special case), parabola, or hyperbola. Schwarzschild’s contribution was centered on the four-dimensional geometry. He showed that test particles move along paths that minimize the four-dimensional distance between two points in spacetime. The Newtonian solution would then emerge as an approximation when the gravitational interaction was taken to be weak. The Schwarzschild metric is

${ds}^2=-c^{2}d{\tau }^2=-\left(1-\frac{r_s}{r}\right)c^2{dt}^2+\frac{{dr}^2}{1-\frac{r_s}{r}}+r^2\left(d{\theta }^2+{sin}^2\theta d{\phi }^2\right)$,

where τ denotes the time measured by a clock in the rest frame of the moving object.¹⁹ The polar coordinates speak for themselves. Consider the quantity r_s, known as the Schwarzschild radius. For a spherical mass M,

$r_s=\frac{2GM}{c^2}$.

It has the dimensions of length, as is evident if the numerator and denominator are multiplied by m. The potential energy multiplied by a distance is GMm. If G is set to zero, the metric becomes the flat spacetime metric in polar coordinates. The 2 in the equation is inexplicable except by calculation.

Despite having a radius of more than 696,000 kilometers, the sun’s Schwarzschild radius—the corresponding value for r_s—is approximately 2.95 kilometers. This is just as well because, as we shall see, if these two values were the other way around, the sun would, in fact, be a black hole. Supermassive black holes have been found at the center of galaxies, including our own, that have masses a billion or more times that of the sun and correspondingly large Schwarzschild radii. At the Schwarzschild radius, the metric is singular—there is no concept of time, and space is infinite. There is also a singularity at the origin. It has been known for many years that it is possible to redefine the coordinates so that the singularity at the Schwarzschild radius disappears. When written in this form, which both Einstein and Oppenheimer used, the physics is more transparent.

When Einstein was writing his 1939 paper, possible orbits for a test mass in the vicinity of a spherical mass that was producing the Schwarzschild metric were well understood. Circular orbits would be possible, provided they were not too close. The limit was 1.5 times the Schwarzschild radius. At this radius, the object would be moving at the speed of light. For any smaller radius, it would have to move at speeds greater than that of light. This is such a curious result that a derivation merits consideration.²⁰ For a circular orbit, the Schwarzschild metric reduces to

${ds}^2=-\left(1-\frac{r_s}{r}\right)c^2{dt}^2+r^2{d\theta }^2$.

This leads to a simple relationship for the angular velocity dθ/dt = ω, namely

${\omega }^2=\frac{\left(\frac{GM}{c^2}\right)}{r^3}$.

This is the Schwarzschild version of Kepler’s third law. Putting this relationship into the metric yields

${ds}^2=\left(1-\frac{3}{r}\frac{GM}{c^2}\right){dt}^2$,

which says that for circular orbits, the radius must always be larger than 1.5 times the Schwarzschild radius.

Einstein makes use of a version of this in his argument. He imagined trying to construct a shell around the central mass by putting successive particles in circular orbits around the sphere at the Schwarzschild radius.

The essential result of this investigation is a clear understanding as to why the “Schwarzschild singularities” do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The “Schwarzschild singularity” does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.²¹

A Schwarzschild black hole is a self-gravitating mass described by a metric with a Schwarzschild singularity.²² Oppenheimer and Snyder showed that such an object can be formed by collapsing matter. Their conclusion was correct. What does this mean for Einstein’s paper? A clue can be found in the sentence: “Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results.” He did not consider paths that were only, or even partially, radial. This makes all the difference.

When Oppenheimer and Snyder wrote their paper, the final fate of stars had already been studied extensively. A star with five times the mass of the sun could undergo a supernova explosion, leaving behind a core consisting almost entirely of neutrons. There would be no counterpressure from nuclear reactions to counterbalance gravitation. In their paper, Oppenheimer and Snyder present the Schwarzschild metric followed by the excerpt below, where r₀ denotes the Schwarzschild radius:

We should now expect that since the pressure of the stellar matter is insufficient to support it against its own gravitational attraction, the star will contract, and its boundary r_b will necessarily approach the gravitational radius r₀. Near the surface of the star, where the pressure must in any case be low, we should expect to have a local observer see matter falling inward with a velocity very close to that of light; to a distant observer this motion will be slowed up by a factor (1 – r₀/r_b). All energy emitted outward from the surface of the star will be reduced very much in escaping, by the Doppler effect from the receding source, by the large gravitational red-shift, (1 – r₀/r_b)^½, and by the gravitational deflection of light which will prevent the escape of radiation except through a cone about the outward normal of progressively shrinking aperture as the star contracts. The star thus tends to close itself off from any communication with a distant observer; only its gravitational field persists. … [A]lthough it takes, from the point of view of a distant observer, an infinite time for this asymptotic isolation to be established, for an observer comoving with this stellar matter this time is finite and may be quite short.²³

In short, Oppenheimer and Snyder’s remarkable paper predicted the existence of black holes. Yet neither of the pair did any further work on the topic, and there is no evidence that they ever discussed it after Snyder’s thesis days. It seems that Oppenheimer dismissed the whole thing as an exercise for students. Einstein died in 1955 and Oppenheimer in 1967, long before the study of black holes became fashionable.

As for Schwarzschild, while serving at the front he contracted pemphigus, a rare autoimmune skin disease for which there was no known cure. He died in Potsdam on May 11, 1916.²⁴

The Best of Friends

In the autumn of 1946, a young physicist became an instructor at the University of Geneva. As part of his duties, Pierre Speziali was charged with maintaining the mathematics library. It was not a position without perks; Speziali recalled often spending more time reading the books than cataloguing them. Each Thursday morning, an elderly man with a grey beard came to read some of the books. On one such occasion, he and Speziali began talking, apparently in Italian, although the old man was also fluent in French and German. His name was Michele Besso. It is unclear whether, at first, Speziali recognized the name. If he had read the closing remarks in Einstein’s 1905 masterpiece on the special theory of relativity, it may have sounded familiar.

In conclusion I wish to say that in working at the problem here dealt with, I have had the loyal assistance of my friend and colleague M. Besso, and I am indebted to him for several valuable suggestions.²⁵

According to Speziali, they spoke for an hour or more about the history of science before the old man borrowed a couple of books and went off to audit a lecture. Besso, as it turned out, was auditing several courses. In July 1953, at the age of 80, he gave a lecture entitled “An Attempt at a Visualization of the Structure of Space-Time.” During the following summer, Speziali visited Besso many times at his house in Geneva. Beneath a giant tree in the garden, the pair discussed some of the great moments in the history of physics. At the end of one visit, Besso walked Speziali to the road and stood still as if he had something else to say. It was the last time they saw each other. Besso died on March 15, 1955.

Some years later it occurred to Speziali that if there was any correspondence between Einstein and Besso it would likely be of interest and could be published as a tribute to Besso. By this time, Speziali had become acquainted with Besso’s only child, Vero. He provided Speziali with 17 letters from Einstein to his father. Six were sent prior to 1918, while the remainder were from the period between 1950 and 1954. Vero thought that there might be some others stored in the basement of his father’s country house. In all, 110 letters from Einstein to Besso were found. Einstein’s secretary, Helen Dukas, had kept the letters from Besso to Einstein, of which there were 119. In 1979, Speziali published the letters after translating them into French from the original German.²⁶ He also included copious footnotes and a biography that offered a glimpse of the deep friendship between Einstein and Besso.

Michele Angelo Besso was born on May 25, 1873, in Riesbach, a district of Zurich. For some generations beforehand, Besso’s family had been living in Trieste. His father, an insurance executive, had moved to Zurich at the behest of his employer.²⁷ The family returned to Trieste in 1879, but Besso was sent to Rome for his secondary education. Speziali’s biography includes a high school report card that shows Besso doing well in mathematics. An uncle suggested that Besso continue his studies at the Swiss Federal Polytechnic School (Eidgenössische Polytechnische Schule) in Zurich. This was the institution that Einstein referred to as the Poly. It later became the Eidgenössische Technische Hochschule.

When they first met in 1896, Besso was 23 and Einstein was 17. Besso had enrolled at the Poly in 1891, and when Einstein arrived five years later, he had essentially graduated and was writing a thesis in order to become an instructor. Exactly how they met is not clear, but they liked each other immediately, and Einstein helped Besso with his thesis.²⁸

In 1902, Einstein began work as a technical expert, class III, at the Federal Office for Intellectual Property in Bern. The following year he married his long-time girlfriend, Mileva Marić, who had been the only female student at the Poly. At the time of their marriage the couple were already parents, although the fate of their first child is an enduring mystery. In 1901, Mileva became pregnant and returned to her native Serbia to have the baby. The child, a girl, affectionately referred to as “Lieserl” by Einstein, disappeared and no trace of her has ever been found. Mileva returned to Switzerland alone and Einstein never met his daughter.

There is no mention of Lieserl in any of Einstein’s letters to Besso; it is unlikely that Besso ever heard of her. Indeed, the first news that Besso received about the marriage was in a letter from January 1903. “I am now a married man,” Einstein wrote, “and my wife and I lead a very agreeable life. She occupies herself perfectly with everything, is a good cook, and is always happy.”²⁹ This would change, and Besso would be in the middle of it. There are a couple more letters from 1903, and then none until 1909. The reason for the lapse in correspondence is that Besso had begun working at the patent office in Bern, a position he likely attained with the help of Einstein. The two men walked to and from work together each day.³⁰

Aside from Besso, it seems that Einstein did not have any other real friends. Philipp Frank, who succeeded him at the German Charles-Ferdinand University in Prague, wrote that “[Einstein] always managed to maintain a certain ‘free space’ around him which protected him from all disturbances.”³¹ Besso was the only person who managed to penetrate that space. He was present through some of Einstein’s most intimate ordeals, including his marital troubles.

On October 31, 1916, Einstein wrote to Besso from Berlin: “As for divorce, I have definitively renounced it. We will now go on to scientific things!”³² His attempt to preoccupy himself with work proved unsuccessful. In 1918, Einstein wrote to Mileva:

The endeavor finally to put my private affairs in some state of order prompts me to suggest the divorce to you for the second time. I am firmly resolved to do everything to make this step possible. In the case of a divorce, I would grant you significant pecuniary advantages through particularly generous concessions. … Now I request being informed whether you agree and are prepared to file a divorce claim against me. I would take care of everything here, so you would have neither trouble nor any inconveniences whatsoever.³³

Throughout the upheaval, Besso remained close to Mileva and the couple’s two sons, Hans Albert and Eduard. He did what he could to mediate.

Some of the most joyous of Einstein’s letters are his accounts of the time he spent with his sons. These occasions also yielded some of his most somber correspondence, such as when he realized that Eduard had mental problems so severe that he would need to be institutionalized. On March 9, 1917, Einstein wrote to Besso:

The state of my youngest son causes me a great deal of concern. It is out of the question that one day he might become a man like the rest. Who knows, perhaps it would have been better if he had left this world before having known life. For the first time in my life, I feel responsible and I blame myself.³⁴

Eduard had musical talents, and for a while he studied psychiatry. But his mental problems became so severe that he was institutionalized in the Burghölzli Hospital in Zurich, where he eventually died in 1965. After migrating to the US in 1933, Einstein never saw Eduard again, but he paid for his treatments and tried to maintain communication. Hans Albert became a professor of hydraulic engineering in California. I have the impression that he and his father were not close.

In 1926, when Besso was on the edge of losing his job because of a “lack of zeal,” Einstein had written to the patent office defending Besso’s abilities. The intervention of Einstein probably saved his job. At the end of 1938, Besso retired at the age of 65. Einstein wrote to offer his congratulations. Their correspondence gradually became more and more one-sided in the years that followed. Besso often wrote several letters before Einstein would reply and apologize for his tardiness.

Besso’s final letter to Einstein was written from Geneva and dated January 29, 1955, less than two months before his death. He writes in part:

What is important to me personally is my father’s absolute refusal to accept any representation or even any denomination of God (which he thought was in complete conformity to the Torah … “You should not make any image or representation…”); so there only remains natural law, which for me amounts to giving meaning to research itself, to recognizing in immediate experience a value of the representation which is free of all contradictions and which represents a spiritual being in which we participate … with an open door to beauty, recognizing joy and goodness and the other genres of truth, goodness, and beauty…³⁵

The letter is signed, “Your old, old, old Michele.” In a footnote, Speziali indicated that he found this letter difficult to translate and was helped by Besso’s son, Vero. Before this, there had been a lifetime of letters that are not difficult to interpret or translate, although some require knowledge of physics to be fully comprehensible.

On March 21, 1955, less than a week after Besso’s death, Einstein wrote to Vero and Besso’s sister, Bice Rusconi. It is one of the most beautiful and poignant letters he ever wrote.

It is really very kind of you to give me so many details of Michele’s death during these so painful days. His end was in harmony with his entire life, with the image of his being surrounded by a circle of his own family. The gift of leading a harmonious life is rarely conjoined with such a sharp intelligence, above all to the degree in which one found in him. But what I admired the most about Michele the man is the fact that he was capable of living for so many years with one woman, not only in peace, but also in constant agreement, an enterprise at which I have lamentably failed twice. …

Now he has gone just before me again, leaving this strange world. It doesn’t mean anything. For us, believing physicists, this separation between past, present, and future retains only the value of an illusion, however tenacious it may be. …

Your
A. Einstein³⁶

Einstein died just a few weeks later on April 18, 1955.