Einstein's thought experiments

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A hallmark of Albert Einstein's career was his use of visualized thought experiments (German: Gedankenexperiment) as a fundamental tool for understanding physical issues and for elucidating his concepts to others. Einstein's thought experiments took diverse forms. In his youth, he mentally chased beams of light. For special relativity, he employed moving trains and flashes of lightning to explain his most penetrating insights. For general relativity, he considered a person falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his debates with Niels Bohr on the nature of reality, he proposed imaginary devices intended to show, at least in concept, how the Heisenberg uncertainty principle might be evaded. In a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement.

Introduction

A thought experiment is a logical argument or mental model cast within the context of an imaginary (hypothetical or even counterfactual) scenario. A scientific thought experiment, in particular, may examine the implications of a theory, law, or set of principles with the aid of fictive and/or natural particulars (demons sorting molecules, cats whose lives hinge upon a radioactive disintegration, men in enclosed elevators) in an idealized environment (massless trapdoors, absence of friction). They describe experiments that, except for some specific and necessary idealizations, could conceivably be performed in the real world.^[2]

As opposed to physical experiments, thought experiments do not report new empirical data. They can only provide conclusions based on deductive or inductive reasoning from their starting assumptions. Thought experiments invoke particulars that are irrelevant to the generality of their conclusions. It is the invocation of these particulars that give thought experiments their experiment-like appearance. A thought experiment can always be reconstructed as a straightforward argument, without the irrelevant particulars. John D. Norton, a well-known philosopher of science, has noted that "a good thought experiment is a good argument; a bad thought experiment is a bad argument."^[3]

When effectively used, the irrelevant particulars that convert a straightforward argument into a thought experiment can act as "intuition pumps" that stimulate readers' ability to apply their intuitions to their understanding of a scenario.^[4] Thought experiments have a long history. Perhaps the best known in the history of modern science is Galileo's demonstration that falling objects must fall at the same rate regardless of their masses. This has sometimes been taken to be an actual physical demonstration, involving his climbing up the Leaning Tower of Pisa and dropping two heavy weights off it. In fact, it was a logical demonstration described by Galileo in Discorsi e dimostrazioni matematiche (1638).^[5]

Einstein had a highly visual understanding of physics. His work in the patent office "stimulated [him] to see the physical ramifications of theoretical concepts." These aspects of his thinking style inspired him to fill his papers with vivid practical detail making them quite different from, say, the papers of Lorentz or Maxwell. This included his use of thought experiments.^{[6]:26–27;121–127}

Special relativity

Pursuing a beam of light

Late in life, Einstein recalled

...a paradox upon which I had already hit at the age of sixteen: If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such thing, however, neither on the basis of experience nor according to Maxwell's equations. From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how should the first observer know or be able to determine, that he is in a state of fast uniform motion? One sees in this paradox the germ of the special relativity theory is already contained.^{[p 1]:52–53}

Einstein's thought experiment as a 16 year old student

Einstein's recollections of his youthful musings are widely cited because of the hints they provide of his later great discovery. However, Norton has noted that Einstein's reminiscences were probably colored by a half-century of hindsight. Norton lists several problems with Einstein's recounting, both historical and scientific:^[7]

1. At 16 years old and a student at the Gymnasium in Aarau, Einstein would have had the thought experiment in late 1895 to early 1896. But various sources note that Einstein did not learn Maxwell's theory until 1898, in university.^[7][8]

2. The second issue is that a 19th century aether theorist would have had no difficulties with the thought experiment. Einstein's statement, "...there seems to be no such thing...on the basis of experience," would not have counted as an objection, but would have represented a mere statement of fact, since no one had ever traveled at such speeds.

3. An aether theorist would have regarded "...nor according to Maxwell's equations" as simply representing a misunderstanding on Einstein's part. Unfettered by any notion that the speed of light represents a cosmic limit, the aether theorist would simply have set velocity equal to c, noted that yes indeed, the light would appear to be frozen, and then thought no more of it.^[7]

Rather than the thought experiment being at all incompatible with aether theories (which it is not), the youthful Einstein appears to have reacted to the scenario out of an intuitive sense of wrongness. He felt that the laws of optics should obey the principle of relativity. As he grew older, his early thought experiment acquired deeper levels of significance: Einstein felt that Maxwell's equations should be the same for all observers in inertial motion. From Maxwell's equations, one can deduce a single speed of light, and there is nothing in this computation that depends on an observer's speed. Einstein sensed a conflict between Newtonian mechanics and the constant speed of light determined by Maxwell's equations.^{[6]:114–115}

Regardless of the historical and scientific issues described above, Einstein's early thought experiment was part of the repertoire of test cases that he used to check on the viability of physical theories. Norton suggests that the real importance of the thought experiment was that it provided a powerful objection to emission theories of light, which Einstein had worked on for several years prior to 1905.^[7][8][9]

Magnet and conductor

In the very first paragraph of Einstein's seminal 1905 work introducing special relativity, he writes:

It is known that the application of Maxwell's electrodynamics, as ordinarily conceived at the present time, to moving bodies, leads to asymmetries which don't seem to be connected with the phenomena. Let us, for example, think of the mutual action between a magnet and a conductor. The observed phenomenon in this case depends only on the relative motion of the conductor and the magnet, while according to the usual conception, a strict distinction must be made between the cases where the one or the other of the bodies is in motion. If, for example, the magnet moves and the conductor is at rest, then an electric field of certain energy-value is produced in the neighbourhood of the magnet, which excites a current in those parts of the field where a conductor exists. But if the magnet be at rest and the conductor be set in motion, no electric field is produced in the neighbourhood of the magnet, but an electromotive force is produced in the conductor which corresponds to no energy per se; however, this causes – equality of the relative motion in both considered cases is assumed – an electric current of the same magnitude and the same course, as the electric force in the first case.^{[p 2]}

Magnet and conductor thought experiment

This opening paragraph recounts well-known experimental results obtained by Michael Faraday in 1831. The experiments describe what appeared to be two different phenomena: the motional EMF generated when a wire moves through a magnetic field (see Lorentz force), and the transformer EMF generated by a changing magnetic field (due to the Maxwell–Faraday equation).^{[9][10][11]:135–157} James Clerk Maxwell himself drew attention to this fact in his 1861 paper On Physical Lines of Force. In the latter half of Part II of that paper, Maxwell gave a separate physical explanation for each of the two phenomena.^{[p 3]}

Although Einstein calls the asymmetry "well-known", there is no evidence that any of Einstein's contemporaries considered the distinction between motional EMF and transformer EMF to be in any way odd or pointing to a lack of understanding of the underlying physics. Maxwell, for instance, had repeatedly discussed Faraday's laws of induction, stressing that the magnitude and direction of the induced current was a function only of the relative motion of the magnet and the conductor, without being bothered by the clear distinction between conductor-in-motion and magnet-in-motion in the underlying theoretical treatment.^{[11]:135–138}

Yet Einstein's reflection on this experiment represented the decisive moment in his long and tortuous path to special relativity. Although the equations describing the two scenarios are entirely different, there is no measurement that can distinguish whether the magnet is moving, the conductor is moving, or both.^[10]

In a 1920 review on the Fundamental Ideas and Methods of the Theory of Relativity (unpublished), Einstein related how disturbing he found this asymmetry:

The idea that these two cases should essentially be different was unbearable to me. According to my conviction, the difference between the two could only lie in the choice of the point of view, but not in a real difference .^{[p 4]:20}

Einstein needed to extend the relativity of motion that he perceived between magnet and conductor in the above thought experiment to a full theory. For years, however, he did not know how this might be done. The exact path that Einstein took to resolve this issue is unknown. We do know, however, that Einstein spent several years pursuing an emission theory of light, encountering difficulties that eventually led him to give up the attempt.^[10]

Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results.^{[p 1]:49}

That decision ultimately led to his development of special relativity as a theory founded on two postulates of which he could be sure.^[10] Expressed in contemporary physics vocabulary, his postulates were as follows:^{[note 1]}

1. The laws of physics take the same form in all inertial frames.

2. In any given inertial frame, the velocity of light c is the same whether the light be emitted by a body at rest or by a body in uniform motion. [Emphasis added by editor]^{[12]:140–141}

Einstein's wording of the second postulate was one with which nearly all theorists of his day could agree. His wording is a far more intuitive form of the second postulate than the stronger version frequently encountered in popular writings and college textbooks.^{[13][note 2]}

Trains, embankments, and lightning flashes

The topic of how Einstein arrived at special relativity has been a fascinating one to many scholars, and it is not hard to understand why: A lowly, twenty-six year old patent officer (third class), largely self-taught in physics and completely divorced from mainstream research, nevertheless in his miracle year of 1905 produces four extraordinary works, only one of which (his paper on Brownian motion) appeared related to anything that he had ever published before.^[8]

Einstein's paper, On the Electrodynamics of Moving Bodies, is a polished work that bears few traces of its gestation. Documentary evidence concerning the development of the ideas that went into it consist of, quite literally, only two sentences in a handful of preserved early letters, and various later historical remarks by Einstein himself, some of them known only second-hand and at times contradictory.^[8]
 

Train and embankment thought experiment

In regards to the relativity of simultaneity, Einstein's 1905 paper develops the concept vividly by carefully considering the basics of how time may be disseminated through the exchange of signals between clocks.^[15] In his popular work, Relativity: The Special and General Theory, Einstein translates the formal presentation of his paper into a thought experiment using a train, a railway embankment, and lightning flashes. The essence of the thought experiment is as follows:

• Observer M stands on an embankment, while observer M' rides on a rapidly traveling train. At the precise moment that M and M' coincide in their positions, lightning strikes points A and B equidistant from M and M'.
• Light from these two flashes reach M at the same time, from which M concludes that the bolts were synchronous.
• The combination of Einstein's first and second postulates implies that, despite the rapid motion of the train relative to the embankment, M' measures exactly the same speed of light as does M. Since M' was equidistant from A and B when lightning struck, the fact that M' receives light from B before light from A means that to M', the bolts were not synchronous. Instead, the bolt at B struck first.^{[p 5]:29–31 [note 3]}

A routine supposition among historians of science is that, in accordance with the analysis given in his 1905 special relativity paper and in his popular writings, Einstein discovered the relativity of simultaneity by thinking about how clocks could be synchronized by light signals.^[15] The Einstein synchronization convention was originally developed by telegraphers in the middle 19th century. The dissemination of precise time was an increasingly important topic during this period. Trains needed accurate time to schedule use of track, cartographers needed accurate time to determine longitude, while astronomers and surveyors dared to consider the worldwide dissemination of time to accuracies of thousandths of a second.^{[16]:132–144;183–187} Following this line of argument, Einstein's position in the patent office, where he specialized in evaluating electromagnetic and electromechanical patents, would have exposed him to the latest developments in time technology, which would have guided him in his thoughts towards understanding the relativity of simultaneity.^{[16]:243–263}

However, all of the above is supposition. In later recollections, when Einstein was asked about what inspired him to develop special relativity, he would mention his riding a light beam and his magnet and conductor thought experiments. He would also mention the importance of the Fizeau experiment and the observation of stellar aberration. "They were enough", he said.^[17] He never mentioned thought experiments about clocks and their synchronization.^[15]

The routine analyses of the Fizeau experiment and of stellar aberration, that treat light as Newtonian corpuscles, do not require relativity. But problems arise if one considers light as waves traveling through an aether, which are resolved by applying the relativity of simultaneity. It is entirely possible, therefore, that Einstein arrived at special relativity through a different path than that commonly assumed, through Einstein's examination of Fizeau's experiment and stellar aberration.^[15]

We therefore do not know just how important clock synchronization and the train and embankment thought experiment were to Einstein's development of the concept of the relativity of simultaneity. We do know, however, that the train and embankment thought experiment was the preferred means whereby he chose to teach this concept to the general public.^{[p 5]:29–31}

General relativity

Falling painters and accelerating elevators

In his unpublished 1920 review, Einstein related the genesis of his thoughts on the equivalence principle:

When I was busy (in 1907) writing a summary of my work on the theory of special relativity for the Jahrbuch für Radioaktivität und Elektronik [Yearbook for Radioactivity and Electronics], I also had to try to modify the Newtonian theory of gravitation such as to fit its laws into the theory. While attempts in this direction showed the practicability of this enterprise, they did not satisfy me because they would have had to be based upon unfounded physical hypotheses. At that moment I got the happiest thought of my life in the following form: In an example worth considering, the gravitational field has a relative existence only in a manner similar to the electric field generated by magneto-electric induction. Because for an observer in free-fall from the roof of a house there is during the fall—at least in his immediate vicinity—no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, independent of their special chemical or physical nature. The observer, therefore, is justified in interpreting his state as being "at rest."^{[p 4]:20–21}

The realization "startled" Einstein, and inspired him to begin an eight-year quest that led to what is considered to be his greatest work, the theory of general relativity. Over the years, the story of the falling man has become an iconic one, much embellished by other writers. In most retellings of Einstein's story, the falling man is identified as a painter. In some accounts, Einstein was inspired after he witnessed a painter falling from the roof of a building adjacent to the patent office where he worked. This version of the story leaves unanswered the question of why Einstein might consider his observation of such an unfortunate accident to represent the happiest thought in his life.^[6]:145
 

A thought experiment used by Einstein to illustrate the equivalence principle

Einstein later refined his thought experiment to consider a man inside a large enclosed chest or elevator falling freely in space. While in free fall, the man would consider himself weightless, and any loose objects that he emptied from his pockets would float alongside him. Then Einstein imagined a rope attached to the roof of the chamber. A powerful "being" of some sort begins pulling on the rope with constant force. The chamber begins to move "upwards" with a uniformly accelerated motion. Within the chamber, all of the man's perceptions are consistent with his being in a uniform gravitational field. Einstein asked, "Ought we to smile at the man and say that he errs in his conclusion?" Einstein answered no. Rather, the thought experiment provided "good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity."^{[p 5]:75–79 [6]:145–147}

Through this thought experiment, Einstein addressed an issue that was so well-known, scientists rarely worried about it or considered it puzzling: Objects have "gravitational mass," which determines the force with which they are attracted to other objects. Objects also have "inertial mass," which determines the relationship between the force applied to an object and how much it accelerates. Newton had pointed out that, even though they are defined differently, gravitational mass and inertial mass always seem to be equal. But until Einstein, no one had conceived a good explanation as to why this should be so. From the correspondence revealed by his thought experiment, Einstein concluded that "it is impossible to discover by experiment whether a given system of coordinates is accelerated, or whether...the observed effects are due to a gravitational field." This correspondence between gravitational mass and inertial mass is the equivalence principle.^[6]:147

An extension to his accelerating observer thought experiment allowed Einstein to deduce that "rays of light are propagated curvilinearly in gravitational fields."^{[p 5]:83–84 [6]:190}

Quantum mechanics

Background: Einstein and the quantum

Many myths have grown up about Einstein's relationship with quantum mechanics. Freshman physics students are aware that Einstein explained the photoelectric effect and introduced the concept of the photon. But students who have grown up with the photon may not be aware of how revolutionary the concept was for his time. The best-known factoids about Einstein's relationship with quantum mechanics are his statement, "God does not play dice" and the indisputable fact that he just didn't like the theory in its final form. This has led to the general impression that, despite his initial contributions, Einstein was out of touch with quantum research and played at best a secondary role in its development.^[18]:1–4 Concerning Einstein's estrangement from the general direction of physics research after 1925, his well-known scientific biographer, Abraham Pais, wrote:

Einstein is the only scientist to be justly held equal to Newton. That comparison is based exclusively on what he did before 1925. In the remaining 30 years of his life he remained active in research but his fame would be undiminished, if not enhanced, had he gone fishing instead.^[19]:43

In hindsight, we know that Pais was incorrect in his assessment.

Einstein was arguably the greatest single contributor to the "old" quantum theory.^{[18][note 4]}

• In his 1905 paper on light quanta,^{[p 6]} Einstein created the quantum theory of light. His proposal that light exists as tiny packets (photons) was so revolutionary, that even such major pioneers of quantum theory as Planck and Bohr refused to believe that it could be true.^{[18]:70–79;282–284 [note 5]} Bohr, in particular, was a passionate disbeliever in light quanta, and repeatedly argued against them until 1925, when he yielded in the face of overwhelming evidence for their existence.^[21]
• In his 1906 theory of specific heats, Einstein was the first to realize that quantized energy levels explained the specific heat of solids.^{[p 7]} In this manner, he found a rational justification for the third law of thermodynamics (i.e. the entropy of any system approaches zero as the temperature approaches absolute zero^{[note 6]}): at very cold temperatures, atoms in a solid don't have enough thermal energy to reach even the first excited quantum level, and so cannot vibrate.^{[18]:141–148 [note 7]}
• Einstein proposed the wave-particle duality of light. In 1909, using a rigorous fluctuation argument based on a thought experiment and drawing on his previous work on Brownian motion, he predicted the emergence of a "fusion theory" that would combine the two views.^{[18]:136–140 [p 8] [p 9]} Basically, he demonstrated that the Brownian motion experienced by a mirror in thermal equilibrium with black body radiation would be the sum of two terms, one due to the wave properties of radiation, the other due to its particulate properties.^[3]
• Although Planck is justly hailed as the father of quantum mechanics, his derivation of the law of black-body radiation rested on fragile ground, since it required ad hoc assumptions of an unreasonable character.^{[note 8]} Furthermore, Planck's derivation represented an analysis of classical harmonic oscillators merged with quantum assumptions in an improvised fashion.^[18]:184 In his 1916 theory of radiation, Einstein was the first to create a purely quantum explanation.^{[p 10]} This paper, well-known for broaching the possibility of stimulated emission (the basis of the laser), changed the nature of the evolving quantum theory by introducing the fundamental role of random chance.^{[18]:181–192}
• In 1924, Einstein received a short manuscript by an unknown Indian professor, Satyendra Nath Bose, outlining a new method of deriving the law of blackbody radiation.^{[note 9]} Einstein was intrigued by Bose's peculiar method of counting the number of distinct ways of putting photons into the available states, a method of counting that Bose apparently did not realize was unusual.^{[note 10]} Einstein, however, understood that Bose's counting method implied that photons are, in a deep sense, indistinguishable. He translated the paper into German and had it published. Einstein then followed Bose's paper with an extension to Bose's work which predicted Bose-Einstein condensation, one of the fundamental research topics of condensed matter physics.^{[18]:215–240}
• While trying to develop a mathematical theory of light which would fully encompass its wavelike and particle-like aspects, Einstein developed the concept of "ghost fields". A guiding wave obeying Maxwell's classical laws would propagate following the normal laws of optics, but would not transmit any energy. This guiding wave, however, would govern the appearance of quanta of energy ${\displaystyle h\nu }$ on a statistical basis, so that the appearance of these quanta would be proportional to the intensity of the interference radiation. These ideas became widely known in the physics community, and through Born's work in 1926, later became a key concept in the modern quantum theory of radiation and matter.^{[18]:193–203 [note 11]}

Therefore, Einstein before 1925 originated most of the key concepts of quantum theory: light quanta, wave-particle duality, the fundamental randomness of physical processes, the concept of indistinguishability, and the probability density interpretation of the wave equation. In addition, Einstein can arguably be considered the father of solid state physics and condensed matter physics.^[24] He provided a correct derivation of the blackbody radiation law and sparked the notion of the laser.

What of after 1925? In 1935, working with two younger colleagues, Einstein issued a final challenge to quantum mechanics, attempting to show that it could not represent a final solution.^{[p 12]} Despite the questions raised by this paper, it made little or no difference to how physicists employed quantum mechanics in their work. Of this paper, Pais was to write:

The only part of this article that will ultimately survive, I believe, is this last phrase [i.e. "No reasonable definition of reality could be expect to permit this" where "this" refers to the instantaneous transmission of information over a distance], which so poignantly summarizes Einstein's views on quantum mechanics in his later years....This conclusion has not affected subsequent developments in physics, and it is doubtful that it ever will.^{[12]:454–457}

In contrast to Pais' negative assessment, this paper, outlining the EPR paradox, is currently among the top ten papers published in Physical Review, and is the centerpiece of the development of quantum information theory,^[25] which has been termed the "third quantum revolution."^{[26] [note 12]}

Einstein's light box

Einstein did not like the direction in which quantum mechanics had turned after 1925. Although excited by Heisenberg's matrix mechanics, Schroedinger's wave mechanics, and Born's clarification of the meaning of the Schroedinger wave equation (i.e. that the absolute square of the wave function is to be interpreted as a probability density), his instincts told him that something was missing.^{[6]:326–335} In a letter to Born, he wrote:

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One.^{[12]:440–443}

The Solvay Debates between Bohr and Einstein began in dining-room discussions at the Fifth Solvay International Conference on Electrons and Photons in 1927. Einstein's issue with the new quantum mechanics was not just that, with the probability interpretation, it rendered invalid the notion of rigorous causality. After all, as noted above, Einstein himself had introduced random processes in his 1916 theory of radiation. Rather, by defining and delimiting the maximum amount of information obtainable in a given experimental arrangement, the Heisenberg uncertainty principle denied the existence of any knowable reality in terms of a complete specification of the momenta and description of individual particles, an objective reality that would exist whether or not we could ever observe it.^{[6]:325–326 [12]:443–446}

Over dinner, during after-dinner discussions, and at breakfast, Einstein debated with Bohr and his followers on the question whether quantum mechanics in its present form could be called complete. Einstein illustrated his points with increasingly clever thought experiments intended to prove that position and momentum could in principle be simultaneously known to arbitrary precision. For example, one of his thought experiments involved sending a beam of electrons through a shuttered screen, recording the positions of the electrons as they struck a photographic screen. Bohr and his allies would always be able to counter Einstein's proposal, usually by the end of the same day.^{[6]:344–347}

On the final day of the conference, Einstein revealed that the uncertainty principle was not the only aspect of the new quantum mechanics that bothered him. Quantum mechanics, at least in the Copenhagen interpretation, appeared to allow action at a distance, the ability for two separated objects to communicate at speeds greater than light. By 1928, the consensus was that Einstein had lost the debate, and even his closest allies during the Fifth Solvay Conference, for example Louis de Broglie, conceded that quantum mechanics appeared to be complete.^{[6]:346–347}
 

Einstein's light box

At the Sixth Solvay International Conference on Magnetism (1930), Einstein came armed with a new thought experiment. This involved a box with a shutter that operated so quickly, it would allow only one photon to escape at a time. The box would first be weighed exactly. Then, at a precise moment, the shutter would open, allowing a photon to escape. The box would then be re-weighed. The well-known relationship between mass and energy ${\displaystyle E=mc^{2}}$ would allow the energy of the particle to be precisely determined. With this gadget, Einstein believed that he had demonstrated a means to obtain, simultaneously, a precise determination of the energy of the photon as well as its exact time of departure from the system.^{[6]:346–347 [12]:446–448}

Bohr was shaken by this thought experiment. Unable to think of a refutation, he went from one conference participant to another, trying to convince them that Einstein's thought experiment couldn't be true, that if it were true, it would literally mean the end of physics. After a sleepless night, he finally worked out a response which, ironically, depended on Einstein's general relativity.^{[6]:348–349} Consider the illustration of Einstein's light box:^{[12]:446–448}

1. After emitting a photon, the loss of weight causes the box to rise in the gravitational field.

2. The observer returns the box to its original height by adding weights until the pointer points to its initial position. It takes a certain amount of time ${\displaystyle t}$ for the observer to perform this procedure. How long it takes depends on the strength of the spring and on how well-damped the system is. If undamped, the box will bounce up and down forever. If over-damped, the box will return to its original position sluggishly (See Damped spring-mass system).^{[note 13]}

3. The longer that the observer allows the damped spring-mass system to settle, the closer the pointer will reach its equilibrium position. At some point, the observer will conclude that his setting of the pointer to its initial position is within an allowable tolerance. There will be some residual error ${\displaystyle \Delta q}$ in returning the pointer to its initial position. Correspondingly, there will be some residual error ${\displaystyle \Delta m}$ in the weight measurement.

4. Adding the weights imparts a momentum ${\displaystyle p}$ to the box which can be measured with an accuracy ${\displaystyle \Delta p}$ delimited by ${\displaystyle \Delta p\Delta q\approx h.}$ It is clear that ${\displaystyle \Delta p$ where ${\displaystyle g}$ is the gravitational constant. Plugging in yields ${\displaystyle gt\Delta m\Delta q>h.}$

5. General relativity informs us that while the box has been at a height different than its original height, it has been ticking at a rate different than its original rate. The red shift formula informs us that there will be an uncertainty ${\displaystyle \Delta t=c^{-2}gt\Delta q}$ in the determination of ${\displaystyle t_{0},}$ the emission time of the photon.

6. Hence, ${\displaystyle c^{2}\Delta m\Delta t=\Delta E\Delta t>h.}$ The accuracy with which the energy of the photon is measured restricts the precision with which its moment of emission can be measured, following the Heisenberg uncertainty principle.

After finding his last attempt at finding a loophole around the uncertainty principle refuted, Einstein quit trying to search for inconsistencies in quantum mechanics. Instead, he shifted his focus to the other aspects of quantum mechanics with which he was uncomfortable, focusing on his critique of action at a distance. His next paper on quantum mechanics foreshadowed his later paper on the EPR paradox.^[12]:448

Einstein was gracious in his defeat. The following September, Einstein nominated Heisenberg and Schroedinger for the Nobel Prize, stating, "I am convinced that this theory undoubtedly contains a part of the ultimate truth."^[12]:448

EPR Paradox

Both Bohr and Einstein were subtle men. Einstein tried very hard to show that quantum mechanics was inconsistent; Bohr, however, was always able to counter his arguments. But in his final attack Einstein pointed to something so deep, so counterintuitive, so troubling, and yet so exciting, that at the beginning of the twenty-first century it has returned to fascinate theoretical physicists. Bohr’s only answer to Einstein’s last great discovery—the discovery of entanglement—was to ignore it.

— Leonard Susskind^[28]

Einstein's fundamental dispute with quantum mechanics wasn't about whether God rolled dice, whether the uncertainty principle allowed simultaneous measurement of position and momentum, or even whether quantum mechanics was complete. It was about reality. Does a physical reality exist independent of our ability to observe it? To Bohr and his followers, such questions were meaningless. All that we can know are the results of measurements and observations. It makes no sense to speculate about an ultimate reality that exists beyond our perceptions.^{[6]:460–461}

Einstein's beliefs had evolved over the years from those that he had held when he was young, when, as a logical positivist heavily influenced by his reading of David Hume and Ernst Mach, he had rejected such unobservable concepts as absolute time and space. Einstein believed:^{[6]:460–461}

1. A reality exists independent of our ability to observe it.

2. Objects are located at distinct points in spacetime and have their own independent, real existence. In other words, he believed in separability and locality.

3. Although at a superficial level, quantum events may appear random, at some ultimate level, strict causality underlies all processes in nature.

EPR paradox thought experiment. (top) The total wave function of a particle pair spreads from the collision point. (bottom) Observation of one particle collapses the wave function.

Einstein considered that realism and localism were fundamental underpinnings of physics. After leaving Nazi Germany and settling in Princeton at the Institute for Advanced Studies, Einstein began writing up a thought experiment that he had been mulling over since attending a lecture by Léon Rosenfeld in 1933. Since the paper was to be in English, Einstein enlisted the help of the 46-year-old Boris Podolsky, a fellow who had moved to the Institute from Caltech; he also enlisted the help of the 26-year-old Nathan Rosen, also at the Institute, who did much of the math.^{[note 14]} The result of their collaboration was the four page EPR paper, which in its title asked the question Can Quantum-Mechanical Description of Physical Reality be Considered Complete?^{[6]:448–450 [p 12]}

After seeing the paper in print, Einstein found himself unhappy with the result. His clear conceptual visualization had been buried under layers of mathematical formalism.^{[6]:448–450}

Einstein's thought experiment involved two particles that have collided or which have been created in such a way that they have properties which are correlated. The total wave function for the pair links the positions of the particles as well as their linear momenta.^{[6]:450–453 [25]} The figure depicts the spreading of the wave function from the collision point. However, observation of the position of the first particle allows us to determine precisely the position of the second particle no matter how far the pair have separated. Likewise, measuring the momentum of the first particle allows us to determine precisely the momentum of the second particle. "In accordance with our criterion for reality, in the first case we must consider the quantity P as being an element of reality, in the second case the quantity Q is an element of reality."^{[p 12]}

Einstein concluded that the second particle, which we have never directly observed, must have at any moment a position that is real and a momentum that is real. Quantum mechanics does not account for these features of reality. Therefore, quantum mechanics is not complete.^[6]:451 It is known, from the uncertainty principle, that position and momentum cannot be measured at the same time. But even though their values can only be determined in distinct contexts of measurement, can they both be definite at the same time? Einstein concluded that the answer must be yes.^[25]

The only alternative, claimed Einstein, would be to assert that measuring the first particle instantaneously affected the reality of the position and momentum of the second particle.^[6]:451 "No reasonable definition of reality could be expected to permit this."^{[p 12]}

Bohr was stunned when he read Einstein's paper and spent more than six weeks framing his response, which he gave exactly the same title as the EPR paper.^{[p 16]} The EPR paper forced Bohr to make a major revision in his understanding of complementarity in the Copenhagen interpretation of quantum mechanics.^[25]

Prior to EPR, Bohr had maintained that disturbance caused by the act of observation was the physical explanation for quantum uncertainty. In the EPR thought experiment, however, Bohr had to admit that "there is no question of a mechanical disturbance of the system under investigation." On the other hand, he noted that the two particles were one system described by one quantum function. Furthermore, the EPR paper did nothing to dispel the uncertainty principle.^{[12]:454–457 [note 15]}

Later commentators have questioned the strength and coherence of Bohr's response. As a practical matter, however, physicists for the most part did not pay much attention to the debate between Bohr and Einstein, since the opposing views did not affect one's ability to apply quantum mechanics to practical problems, but only affected one's interpretation of the quantum formalism. If they thought about the problem at all, most working physicists tended to follow Bohr's leadership.^[25][30][31]

So stood the situation for nearly 30 years. Then, in 1964, John Stewart Bell made the groundbreaking discovery that Einstein's local realist world view made experimentally verifiable predictions that would be in conflict with those of quantum mechanics. Bell's discovery shifted the Einstein–Bohr debate from philosophy to the realm of experimental physics. Bell's theorem showed that, for any local realist formalism, there exist limits on the predicted correlations between pairs of particles in an experimental realization of the EPR thought experiment. In 1972, the first experimental tests were carried out. Successive experiments improved the accuracy of observation and closed loopholes. To date, it is virtually certain that local realist theories have been falsified.^[32]

So Einstein was wrong. But it has several times been the case that Einstein's "mistakes" have foreshadowed and provoked major shifts in scientific research. Such, for instance, has been the case with his proposal of the cosmological constant, which Einstein considered his greatest blunder, but which currently is being actively investigated for its possible role in the accelerating expansion of the universe. In his Princeton years, Einstein was virtually shunned as he pursued the unified field theory. Nowadays, innumerable physicists pursue Einstein's dream for a "theory of everything."^[33]

The EPR paper did not prove quantum mechanics to be incorrect. What it did prove was that quantum mechanics, with its "spooky action at a distance," is completely incompatible with commonsense understanding.^[34] Furthermore, the effect predicted by the EPR paper, quantum entanglement, has inspired approaches to quantum mechanics different from the Copenhagen interpretation, and has been at the forefront of major technological advances in quantum computing, quantum encryption, and quantum information theory.