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 theory. 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 that attempted to show, at least in concept, how the Heisenberg uncertainty principle might be evaded. In a 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. 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." 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. 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). 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.
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.
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:
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. 2. 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. 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. 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.
Magnet and conductor
In the first paragraph of Einstein's 1905 work introducing special relativity, he writes: It is well known that Maxwell's electrodynamics—as usually understood at present—when applied to moving bodies, leads to asymmetries that do not seem to attach to the phenomena. Let us recall, for example, the electrodynamic interaction between a magnet and a conductor. The observable phenomenon depends here only on the relative motion of conductor and magnet, while according to the customary conception the two cases, in which, respectively, either the one or the other of the two bodies is the one in motion, are to be strictly differentiated from each other. For if the magnet is in motion and the conductor is at rest, there arises in the surroundings of the magnet an electric field endowed with a certain energy value that produces a current in the places where parts of the conductor are located. But if the magnet is at rest and the conductor is in motion, no electric field arises in the surroundings of the magnet, while in the conductor an electromotive force will arise, to which in itself there does not correspond any energy, but which, provided that the relative motion in the two cases considered is the same, gives rise to electrical currents that have the same magnitude and the same course as those produced by the electric forces in the first-mentioned case.
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). 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. 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. 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. 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 <in the reality of nature>. 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.
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. That decision ultimately led to his development of special relativity as a theory founded on two postulates. Einstein's original expression of these postulates was:
"The laws governing the changes of the state of any physical system do not depend on which one of two coordinate systems in uniform translational motion relative to each other these changes of the state are referred to. Each ray of light moves in the coordinate system "at rest" with the definite velocity V independent of whether this ray of light is emitted by a body at rest or a body in motion." In their modern form:
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] Einstein's wording of the first postulate was one with which nearly all theorists of his day could agree. His second postulate expresses a new idea about the character of light. Modern textbooks combine the two postulates. One popular textbook expresses the second postulate as, "The speed of light in free space has the same value c in all directions and in all inertial reference frames."
Trains, embankments, and lightning flashes
The topic of how Einstein arrived at special relativity has been a fascinating one to many scholars: A twenty-six year old patent officer (third class), largely self-taught in physics and completely divorced from mainstream research, nevertheless in the year 1905 produced four extraordinary works (Annus Mirabilis papers), only one of which (his paper on Brownian motion) appeared related to anything that he had ever published before. 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.
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. 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. 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. 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. 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. 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. He never mentioned thought experiments about clocks and their synchronization. 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. 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.
Relativistic center-of-mass theorem
Einstein proposed the equivalence of mass and energy in his final Annus Mirabilis paper. Over the next several decades, the understanding of energy and its relationship with momentum were further developed by Einstein and other physicists including Max Planck, Gilbert N. Lewis, Richard C. Tolman, Max von Laue (who in 1911 gave a comprehensive proof of M0 = E0/c2 from the stress–energy tensor), and Paul Dirac (whose investigations of negative solutions in his 1928 formulation of the energy–momentum relation led to the 1930 prediction of the existence of antimatter).
Einstein's relativistic center-of-mass theorem of 1906 is a case in point. In 1900, Henri Poincaré had noted a paradox in modern physics as it was then understood: When he applied well-known results of Maxwell's equations to the equality of action and reaction, he could describe a cyclic process which would result in creation of a reactionless drive, i.e. a device which could displace its center of mass without the exhaust of a propellant, in violation of the conservation of momentum. Poincaré resolved this paradox by imagining electromagnetic energy to be a fluid having a given density, which is created and destroyed with a given momentum as energy is absorbed and emitted. The motions of this fluid would oppose displacement of the center of mass in such fashion as to preserve the conservation of momentum. Einstein demonstrated that Poincaré's artifice was superfluous. Rather, he argued that mass-energy equivalence was a necessary and sufficient condition to resolve the paradox. In his demonstration, Einstein provided a derivation of mass-energy equivalence that was distinct from his original derivation. Einstein began by recasting Poincaré's abstract mathematical argument into the form of a thought experiment: Einstein considered (a) an initially stationary, closed, hollow cylinder free-floating in space, of mass M {\displaystyle M} and length L {\displaystyle L} , (b) with some sort of arrangement for sending a quantity of radiative energy (a burst of photons) E {\displaystyle E} from the left to the right. The radiation has momentum E / c . {\displaystyle E/c.} Since the total momentum of the system is zero, the cylinder recoils with a speed v = − E / ( M c ) . {\displaystyle v=-E/(Mc).} (c) The radiation hits the other end of the cylinder in time Δ t = L / c , {\displaystyle \Delta t=L/c,} (assuming v << c {\displaystyle v<<c} ), bringing the cylinder to a stop after it has moved through a distance
Δ x = − E L M c 2 . {\displaystyle \Delta x=-{\frac {EL}{Mc^{2}}}.}
(d) The energy deposited on the right wall of the cylinder is transferred to a massless shuttle mechanism k , {\displaystyle k,} (e) which transports the energy to the left wall (f) and then returns to re-create the starting configuration of the system, except with the cylinder displaced to the left. The cycle may then be repeated. The reactionless drive described here violates the laws of mechanics, according to which the center of mass of a body at rest cannot be displaced in the absence of external forces. Einstein argued that the shuttle k {\displaystyle k} cannot be massless while transferring energy from the right to the left. If energy E {\displaystyle E} possesses the inertia m = E / c 2 , {\displaystyle m=E/c^{2},} the contradiction disappears. Modern analysis suggests that neither Einstein's original 1905 derivation of mass-energy equivalence nor the alternate derivation implied by his 1906 center-of-mass theorem are definitively correct. For instance, the center-of-mass thought experiment regards the cylinder as a completely rigid body. In reality, the impulse provided to the cylinder by the burst of light in step (b) cannot travel faster than light, so that when the burst of photons reaches the right wall in step (c), the wall has not yet begun to move. Ohanian has credited von Laue (1911) as having provided the first truly definitive derivation of M0 = E0/c2.
Impossibility of faster-than-light signaling
In 1907, Einstein noted that from the composition law for velocities, one could deduce that there cannot exist an effect that allows faster-than-light signaling. Einstein imagined a strip of material that allows propagation of signals at the faster-than-light speed of W {\displaystyle W} (as viewed from the material strip). Imagine two observers, A and B, standing on the x-axis and separated by the distance L {\displaystyle L} . They stand next to the material strip, which is not at rest, but rather is moving in the negative x-direction with speed v {\displaystyle v} . A uses the strip to send a signal to B. From the velocity composition formula, the signal propagates from A to B with speed ( W − v ) / ( 1 − ( W v / c 2 ) ) {\displaystyle {(W-v)/(1-(Wv/c^{2}))}} . The time T {\displaystyle T} required for the signal to propagate from A to B is given by
T = L 1 − ( W v / c 2 ) W − v . {\displaystyle T=L{1-(Wv/c^{2}) \over W-v}.}
The strip can move at any speed v < c {\displaystyle v<c} . Given the starting assumption W > c {\displaystyle W>c} , one can always set the strip moving at a speed v {\displaystyle v} such that T < 0 {\displaystyle T<0} . In other words, given the existence of a means of transmitting signals faster-than-light, scenarios can be envisioned whereby the recipient of a signal will receive the signal before the transmitter has transmitted it. About this thought experiment, Einstein wrote:
Even though this result, in my opinion, does not contain any contradiction from a purely logical point of view, it conflicts with the character of all our experience to such an extent that this seems sufficient to prove the impossibility of the assumption W > c {\displaystyle W>c} .
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 der 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." 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.
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." 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. An extension to his accelerating observer thought experiment allowed Einstein to deduce that "rays of light are propagated curvilinearly in gravitational fields."
Early applications of the equivalence principle Einstein's formulation of special relativity was in terms of kinematics (the study of moving bodies without reference to forces). Late in 1907, his former mathematics professor, Hermann Minkowski, presented an alternative, geometric interpretation of special relativity in a lecture to the Göttingen Mathematical society, introducing the concept of spacetime. Einstein was initially dismissive of Minkowski's geometric interpretation, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). As with special relativity, Einstein's early results in developing what was ultimately to become general relativity were accomplished using kinematic analysis rather than geometric techniques of analysis. In his 1907 Jahrbuch paper, Einstein first addressed the question of whether the propagation of light is influenced by gravitation, and whether there is any effect of a gravitational field on clocks. In 1911, Einstein returned to this subject, in part because he had realized that certain predictions of his nascent theory were amenable to experimental test. By the time of his 1911 paper, Einstein and other scientists had offered several alternative demonstrations that the inertial mass of a body increases with its energy content: If the energy increase of the body is E {\displaystyle E} , then the increase in its inertial mass is E / c 2 . {\displaystyle E/c^{2}.}
Einstein asked whether there is an increase of gravitational mass corresponding to the increase in inertial mass, and if there is such an increase, is the increase in gravitational mass precisely the same as its increase in inertial mass? Using the equivalence principle, Einstein concluded that this must be so.
To show that the equivalence principle necessarily implies the gravitation of energy, Einstein considered a light source S 2 {\displaystyle S_{2}} separated along the z-axis by a distance h {\displaystyle h} above a receiver S 1 {\displaystyle S_{1}} in a homogeneous gravitational field having a force per unit mass of 1 g . {\displaystyle g.} A certain amount of electromagnetic energy E {\displaystyle E} is emitted by S 2 {\displaystyle S_{2}} towards S 1 . {\displaystyle S_{1}.} According to the equivalence principle, this system is equivalent to a gravitation-free system which moves with uniform acceleration g {\displaystyle g} in the direction of the positive z-axis, with S 2 {\displaystyle S_{2}} separated by a constant distance h {\displaystyle h} from S 1 . {\displaystyle S_{1}.}
In the accelerated system, light emitted from S 2 {\displaystyle S_{2}} takes (to a first approximation) h / c {\displaystyle h/c} to arrive at S 1 . {\displaystyle S_{1}.} But in this time, the velocity of S 1 {\displaystyle S_{1}} will have increased by v = g h / c {\displaystyle v=gh/c} from its velocity when the light was emitted. The energy arriving at S 1 {\displaystyle S_{1}} will therefore not be the energy E 2 , {\displaystyle E_{2},} but the greater energy E 1 {\displaystyle E_{1}} given by
E 1 ≈ E 2 ( 1 + v c ) = E 2 ( 1 + g h c 2 ) . {\displaystyle E_{1}\approx E_{2}\left(1+{\frac {v}{c}}\right)=E_{2}\left(1+{\frac {gh}{c^{2}}}\right).}
According to the equivalence principle, the same relation holds for the non-accelerated system in a gravitational field, where we replace g h {\displaystyle gh} by the gravitational potential difference Φ {\displaystyle \Phi } between S 2 {\displaystyle S_{2}} and S 1 {\displaystyle S_{1}} so that
E 1 = E 2 + E 2 c 2 Φ . {\displaystyle E_{1}=E_{2}+{\frac {E_{2}}{c^{2}}}\Phi .}
The energy E 1 {\displaystyle E_{1}} arriving at S 1 {\displaystyle S_{1}} is greater than the energy E 2 {\displaystyle E_{2}} emitted by S 2 {\displaystyle S_{2}} by the potential energy of the mass E 2 / c 2 {\displaystyle E_{2}/c^{2}} in the gravitational field. Hence E / c 2 {\displaystyle E/c^{2}} corresponds to the gravitational mass as well as the inertial mass of a quantity of energy.
To further clarify that the energy of gravitational mass must equal the energy of inertial mass, Einstein proposed the following cyclic process: (a) A light source S 2 {\displaystyle S_{2}} is situated a distance h {\displaystyle h} above a receiver S 1 {\displaystyle S_{1}} in a uniform gravitational field. A movable mass M {\displaystyle M} can shuttle between S 2 {\displaystyle S_{2}} and S 1 . {\displaystyle S_{1}.} (b) A pulse of electromagnetic energy E {\displaystyle E} is sent from S 2 {\displaystyle S_{2}} to S 1 . {\displaystyle S_{1}.} The energy E ( 1 + g h / c 2 ) {\displaystyle E(1+gh/c^{2})} is absorbed by S 1 . {\displaystyle S_{1}.} (c) Mass M {\displaystyle M} is lowered from S 2 {\displaystyle S_{2}} to S 1 , {\displaystyle S_{1},} releasing an amount of work equal to M g h . {\displaystyle Mgh.} (d) The energy absorbed by S 1 {\displaystyle S_{1}} is transferred to M . {\displaystyle M.} This increases the gravitational mass of M {\displaystyle M} to a new value M ′ . {\displaystyle M'.} (e) The mass is lifted back to S 2 {\displaystyle S_{2}} , requiring the input of work M ′ g h . {\displaystyle M'gh.} (e) The energy carried by the mass is then transferred to S 2 , {\displaystyle S_{2},} completing the cycle. Conservation of energy demands that the difference in work between raising the mass and lowering the mass, M ′ g h − M g h {\displaystyle M'gh-Mgh}