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For most people, E=mc² is synonymous with two things: Albert Einstein and the theory of relativity. Few are aware, however, that historically speaking, the celebrated mass-energy equivalent relation wasn't actually first discovered by Einstein. But even more surprising is the fact that this equation, although touted to be the most significant consequence of special relativity, can be derived without any reference to the postulates of relativity whatsoever. Indeed, discovering how it emerges from purely classical considerations of electromagnetism yields some pretty incredible insights. This is dialect with where E=mc² really comes from.
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In the year 1900, the French mathematician Henri Poincaré published a groundbreaking paper entitled "The Theory of Lorentz and the Principle of Reaction." At the time, electromagnetism was all the rage, with different physicists such as Hendrik Lorentz jostling to discover new ways to interpret and implement Maxwell's equations. And although works like Lorentz's "Theory of the Electron" were revolutionary in their own right, other physicists like Poincaré felt they were highly problematic. In particular, Poincaré understood they suffered from a crucial problem, which was that they didn't obey Newton's third law: the law of action and reaction.
Indeed, if no external forces are acting on a system, Newton's third law states that when one body acts upon another, that body has to react equally and oppositely. Now, in Newtonian physics, force fields like gravity act instantaneously across a distance, so bodies separated by such distances can act and react upon one another without issue. But in electromagnetism, information and energy are transmitted in a local fashion via disturbances in the fields themselves. Hence, the action of one body isn't simultaneously compensated by the reaction of another body.
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Thus, as Poincaré and others came to realize, in order for Newton's third law of action and reaction to be preserved for electromagnetism, a body can't simply act upon other bodies, but rather, it has to act upon the electromagnetic field itself. Indeed, for one electromagnetic particle to act and exert a force upon another electromagnetic particle, it must first act upon the field, which in turn must react equally and oppositely upon the particle. This led naturally to the idea that electromagnetic fields themselves must carry momentum.
But how could such momentum be described? In his 1900 paper, Poincaré answered this question by examining the expression for the total electromagnetic force, the Lorentz force. His strategy was to recast this expression in terms of the rate of change of some quantity. That is, he knew if he could rewrite Lorentz's force as the time derivative of some combination of electromagnetic field quantities, then he could interpret this new quantity as being the momentum of the electromagnetic field itself.
Now, after some extensive manipulations of Maxwell's equations, Poincaré found the desired quantity was this expression: 1/c² * the integral of S dV, where S is the Poynting vector (or 1/μ * E x B) and V refers to the volume of the enclosed field. Thus, it became clear that the momentum of the electromagnetic field was proportional to the strength of its electric and magnetic fields, a fact Poincaré subsequently exploited in order to achieve what would be the first explicit derivation of E=mc².
Now, historically speaking, multitudes of other physicists had already suggested the idea of mass-energy equivalence, and many during this time were closing in on various forms of the relation. But Poincaré's derivation in his 1900 paper stands as the first true formulation of the equation, despite the fact that, however ironically, his derivation didn't require relativity at all. Rather, it simply required combining his derived expression for the momentum of the electromagnetic field with the expression for another well-known electromagnetic quantity: that of the energy of the electromagnetic field.
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To understand Poincaré's derivation, we can consider the simplest version of it involving a plane wave. This is the type of wave you're probably used to encountering in visuals, where the electric and magnetic fields oscillate in tandem. Indeed, for such a plane wave solution to Maxwell's equations, the magnitude of the magnetic field equals the magnitude of the electric field over the speed of light, c.
Now, plugging that relation into the expressions for energy density and momentum density, the result is that we have a simple way of relating the momentum of an electromagnetic wave to its energy. And that relation proves to be the following: the momentum of an electromagnetic wave, that is, the momentum of light, is equal to its energy divided by its speed, c. It's this relation which immediately yields E=mc².
This is because momentum also equals mv, or mass times velocity. Meaning, since the speed of a light wave equals c, we can also express its momentum as mc, where m stands in for the mass of the light wave. But hold on a minute, light doesn't have mass, right? Well, no, that's not true. Light doesn't have rest mass. But based on the definition of mass as the ratio of an entity's momentum to its velocity, we can assign it an equivalent mass. That is, we can treat a given volume of light as having mass, but with the qualification that such mass only travels at speed c. Which means we now have two expressions for the momentum of light: p = E/c and p = mc. Setting these equal to each other, we multiply both sides by c, and voila: E=mc².
And that's it! The entire mystery of E=mc² distilled down to a few simple steps. But of course, this equation is not yet so much telling us about mass-energy equivalence as it's contextualizing the mass of light in terms of its energy. And indeed, in his 1900 paper, Poincaré arrived at this exact conclusion, writing that "the electromagnetic field should be regarded as a sort of fluid which, when set in motion, possesses its own inertia, that is, its own mass." The value of this inertial density, he wrote down as K * J, where K here refers to 1/c² and J represents energy density. Thus, in this part of his paper, Poincaré explicitly demonstrates that the mass density of the electromagnetic field is equal to its energy density over c².
Now, the same year that this paper was published, a certain 21-year-old by the name of Albert Einstein had just completed his studies at the Polytechnic Institute in Zurich. Now, like the other graduates of his class, he expected that he would be hired by one of his former professors for an assistantship position upon graduation. However, unlike the other graduates, he had, due to his disinterest in the subject material being taught, ditched the majority of his lectures, passing his exams only by borrowing his friends' notes and ultimately receiving marks just sufficient enough to graduate. Unsurprisingly, his former professors all refused to hire him, and subsequently, the young man was left with a lot of free time on his hands. Free time, which he used to continue his own independent study of electromagnetic theory.
In particular, Einstein was drawn to the works of Poincaré, with he and his friends even forming a Poincaré reading group around that time. Not coincidentally, several years later, Einstein would wind up publishing a paper on the theory of special relativity, which traded heavily in Poincaré's ideas. In particular, the idea of the principle of relativity, the notion of local times, the one-way synchronization issue for spatially separated clocks, and the notion of the frame-invariant behavior of the speed of light were all directly lifted from various works of Poincaré.
Shortly thereafter, after this first special relativity paper in 1905, Einstein would also publish a second paper, this time on E=mc², which again proved to be directly lifted from Poincaré's work. Indeed, towards the end of his 1900 paper, Poincaré had considered the case of a body of mass which emits a burst of radiation and recoils. Poincaré then employed the principle of relativity to consider how the energy of such radiation might appear to two observers in different frames of reference.
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Likewise, Einstein in 1905 also considered the case of a body of mass which emits radiation. Though here, he modified Poincaré's premise by assuming the radiation to have been emitted in both directions, thus avoiding the recoil of the mass, just as Poincaré did. Einstein then employed the principle of relativity to consider how the energy would be measured in different frames of reference.
Now, in 1900, Poincaré's considerations primarily concluded by noting that the principle of relativity and the concept of frame-invariant energy were incompatible, meaning that observers in different reference frames would be required to measure different values for the energy of the emitted radiation. Einstein wanted to take this thought experiment a step further, however, and extract the notion of mass-energy equivalence from it. Unfortunately, his proof was far from rigorous and proved to be rather problematic. While, although thanks to his "relative Doppler" derivation, he begins with the correct expressions for relativistic energy, he proceeds to assert that the change in kinetic energy of the body in the moving frame equals the difference between the energy measurements made by the two observers in their different frames of reference, without making it clear why he assumes this. Such an assertion has produced a great deal of scholarly debate over the years, with some arguing that it assumes the very notion of mass-energy equivalence that it's trying to prove.
But equally confusing is the fact that in order to arrive at his famous result, Einstein winds up taking a low-velocity approximation for the change in kinetic energy without explaining why he does so. This is particularly strange, given he had derived the correct, full expression for relativistic kinetic energy in his last paper and shouldn't have needed to take a low-velocity approximation at all. Regardless, to the astute relativist here, it's clear that at this point in time, Einstein was still fumbling his way around a formalism he hadn't so much invented as appropriated from others.
Indeed, Einstein's 1905 paper soon came under heavy criticism from Max Planck. Sensing he had done an unsatisfactory job, Einstein returned to make a second effort at proving E=mc² in 1906. This time around, he realized that the implication of mass-energy equivalence that he had been seeking wasn't contained within the principle of relativity portion of Poincaré's paper, but rather implied in a different section.
Indeed, in possibly the most crucial part of Poincaré's paper, the French mathematician had written that for a system of bodies with no external forces acting on them, the only way for Newton's third law to be valid was if the total momentum of the masses within the system plus the total momentum of the intervening electromagnetic fields was conserved. Specifically, he wrote that if this condition was met, then the center of gravity of the system (aka its center of mass) would always move in a straight line.
This clue propelled Einstein to a new and much more clever proof of mass-energy equivalence, which he detailed in his 1906 paper entitled "The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy," and for which, this time around, he did finally credit Poincaré.
In this experiment, Einstein imagined a rigid, empty box of mass which emits a burst of radiation from its left end.
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Since this radiation carries momentum, the box simultaneously recoils to the left. The radiation then travels to the right end of the box, where it is subsequently reabsorbed, bringing the box to a halt. Now, to an outside observer, it would appear as though during this entire process, the center of mass of the box had somehow spontaneously shifted. That is, despite no external forces acting on the box, it somehow underwent a translation. Such a spontaneous translation of mass, Einstein argued, would violate the laws of mechanics.
The only way to resolve this problem, then, would be to assign a quantity of mass to the radiation such that when this radiation was emitted, the box lost that quantity of mass. To understand this, let's imagine that the radiation is like a billiard ball, initially attached to the left side frame of the box. When the Box emits this radiation, the billiard ball travels to the right, while the box travels to the left. The billiard ball then hits the right end of the box, and the box stops moving. During this whole process, because the billiard ball has mass, the center of mass of the collective system remains stationary, as a portion of the total mass of the system is merely shifted from the left side to the right side.
But of course, the left side of the box has to lose this amount of mass, and the right side has to gain it. This means that if radiation, i.e., light, likewise carries mass, then when the box emits this radiation, it must lose some of its mass so that the center of mass of the system remains stationary and there is no spontaneous motion.
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This was the clear implication of mass-energy equivalence Einstein had been seeking all along. Indeed, to calculate the amount of mass lost by the box, Einstein simply followed Poincaré's lead and imposed the condition that the center of mass of the system, that is, the mass of the box plus whatever mass was carried away by the radiation, should remain stationary. The resultant math showed that this could only be accomplished if the amount of mass carried away by the emitted radiation was none other than a quantity equal to that of its energy / c².
Now, through this proof, Einstein made explicit the transactional relationship between mass and energy, which was already implicit in Poincaré's work. But alas, although it cemented the notion of mass-energy equivalence, it still hadn't achieved what Einstein wanted. It still wasn't a relativistic proof of E=mc², since it relied on no axioms of special relativity whatsoever. Moreover, it was only technically valid for low-velocity considerations, since the recoil velocity of the box was assumed to be much lower than the speed of light.
Interestingly, over the course of the next 40 years, Einstein would go on to attempt to prove E=mc² five more times, with some scholars arguing that none of these attempts proved to be wholly satisfactory. Today, mass-energy equivalence is demonstrated by first introducing the notion of relativistic momentum (or relativistic mass) and then deriving E=mc² from there. Relativistic momentum, in turn, is derived from the principle of relativity. So, the complete modern derivation of E=mc² would still appear to be a relativistic one.
Yet, in perhaps a final twist of ironic fate, it's within these early proofs of Einstein's that a significant clue to the classical meaning behind relativistic momentum can be found. That is, with some slight modifications to Einstein's proofs, it can be shown that the concept of relativistic mass is derivable through entirely classical concepts, without any reference to the principle of relativity. And well, for those of you who have been following our channel for a while now, you know what that means. This has been Dialect. Stay tuned.
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