No equation is more famous than E = mc2, and few are simpler. Indeed, the immortal equation’s fame rests largely on that utter simplicity: the energy E of a system is equal to its mass m multiplied by c2, the speed of light squared. The equation’s message is that the mass of a system measures its energy content. Yet E = mc2 tells us something even more fundamental. If we think of c, the speed of light, as one light year per year, the conversion factor c2 equals 1. That leaves us with E = m. Energy and mass are the same.
According to scientific folklore, Albert Einstein formulated this equation in 1905 and, in a single blow, explained how energy can be released in stars and nuclear explosions. This is a vast oversimplification. Einstein was neither the first person to consider the equivalence of mass and energy, nor did he actually prove it.
Anyone who sits through a freshman electricity and magnetism course learns that charged objects carry electric fields, and that moving charges also create magnetic fields. Hence, moving charged particles carry electromagnetic fields. Late 19th-century natural philosophers believed that electromagnetism was more fundamental than Isaac Newton’s laws of motion and that the electromagnetic field itself should provide the origin of mass. In 1881 J. J. Thomson, later a discoverer of the electron, made the first attempt to demonstrate how this might come about by explicitly calculating the magnetic field generated by a moving charged sphere and showing that the field in turn induced a mass into the sphere itself.
The effect is entirely analogous to what happens when you drop a beach ball to the ground. The force of gravity pulls the ball downward; buoyancy and drag forces from the air impede the ball’s fall. But this is not the whole story. Drag or no drag, in order to fall the ball must push the air ahead of it out of the way and this air has mass. The “effective” mass of the falling beach ball is consequently larger than the mass of the ball at rest. Thomson understood that the field of the sphere should act like the air before the beach ball; in his case the effective mass of the sphere was the entire mass induced by the magnetic field.
Thomson’s slightly complicated result depended on the object’s charge, radius and magnetic permeability, but in 1889 English physicist Oliver Heaviside simplified his work to show that the effective mass should be m = (4⁄3) E / c2, where E is the energy of the sphere’s electric field. German physicists Wilhelm Wien, famous for his investigations into blackbody radiation, and Max Abraham got the same result, which became known as the “electromagnetic mass” of the classical electron (which was nothing more than a tiny, charged sphere). Although electromagnetic mass required that the object be charged and moving, and so clearly does not apply to all matter, it was nonetheless the first serious attempt to connect mass with energy.
It was not, however, the last. When Englishman John Henry Poynting announced in 1884 a celebrated theorem on the conservation of energy for the electromagnetic field, other scientists quickly attempted to extend conservation laws to mass plus energy. Indeed, in 1900 the ubiquitous Henri Poincaré stated that if one required that the momentum of any particles present in an electromagnetic field plus the momentum of the field itself be conserved together, then Poynting’s theorem predicted that the field acts as a “fictitious fluid” with mass such that E = mc2. Poincaré, however, failed to connect E with the mass of any real body.
The scope of investigations widened again in 1904 when Fritz Hasenöhrl created a thought experiment involving heat energy in a moving cavity. Largely forgotten today except by Einstein detractors, Hasenöhrl was at the time more famous than the obscure patent clerk. Then one of Austria’s leading physicists, he wrote a prize-winning trilogy of papers, “On the theory of radiation in moving bodies,” the last two of which appeared in the Annalen der Physik in 1904 and early 1905. In the first he imagined a perfectly reflecting cylindrical cavity in which the two end disks—which served as heaters—were suddenly switched on, filling the cavity with ordinary heat, or in physicist lingo, blackbody radiation. Newton’s third law (“for every action there is an equal and opposite reaction”) tells us in modern language that any photons emitted from the heaters must exert a reaction force against the heaters themselves, and so to keep them in place one must exert an external force against each of them (we imagine that these external forces are what keep the disks attached to the cylinder). But because identical photons are emitted from each end, the forces are equal in magnitude, at least as observed by someone sitting inside the cavity.
Hasenöhrl, though, next asked what the system looks like as it moves at a fixed velocity with respect to an observer sitting in a laboratory. Basic physics tells us that light emitted from a source moving toward you becomes bluer, and gets redder from a source moving away from you—the famous Doppler shift. Photons from one end disk will therefore appear Doppler blue-shifted to the laboratory observer and those from the other end will be red-shifted. Blue photons carry more momentum than red photons and hence, in order to keep the cavity moving at constant velocity the two external forces must now be different. A simple application of the “work–energy theorem,” which equates the difference in work produced by the forces to the cavity’s kinetic energy, allowed Hasenöhrl to conclude that blackbody radiation has mass m = (8⁄3) E / c2. In his second paper Hasenöhrl considered a slowly accelerating cavity already filled with radiation and got the same answer. After a communication from Abraham, however, he uncovered an algebraic error and in his third paper corrected both results to m = (4⁄3) E / c2.
In considering the mass inherent in heat Hasenöhrl extended the previous deliberations beyond the electromagnetic field of charged objects to a broader thought experiment very similar to Einstein’s own of the following year, which gave birth to E = mc2. Of course Hasenöhrl was writing pre-relativity and one might think an incorrect result was inevitable. The matter is not so simple. Astronomer Stephen Boughn and I have closely analyzed Hasenöhrl’s trilogy and the usual claim that “he forgot to take into account the forces the shell itself exerts to hold the end caps in place” is not the issue. The main error in Hasenöhrl’s first thought experiment is that he did not realize that if the end caps are emitting heat, they must be losing mass—an ironic oversight given that it is exactly the equivalence of mass and energy he was attempting to establish. Nevertheless, Hasenöhrl was correct enough that Max Planck could say in 1909, “That the black body radiation possesses inertia was first pointed out by F. Hasenöhrl.” Black body radiation—heat—has mass.
A greater surprise is that with his second experiment, in which the cavity is already filled with radiation and the caps are not radiating, Hasenöhrl’s answer is not obviously wrong, even according to relativity. Einstein’s famous 1905 E = mc2 paper, “Does the inertia of a body depend on its energy content?” considers only a point particle emitting a burst of radiation and asks, as Hasenöhrl did, how the system looks from a moving reference frame. In considering a cavity of finite length Hasenöhrl was being much more audacious, or reckless. Extended bodies have produced numerous and prolonged headaches in special relativity, such as the fact that the mass of the classical electron also comes out to m = (4⁄3) E / c2. That is, using relativistically correct mathematics one gets a result that at first blush contradicts the answer everyone expects and loves. Arguments about how to properly resolve the issue persist to this day.
Equally surprising is that although Einstein was the first to propose the correct relationship, E = mc2, he didn’t actually prove it, at least according to his own special relativity. Einstein began by employing the relativistic relationships (the relativistic Doppler shift) he had derived a few months earlier but finally approximated away the relativistic bits, leaving an answer one can get from purely classical physics and which may or may not remain true at the higher velocities where relativity comes into play. Moreover, although he stated that his conclusions apply to all bodies and all forms of energy, Einstein certainly made no attempt to prove it. He was aware of the shortcomings of his derivation and wrote a half dozen more papers over the next 40 years trying to patch things up but arguably never succeeded. Of course, countless experiments since then have convinced us of the correctness of Einstein’s result.
One naturally wonders whether Einstein knew of Hasenöhrl’s work. It is difficult to believe that he did not, given that the bulk of the prize-winning trilogy appeared in the most prominent journal of the day. Certainly at some point he learned of Hasenöhrl: a famous photo of the first Solvay Conference in 1911 shows both men gathered around the table with the other illustrious attendees.
And so, although Einstein achieved a definite conceptual advance in equating the mass of an object with its total energy content—whether or not it is moving, whether or not it has an electromagnetic field—we can also credit Hasenöhrl for unambiguously recognizing that heat itself possess an equivalent mass, and physicists before him for providing a chain of shoulders on which to stand. E = mc2 is the short punch line to a long and winding scientific story.