The great physicist was not the first to equate forms of mass to energy, nor did he definitively prove the relationship

No equation is more famous than

*E = mc2*, và few are simpler. Indeed, the immortal equation’s fame rests largely on that utter simplicity: the energy

*E*of a system is equal to lớn its mass

*m*multiplied by

*c*2, the tốc độ 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 tốc độ 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.

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According lớn scientific folklore, Albert Einstein formulated this equation in 1905 and, in a single blow, explained how energy can be released in stars & nuclear explosions. This is a vast oversimplification. Einstein was neither the first person khổng lồ consider the equivalence of mass and energy, nor did he actually prove it.

Anyone who sits through a freshman electricity & 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 lớn demonstrate how this might come about by explicitly calculating the magnetic field generated by a moving charged sphere & 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 khổng lồ the ground. The force of gravity pulls the ball downward; buoyancy & 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 & 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 và magnetic permeability, but in 1889 English physicist Oliver Heaviside simplified his work to lớn 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, & 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 lớn all matter, it was nonetheless the first serious attempt khổng lồ 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 lớn extend conservation laws khổng lồ 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 khổng lồ 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 & early 1905. In the first he imagined a perfectly reflecting cylindrical cavity in which the two kết thúc 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 & opposite reaction”) tells us in modern language that any photons emitted from the heaters must exert a reaction force against the heaters themselves, & so khổng lồ 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 lớn 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 lớn an observer sitting in a laboratory. Basic physics tells us that light emitted from a source moving toward you becomes bluer, và 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 lớn the laboratory observer & those from the other over will be red-shifted. Blue photons carry more momentum than red photons & hence, in order to lớn 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 khổng lồ the cavity’s kinetic energy, allowed Hasenöhrl khổng lồ 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 & got the same answer. After a communication from Abraham, however, he uncovered an algebraic error and in his third paper corrected both results lớn *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 lớn a broader thought experiment very similar lớn Einstein’s own of the following year, which gave birth to lớn *E = mc2*. Of course Hasenöhrl was writing pre-relativity và 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 lớn take into account the forces the shell itself exerts to hold the kết thúc 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 kết thúc 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 toàn thân radiation possesses inertia was first pointed out by F. Hasenöhrl.” Black body toàn thân radiation—heat—has mass.

A greater surprise is that with his second experiment, in which the cavity is already filled with radiation & the caps are not radiating, Hasenöhrl’s answer is not obviously wrong, even according to lớn relativity. Einstein’s famous 1905 *E = mc2*paper, “Does the inertia of a body toàn thân 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 và prolonged headaches in special relativity, such as the fact that the mass of the classical electron also comes out khổng lồ *m = (4⁄3) E / c2*. That is, using relativistically correct mathematics one gets a result that at first blush contradicts the answer everyone expects và loves. Arguments about how lớn properly resolve the issue persist lớn this day.

Equally surprising is that although Einstein was the first khổng lồ propose the correct relationship, *E = mc2*, he didn’t actually prove it, at least according to lớn 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 lớn all bodies và all forms of energy, Einstein certainly made no attempt khổng lồ 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 lớn 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, & physicists before him for providing a chain of shoulders on which to lớn stand. *E = mc2*is the short punch line to a long and winding scientific story.

For hundreds of years, there was an immutable law of physics that was never challenged: that in any reaction occurring in the Universe, mass was conserved. That no matter what you put in, what reacted, và what came out, the sum of what you began with & the sum of what you ended with would be equal. But under the laws of special relativity, mass simply couldn"t be the ultimate conserved quantity, since different observers would disagree about what the energy of a system was. Instead, Einstein was able to derive a law that we still use today, governed by one of the simplest but most powerful equations ever to be written down,*E = mc2*.

A nuclear-powered rocket engine, preparing for testing in 1967. This rocket is powered by... <+> Mass/Energy conversion, and E=mc^2. ECF (Experimental Engine Cold Flow) experimental nuclear rocket engine, NASA, 1967

**, or energy, which is the entirety of one side of the equation, & represents the total energy of the system.**

*E***, or mass, which is related khổng lồ energy by a conversion factor. And**

*m**, which is the tốc độ of light squared: the right factor we need khổng lồ make mass và energy equivalent.*

**c2**Niels Bohr and Albert Einstein, discussing a great many topics in the trang chủ of Paul Ehrenfest in... <+> 1925. The Bohr-Einstein debates were one of the most influential occurrences during the development of quantum mechanics. Today, Bohr is best known for his quantum contributions, but Einstein is better-known for his contributions khổng lồ relativity và mass-energy equivalence.

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Paul EhrenfestWhat this equation means is thoroughly world-changing. As Einstein himself put it:

It followed from the special theory of relativity that mass & energy are both but different manifestations of the same thing — a somewhat unfamiliar conception for the average mind.

Here are the three biggest meanings of that simple equation.

The quarks, antiquarks, & gluons of the standard mã sản phẩm have a màu sắc charge, in addition to all the... <+> other properties like mass & electric charge. Only the gluons và photons are massless; everyone else, even the neutrinos, have a non-zero rest mass. E. Siegel / Beyond The Galaxy

**Even masses at rest have an energy inherent to them**. You"ve learned about all types of energies, including mechanical energy, chemical energy, electrical energy, as well as kinetic energy. These are all energies inherent to lớn moving or reacting objects, và these forms of energy can be used to vị work, such as run an engine, power nguồn a light bulb, or grind grain into flour. But even plain, old, regular mass at rest has energy inherent khổng lồ it: a tremendous amount of energy. This carries with it a tremendous implication: that gravitation, which works between any two masses in the Universe in Newton"s picture, should also work based off of energy, which is equivalent to mass via *E = mc2*.

The production of matter/antimatter pairs (left) from pure energy is a completely reversible... <+> reaction (right), with matter/antimatter annihilating back khổng lồ pure energy. This creation-and-annihilation process, which obeys E = mc^2, is the only known way khổng lồ create và destroy matter or antimatter. Dmitri Pogosyan / University of Alberta

**Mass can be converted into pure energy**. This is the second meaning of the equation, where *E = mc2*tells us exactly how much energy you get from converting mass. For every 1 kilogram of mass you turn into energy, you get 9× 1016 joules of energy out, which is the equivalent of 21 Megatons of TNT. When we experience a radioactive decay, or a nuclear fission or fusion reaction, the mass of what we started with is greater than the mass we wind up with; the law of conservation of mass is invalid. But the amount of the difference is how much energy is released! That"s true for everything from decaying uranium to lớn fission bombs lớn nuclear fusion in the Sun to lớn matter-antimatter annihilation. The amount of mass you destroy becomes energy, and the amount of energy you get is given by *E = mc2*.

The particle tracks emanating from a high energy collision at the LHC in 2014. Composite particles... <+> are broken up into their components & scattered, but new particles are also created from the available energy in the collision.

**Energy can be used lớn make mass out of nothing... Except pure energy**. The final meaning is the most profound. If you take two billiard balls và smash them together, you get two billiard balls out. If you take a photon and and electron and smash them together, you get a photon và an electron out. But if you smash them together with enough energy, you"ll get a photon, & electron, and a new matter-antimatter pair of particles out. In other words, you will have created two new massive particles:

whose existence can only arise if you put in enough energy khổng lồ begin with. This is how particle accelerators, lượt thích the LHC at CERN, tìm kiếm for new, unstable, high-energy particles (like the Higgs boson or the top quark) in the first place: by making new particles out of pure energy. The mass you get out comes from the available energy: *m = E/c2*. It also means that if your particle has a finite lifetime, then due lớn Heisenberg uncertainty, there"s an inherent unknowability khổng lồ its mass, since∆*E*∆*t* ~*ħ*, and therefore there"s a corresponding∆*m* from Einstein"s equation, too. When physicists talk about a particle"s width, this inherent mass uncertainty is what they"re talking about.

The warping of spacetime, in the General Relativistic picture, by gravitational masses. LIGO/T. Pyle

The fact of mass-energy equivalence also led Einstein to his greatest achievement: General Relativity. Imagine that you"ve got a particle of matter and a particle of antimatter, each with the same rest mass. You can annihilate them, và they"ll produce photons of a specific amount of energy, of the exact amount given by *E = mc2*. Now, imagine you had this particle/antiparticle pair moving rapidly, as though they had fallen from outer space, & then annihilated close to lớn the surface of Earth. Those photons would now haveextra energy: not just the*E* from *E = mc2*, but the additional*E* from the amount of kinetic energy they gained by falling.

If two objects of matter and antimatter at rest annihilate, they produce photons of an extremely... <+> specific energy. If they produce those photons after falling deeper into a gravitational field, the energy should be higher. This means there must be some sort of gravitational redshift/blueshift, the kind not predicted by Newton"s gravity, otherwise energy wouldn"t be conserved. Ray Shapp / Mike Luciuk; modified by E. Siegel

If we want to conserve energy, we have to lớn understand that gravitational redshift (and blueshift) must be real. Newton"s gravity has no way to account for this, but in Einstein"s General Relativity, the curvature of space means that falling into a gravitational field makes you gain energy, & climbing out of a gravitational field makes you thua trận energy. The full và general relationship, then, for any moving object, isn"t just *E = mc2*, but that *E2 = m2c4*+ *p2c2*. (Where *p* is momentum.)Only by generalizing things lớn include energy, momentum, & gravity can we truly describe the Universe.

When a quantum of radiation leaves a gravitational field, its frequency must be redshifted to... <+> conserve energy; when it falls in, it must be blueshifted. Only if gravitation itself is linked lớn not only mass but energy, too, does this make sense. Vlad2i and mapos / English Wikipedia

Einstein"s greatest equation, *E = mc2*, is a triumph of the power & simplicity of fundamental physics. Matter has an inherent amount of energy khổng lồ it, mass can be converted (under the right conditions) khổng lồ pure energy, và energy can be used lớn create massive objects that did not exist previously. Thinking about problems in this way enabled us khổng lồ discover the fundamental particles that make up our Universe, to invent nuclear power and nuclear weapons, and to discover the theory of gravity that describes how every object in the Universe interacts. And the key to figuring the equation out? A humble thought experiment, based on one simple notion: that energy and momentum are both conserved. The rest? It"s just an inevitable consequence of the Universe working exactly as it does.

I am a Ph.D. Astrophysicist, author, và science communicator, who professes physics and astronomy at various colleges. I have won numerous awards for science writing since 2008 for my blog, Starts With A Bang, including the award for best science blog by the Institute of Physics. My two books, Treknology: The Science of Star Trek from Tricorders lớn Warp Drive, Beyond the Galaxy: How humanity looked beyond our Milky Way và discovered the entire Universe, are available for purchase at Amazon. Follow me on Twitter