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An Elementary Derivation of E = mc2†

The following derivation of E = mc2, which, according to Max Born, is due to Einstein [1], but which Einstein attributes to Poincaré [2], does not make use of anything at all from special relativity, but rather, it only makes use of very elementary concepts of classical mechanics and electromagnetism. (Although the result, i.e. E = mc2, is exact it turns out that the derivation has only approximate validity, since it uses non-relativistic formulae like the non-relativistic formula for the three momentum of the recoiling body.)

The proof presented here is based on the fact that radiation exerts a pressure on all material objects upon which it impinges. Maxwell showed that when an amount of electromagnetic radiation is absorbed by an object it exerts a ``radiation pressure’’ p on the object which is equal to

I p  (1) c where I is the intensity of the absorbed radiation. (I is equal to , the time averaged magnitude of the Poynting vector for a plane electromagnetic wave impinging upon a surface.) Similarly, Poincaré showed, using Maxwell’s equations together with Poynting’s theorem, that a light wave which falls upon an absorbing body imparts a  momentum p to it, the magnitude of Poincaré’s momentum of radiation being given by [3]:

 E | p | = (2) c where E is the energy of the light wave, and c is the speed of light.

Now imagine a cylinder of length L as shown in the FIGURE. At the ends of the cylinders are two identical bodies A and B, except that A carries an additional amount of energy E (in the form of heat say). Suppose that a certain time the energy E is sent from A to B in the form of light (electromagnetic radiation). We assume that the spatial extent of this light flash is incomparably small to the length L of the FIGURE: A cylinder with two identical bodies A and B at its ends, except that A carries an additional energy E which is sent from A to B in the form of a light flash with velocity c. The recoil produces a velocity v of the tube. When E is absorbed by B, the tube must again be at rest, but displaced by a distance x. cylinder. The (non-relativistic) law of conservation of momentum together with equation (2) gives:

E M v = (3) c where M is the mass of the cylinder. After a time t the light wave reaches the other end of the cylinder and the body B of mass M/2* absorbs the radiation, thus imparting an additional velocity – v to the cylinder which is moving as a whole to the right with speed v, so that the cylinder comes to rest again. Since the light wave travels across the length of the cylinder with speed c, we have

L t = (4) c and, since the cylinder travels with a constant speed v during the the time t, we also have

L x = v t = v . (5) c

If we solve equation (3) for v and substitute this into equation (5), we obtain

EL X = 2 . (6) Mc

Now the bodies A and B may be exchanged without any external influences. (Imagine 2 men are in the cylinder, and one of the men puts A in the place of B, and the other man puts B in the place of A. Afterwards the two men return to their original positions.) Clearly, everything has returned to the original state of affairs: two identical bodies, A and B, with A on the left side of the cylinder containing an additional amount of energy E. Yet the center of mass of the whole cylinder has moved a distance x and come back to rest without any ______

*We assume that (essentially) all of the mass of the cylinder is shared equally by the two bodies, so that the mass of body B is M/2. net external force acting on it! This gives rise to a “perpetuum mobile,” since we can repeat the process again and again creating perpetual motion without external work being done on the system!

The only reasonable way out of this absurd state of affairs is to assume that when the two bodies (A and B) are exchanged, these two bodies are not identical but that B now has a mass M/2 + m as a consequence of it absorbing an energy E. Now the situation is quite different with regards to the process involving the exchange of the two bodies: the additional mass m is moved by one of the men from right to left by a distance L. At the same time, the whole cylinder is displaced a distance x in the reverse direction, so that now the cylinder has moved back to its original position, as it must, in order to prevent the possibility of violation of Newton’s second law.

Quantitatively, we must have for the process involving the men the following: the total momentum, which consists of that of the cylinder, x L M , and that of the transported mass –m , must be zero. (t’ is t' t' the time taken for the men to exchange the two bodies with each other.) Thus:

Mx = m L (7) from which it follows that

mL X = . (8) M

Finally, we compare equations (6) and (8) to arrive at E=mc2.

† [1] This handout is based on the treatment given on pages 283 to 286 of the book, Einstein’s Theory of Relativity, by Max Born, Dover Publications, New York (1965). In the treatment given in this handout, the cylinder is treated as a perfectly rigid body, but Born shows how this derivation can be slightly modified to get around this, and satisfy the demands of special relativity that there be no perfectly rigid body. [2] A. Einstein, Annalen der Physik, 18, (1906) Specifically, citing ref. [3] Einstein writes: “Although the simple formal considerations . . .are in the main already contained in a work by H. Poincaré, for the sake of clarity I shall not base myself upon that work.” [3] H. Poincaré, , Archives nederlandaises des Sciences exactes et naturelles, 20series, t. 5, p. 252-278 (1900).

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