Consider an object at rest in the laboratory frame of reference. It emits two bundles of radiant energy, one to the west (left in the figure), one to the east (right in the figure), each labeled γ. For convenience, I will call these energy bundles photons, although Einstein deliberately avoided doing so.1
Laboratory frame of reference. An object emits oppositely directed photons of equal energy.
Since equally energetic photons have equal magnitudes of momentum, the total momentum carried away by the photons is zero (a vector sum). The motionless object, having lost no momentum, remains motionless. It is as if a rifle could fire bullets simultaneously in opposite directions and thereby suffer no recoil.
Our object does lose energy, however. Its energy loss is the sum of the energies of the two photons, or twice the energy of each. The fact that some energy has been carried away from the object does not by itself reveal that the object must have diminished its mass, for the energy might have been supplied by chemical change, nuclear transformation, or the random kinetic energy of molecular vibrations. There is no shortage of possible internal sources of energy to account for the emission of the radiation. That a mass change accompanies the energy change can be discovered only by reexamining the same process in a moving frame of reference.
Imagine yourself now moving through the laboratory at constant speed v, westbound, let us say, to be definite. From your point of view the object (and, of course, the whole laboratory, but that is irrelevant) is moving eastward at a fixed speed v before it emits the pair of photons, and at the same speed v after it emits them. Since it experienced no recoil in the laboratory frame of reference, it experiences no change of velocity in the moving frame of reference. This is a key point.
Moving frame of reference. The object does not change speed as it emits photons of different energy.
However, in the moving frame of reference the two bundles of radiation do not possess the same momenta or the same energies as in the laboratory frame. This is not surprising. If a baseball pitcher threw a ball from a platform moving toward the batter, the ball would arrive over the plate with more energy and more momentum than if he threw the ball from a stationary platform or from a receding platform. Similarly, the eastbound photon, thrown forward from the moving observers point of view, has more energy than the same photon in the laboratory frame of reference, more by a factor
The backward-thrown photon has less energy than in the laboratory frame, less by a factor
Where did these factors come from? Within Maxwell’s electromagnetic theory, as Einstein deduced, is the implication that the energy in a bundle of radiation depends on the frame of reference in which it is viewed and that the factor of change from one frame to another is given by these factors. Einstein could use this “classical” result because he knew that electromagnetic theory (unlike, say, Newtonian theory) must be consistent with special relativity, since it deals with energy moving at the speed of light.2
The importance of these factors lies in the fact that their sum is not unity. Suppose that the energy of one photon in the laboratory frame is called 1/2ΔE. Then, in the moving frame, the energies of the two photons are
The sum of these two energies is
By contrast, the total photon energy in the laboratory frame is
ΔE (total energy loss in laboratory frame).
The same pair of photons have more energy in the moving frame than in the laboratory frame. Accordingly the object has lost more energy from the moving observers point of view than from the laboratory observers point of view. Apparently the law of energy conservation does not possess the invariance required by the Principle of Relativity. Can it be saved? Yes, but only at the expense of what was one of the solidest laws of nineteenth-century physical science, the law of mass conservation.
Einstein argued that the place to find the extra energy needed to make up the apparent discrepancy between the two observers is in the bulk kinetic energy of the object that emits the photons. In the laboratory frame the object remains motionless, with zero kinetic energy both before and after the photons are emitted. In the moving frame, it possesses kinetic energy which, according to old ideas, remains constant because its speed remains constant. But, said Einstein, suppose that a decrease of mass accompanies the emission of the photons. Then, in the moving frame of reference, there is a loss of kinetic energy even though there is no change of speed. This change of kinetic energy can provide the extra energy content of the photons in the moving frame of reference and preserve the invariance of the law of energy conservation. It was Einstein s genius to recognize the more fundamental nature of energy conservation than of mass conservation, and to sacrifice the one in order to save the other.
Once the change of mass has been pinpointed conceptually as the key to preserving energy conservation, it is not a difficult matter to derive mathematically the relationship between the energy ΔE of the emitted photons in the laboratory frame and the mass change Δm of the emitting object. One may do this by postulating that the object emitting the photons is moving slowly (that is, v << c) so that its kinetic energy is given accurately by K.E. = 1/2mv2. If its mass changes as a result of the emission of the photons, but its speed doesn’t, its change of kinetic energy will be
ΔK.E. = ½Δmv2.
This can, in turn, be equated to the difference between the two expressions above for the loss of energy in the stationary and in the moving frame. In taking that difference, one may use the approximations √1 – x ≅ 1 – (½)x and 1/√(1 – x) ≅ 1 + (½)x, valid for small x. When this is done, one finds the change of energy of the object to be
Change of energy = ΔE (1/2) (v2/c2).
Equating these two expressions for energy change then gives
Δm = ΔE/c2,
Einstein’s famous formula. In words: If the object suffers a loss of mass equal to the total radiated energy divided by the square of the speed of light, the law of energy conservation will be preserved in all inertial frames of reference.
Is our derivation faulty—or limited in its range of validity—because we assumed the object to be moving slowly relative to the speed of light? No. Einstein himself made that approximation, yet realized, once he had that result, that it was generally valid and led to a self-consistent overall account of mass and energy.
Einstein realized, too, that mass changes in ordinary processes, even violent chemical ones, would be too small to measure. He suggested that radioactivity, where much greater energy changes per unit mass are displayed, might lend itself to verifying the mass-energy equivalence. He probably did not envision a 100-percent transformation of mass to energy (although I dont rule out that this thought crossed his mind). In any event, exactly such a process is now known. When a neutral pion decays into two photons, all of its mass disappears and the photons carry away an energy exactly equal to mc2.
1 Einstein’s E = mc2 paper, under discussion here, was published in September 1905, several months after he had published the idea of quantized bundles of radiant energy—later called photons—and showed how each bundle’s energy is related to its frequency (E = hν). Why, in the September paper, did Einstein choose not to introduce the photon idea, even though it would have made the derivation of E = mc2 easier? Perhaps he thought it best not to base one revolutionary idea (mass-energy) on the back of another revolutionary idea (photons).
2 These factors of energy change, are, as Einstein knew, also the factors by which the radiant energy bundles change their frequency—consistent with the photon formula E = hν. That is, they are Doppler-effect formulas.