Deducing Mass-Energy Equivalence

Consider an object that contains some bound energy E (say, in a capacitor, or any other form of stored energy) that can be released as radiant electromagnetic energy at any selected time. The object is initially at rest in terms of a system S of inertia-based coordinates. Einstein’s original September 1905 argument for mass-energy equivalence can be expressed in terms of two separate processes that we might perform on this object. (We simplify Einstein’s presentation by focusing just on the purely transverse case.)

First possible process: Apply a gentle force for a limited time, causing the object to change its state of motion such that, after the temporary application of the force, the object has an arbitrarily small speed v in terms of S, so it is now at rest in another inertial system S′. We then release the energy E (in terms of S′) from the object, in opposite directions transverse to the direction of motion, so that the object remains at rest in S′.

Second possible process: This is the same as the first process, but carried out in reverse order. We first release the radiant energy E (in opposite directions in terms of S) while the object is still at rest in S, and then we apply a force for a limited time to accelerate the object to speed v (in a direction transverse to the direction of the emitted energy), so it comes to rest in S′.

The principle of relativity implies that, in the first process, the description of the object in terms of S′ at equilibrium after acceleration is congruent to the description of the object in terms of S before the acceleration, and these relations are reciprocal. Also, by conservation of energy, the difference between work done and energy emitted must be the same for these two processes (in terms of either system of coordinates), because they begin and end in the same states, meaning they both begin at rest in S while containing the stored energy, and they both end at rest in S′ after having discharged the stored energy.

However, from the Lorentz invariance established in Einstein’s June 1905 paper, we know that in terms of S, the quantity of transverse radiant energy that was emitted in the first process was E/(1 − v²)^1/2 because the object was moving at speed v when the emission occurred, whereas for the second process, the quantity of radiant energy emitted was simply E, because it was emitted while the object was at rest. (Note that this transverse Doppler effect is unique to special relativity, and the derivation explicitly relies on this.)

Therefore, since the emitted energy is less in the second process (in terms of S) than in the first process, the work done on the object to accelerate it to speed v in the second process must also be less than in the first process. Thus it takes less work to accelerate the object after the energy E has been released than it does while that energy is still stored in the object.

Since v is arbitrarily small, the work required to accelerate an object of rest mass m to speed v is essentially (1/2)mv², (the product of force times applied distance), so the difference in the work for the two processes is (1/2)Δmv², where Δm is the difference in rest mass, and the difference in energy between E and E/(1 − v²)^1/2 is essentially (1/2)Ev², and these differences are equal, so we have Δm = E. Of course, if desired, we can eliminate the restriction to low speeds, making use of the full relativistic expressions for kinetic energy and radiant energy derived in Einstein’s June 1905 paper. Using those formulas, we have

Thus we have again the result Δm = E.

One of the most common misconceptions among anti-relativityists is the idea that Einstein’s 1905 deduction of mass-energy equivalence relied on circular reasoning. This misconception can be traced back to the inventor and anti-relativityist Herbert Ives, who in 1937 published (in the Journal of the Optical Society of America) a specious criticism of Einstein’s paper on the inertia of energy. (Ives claimed that Max Planck had made the same criticism, which is untrue.) This was then naively accepted by historians such as Miller and Jammer, and from their writings the misconception entered the mainstream literature… despite the fact that the criticism has been conclusively disproved. In essence, Ives perceived that Einstein’s conclusion is actually implied by his premises, and from this Ives concluded that Einstein was guilty of circular reasoning (a petitio principii), by assuming what was to be proven. Needless to say (or so one would have thought), this is a senseless criticism, since the conclusion of a derivation is necessarily implicit in its premises. The conclusion of a tightly-reasoned logical argument does not appear by magic from nowhere, it is implicit in the premises, and the purpose of the argument is to demonstrate this implication. If someone wants to question our premises, they can do so, but merely pointing out that our conclusion is implied by our premises (without challenging our premises) is not a valid criticism, it is a confirmation.

No one disputes that, for sufficiently small non-zero v, the work required to accelerate from rest an object with rest mass m to the speed v approaches (1/2)mv², because Newton’s laws hold good in the low speed limit. (We can easily generalize to arbitrary speeds using the relations from the June 1905 paper, so the allegation that Einstein’s argument depended on an invalid approximation is unfounded. In any case, the case of arbitrarily small v suffices to complete the derivation.) Also, Einstein had already shown in his June 1905 paper that if an electromagnetic pulse in the y direction has energy E in terms of S, then it’s energy in terms of S′ moving transversely at an aritrarily small speed v approaches (1/2)Ev². This is due to the transverse Doppler effect which is unique to special relativity, corresponding to the fact that inertia-based coordinate systems are related by Lorentz transformations, all of which had been established in the June paper. Yes, the inertia E/c² of energy E is a fairly immediate consequence of the June paper (indeed for electromagnetic energy it is already a consequence of Maxwell’s equations – which are Lorentz invariant), but it was certainly worth highlighting this extraordinarily significant result explicitly in the September paper.

Students are sometimes confused when they learn that we can predict (to good accuracy) the speeds of the products of nuclear fission purely by the electrostatic forces of repulsion acting on the products, without invoking the mass-energy relation. They think this somehow implies that the reaction does not exhibit the inertia of energy. They make the same mistake regarding chemical reactions. The point is that if we consider two identical masses M and a spring with rest mass m_s, the aggregate of these three entities has rest mass 2M + m_s, but if we force the two masses together by compressing the spring between them, doing an amount of work equal to E, and latch them in this configuration, the rest mass of the aggregate is now 2M + m_s + E/c².  If we then release the latch we can calculate (with good accuracy), purely from the repulsive force of the spring, how the two masses will recoil from each other, but this does not mean that the equivalence of mass and energy is not involved. When the latch is released and the spring imparts its compression energy to the masses, each of the two masses acquires a relativistic mass-energy such that the total mass-energy is conserved. If we allow the masses to perform work on their surroundings to bring them to rest, they will have their original rest masses, and the amount of work they performed on the surroundings will be E/c². Again, the fact that we can calculate the approximate recoil speeds without noticing that the rest mass of the aggregate has changed by the amount of the energy released is irrelevant. The same applies to chemical reactions and nuclear reactions.

There is a tendency to think that if we can start a fire or initiate a nuclear reaction or unlatch a coiled spring without understanding or perceiving the relativistic relations between the energies and rest masses before and after the reaction, this somehow implies that the relativistic relations are not actually involved. That reasoning is not valid. The inertia of energy is fundamental to the Lorentz invariance of all physical laws, without which nothing like the world we inhabit could exist.

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