Einstein’s Box 

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In 1906 Einstein published a paper entitled The Principle of Conservation of the Center of Gravity and the Inertia of Energy [Ref 1]. In this article Einstein utilizes that center-of-mass theorem which states that the center of mass of a closed system, which is initially at rest, will remain at rest. Einstein’s paper consists of two sections. Section 1 is entitled A special case. The second section is entitled On the Principle of the Conservation of the Center of the motion of the center of gravity. It is the second section that gives the more general version of the center of mass theorem. Descried below is the first section. This section of Einstein’s paper provides the springboard to the derivations given in the second section.
    Consider a hollow cylinder of mass M, which is initially at rest in an inertial frame of reference S. The cylinder has two walls at each end of equal mass. Einstein considers a six step process: The first step consists of a finite amount of radiation (photons) of total energy E is emitter from end A towards end B parallel to the axis of the cylinder as shown in Fig. 1 below

The spatial extent of the radiation is small compared to the length of the cylinder. In modern language we can think of this radiation as a single photon of the same energy. The relationship between the energy of an amount of radiation of energy E has an amount of radiation of magnitude p is

This is the phenomenon of radiation pressure. In order for momentum to be conserved momentum must be transferred to the cylinder, which then starts to move with velocity v. Conservation of momentum then demands that

The speed of the cylinder is then

The speed of the box will remain constant until the radiation arrives at end B. The wall then absorbs the radiation and the cylinder comes to rest. The flight time, T, of the radiation is related to the speed of light and the distance traveled. This distance is approximately L, the length of the cylinder and thus

During this time the cylinder moves a distance d where

Next an amount of energy E is transferred from wall B into a massless carrier, k, collocated at wall B. The carrier then moves to wall A wherein the energy E is transferred from the carrier to wall A. The carrier then moves back to wall B. The whole system has undergone a complete cycle which may be repeated an arbitrary number of times. These steps are shown in Fig. 2 below

If the carrier remains massless even after it has absorbed an amount of energy E then the returning of the energy to wall A will not result in a change in position of the cylinder. This implies that the only result of this six-step cycle is to displace the body by an amount d. This shift can be made arbitrarily large by repeating the process. Thus if the assumption that the carrier remains massless is valid then the system, which was initially at rest, will have its center of mass shifted without any external forces acting on the system. This clearly violates the law of mechanics which states that a body, which is initially at rest, cannot undergo translational motion unless there is an external force acting on the body. Einstein therefore concludes

…, if one assumes that any energy E possesses the inertia E/c², then the contradiction with the principle of mechanics disappears. For according to the this assumption the carrier body of mass E/c² while it transports the amount of energy E from B to A; and since the center of gravity of the entire system must be at rest during this process according to the center-of-mass theorem the cylinder K undergoes during it a total shift S’ to the right..

where the total shift S’ is given by d. This result follows by examining the center of mass of this system, which demands that an amount of mass, m, be transported to the right, i.e.

Substituting the expression for d from Eq. (5) above we find

Canceling like factors of M and L we arrive at the general result

In section 2 Einstein uses this expression by assigning mass to the energy of radiation. Einstein makes the substitution of the energy density of radiation with a mass density when he stated that If we assign the electromagnetic field too a mass density (re)..

Einstein’s derivation has some conceptual problems with it, namely that Einstein treats the cylinder as a rigid body, a concept that is inconsistent with the principles of special relativity. Other variations have been offered since Einstein’s article was published. French circumvents the rigid body problem and considers only the two walls, A and B, of the box arriving at the same result utilizing the center-of-mass theorem [2]. Antipa presented the idea of considering instead an atom that emits a photon and applies the center-of-mass theorem to the atom-photon system [3]. The following is a variation of that approach, the only exception being a simplification which is effected by employing the principle of the conservation of mass. Consider an atom of mass M initially at rest in the inertial frame S at x = Xcm. A photon of energy E and mass m is emitted in the +x-direction as shown below in Fig. 3.

The mass of the atom is decreased to the new value m. The magnitude of the photon’s momentum is given by Eq. (1) as p = E/c. Since the initial total momentum is zero the atom must have a momentum of the same magnitude and having the opposite direction. The atom must therefore be kicked in the x-direction and moves with speed v as illustrated in Fig. 3. The magnitude of the momentum of the atom is mv and has the same magnitude as the momentum of the photon, i.e. mv = p = E/c. Conservation of mass demands that

The x-component of the center of mass vector is given by as

where the x-coordinate of the atom at time t is x1 and the x-coordinate of the photon at time t is x2. x-coordinate can be read off from Fig. 1. They have the values

Substituting Eq. (11) into Eq. (10) and multiplying through by M, gives

Substituting in Eq. (9) and expanding gives

Canceling like terms reduces Eq. (13) to

Solving for m and substituting mv = E/c gives

Note that m is the mass of the photon (not the proper mass). This is simply the statement of mass-energy equivalence found in Eq. (8) above. 

References: 

[1] The Principle of Conservation of the Center of Gravity and the Inertia of Energy, Albert Einstein, Annalen der Physik, 20 (1906): 626-633.
[2] Special Relativity, A.P. French, The MIT Intro. Series,. W.W. Norton & Co., (1968), page 16
[3] Inertia of Energy and the Liberated Photon, Adel Antipa, Am. J. Phys. 44(9), Sept 1976 (1976) 

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