On Thursday, October 1, 2015 1:04 PM, Jeffrey Schnick <JSchnick@Anselm.Edu> wrote (quoted in italics):
"... I have been thinking about essentially the same example, one discussed earlier in this thread, a control mass system originally consisting of an electron and a positron, each of which is at rest in the systems center of mass..."
It is important to note that condition of being at rest in the system's CM is not enough to determine its rest energy. If the parts of the system interact with one another, then the distance between them is very important. An electron (positron) energy can be considered as stored in its EM field. The field energy of e-p pair is much greater when the members of the pair are widely separated than when brought to rest close together. An electron and positron both at rest in their CM frame will have net energy close to E=2mc^2 only when they are far away from each other. By bringing them close together while still keeping them at rest (call this process "adiabatic contraction") we can make their net EM field negligible. It will be the electric field of a dipole d=qs with vanishing separation s . The initial energy of the system is pumped out of the box in such process. Now, if they annihilate, the rest energy of the resulting photon pair will be E<< 2mc^2. It may be a pair of IR photons instead of the gamma-photons. As we know, generally the parts of the system do not need to be at rest for the system to have rest energy. Neither of the photons in the previous example is at rest, but their system has rest energy and non-zero rest mass.
"Suppose the electron-positron system were at the surface of a planet with no atmosphere (and initially at rest relative to the planet). Consider the separation between the two particles to be negligible. With no annihilation the pair falls with the acceleration due to gravity of the planets. If they annihilate while still at rest relative to the planet, and the photons shoot out horizontally, assuming a small planet, the gravitational deflection of the light is miniscule and the center of mass of the system of two particles remains essentially fixed."
The CM of the system cannot be fixed (stationary) in the field of gravity. As Jeffrey says quite correctly, it is accelerating towards the center of the planet before annihilation, but then it cannot be at rest relative to the planet. And this fall must continue after the annihilation as well! And it is precisely the gravitational deflection that takes care of the continuity of the process, so we cannot say that the CM will remain "essentially fixed" if it was initially falling.
"Two years later, the center of mass is still at the initial position of the electron-positron pair."
If we track the trajectories of both photons, we will see that two years later their CM is far away from its initial position.
"This tells me the mass of the system of two photons is not at the center of mass of the system. In fact it tells me that it is, for the vast majority of the two-year time interval, the mass is very far from there. Suppose that two years after the annihilation event, they each hit a rock. Each of the two rocks has always been at rest
relative in the only frame in this discussion. In each case the photon is absorbed by an atom in the rock--an electron in that atom gets knocked up to a higher atomic energy level. Considering the rocks to be so massive that upon acquiring the momentum of the photon, the velocity of the rock is negligible, the mass of each rock increases by q (where q is the energy associated with the motion of each photon). This tells me that half the mass of the system of two photons is at the location of one of the photons and the other half is at the location of the other photon".
This is a very good argument, and I agree with it 100%! But the drawn conclusion below does not make sense to me:
"These two contributions to the total mass of the system are not the masses of the photons, they cannot be, the photons are massless."
This again raises the question: If each photon is really "massless", where the non-zero rest mass of their composite system comes from? From their individual energies Ei? So it is legal to write M=(2Ei)/c^2, but illegal to write M=2mi with mi=Ei/c^2 (the photon's relativistic mass)? None of the known "explanations" why it is illegal sounds reasonable. The frequently exploited word "modern" is not a physical explanation. Not everything modern is good.
"...the system mass at the location of the photon is not the mass of any particle, the photon is massless."
So the system does have mass at the location of each photon, after all? Again, I agree with this 100%. But if it is not the mass of any particle, then mass of what is it?
Moses Fayngold,NJIT