The math is trivial once we accept these, but the relativity principle deserves as much scrutiny as the source and meaning of the gamma factor. His failure to define the concept ‘energy’ and relate it without contradiction to the concept ‘light’ constitutes the root problem in this work.
He begins by stating his relativity principle: the laws regarding the changes in the conditions (Zustände) of a physical system are independent of which of two coordinate systems, that are in relative linear motion, we use to describe these facts, or conditions.
In trying to understand his relativity principle consider this: Suppose we have a gun at rest on the earth that we want to fire at a target ten feet away. But now we want to relate this situation to a train moving at twenty feet per second from left to right. Are the gun and the target to be now considered to be on the train, or is the train, with the observer in it, just going by and we are observing the situation while seated on the train? Which situation represents his relativity principle.
He continues by referring to a body at rest in one of these systems and asserting it has energy Eo, and then he wants to find its energy, in the other system. The other system is moving with velocity v with respect to the first system.
Energy, in so far as it is classical kinetic energy, relates to mass and motion. In the case of a gun at rest on the ground we would have to know the velocity of the bullet with respect to the target and the mass of the bullet, and then we could compute mv². But we still would need to know which scenario to use in the case that we are on a moving train. Nothing in his math that follows his remarks clarifies the scenario.
If a body is at rest, and we want to associate with it some quantity or type of energy, independent of its movement or relationship to another body, it is not its mass but its atomic or nuclear character that can yield such energy or energies. To the extent that a substance is destroyed and its mass, m, is carried off by radiation and nuclear fragments moving at close to the speed of light, to that extent we can hope to generate classical kinetic energy approaching mc², but that will not happen to a mass of clay or iron, so it is the nature and character of the mass that is relevant. We are outside the domain of classical physics – and outside the domain of special relativity.
He speaks of a body at rest in the first system as emitting light energy. What is the meaning of the phrase ‘to emit light energy’? In classical physics we transfer energy from one body to another by means of matter – for example from the bullet to the target. So what is the light energy that is ‘emitted’? (On page 640 he talks about light waves emitted from a resting body that have energy L/2.) According to Einstein there is energy residing in the light – but also, according to Einstein, light has no mass. But, as we know, energy requires, and is coupled to mass. There is an internal contradiction in his calculations and concepts, and it appears his idea of energy is clouded or confused. But out of nowhere, by using an approximation derived from the gamma factor, he comes up with the assertion that this loss of energy represents a loss of mass, and that gets him
E = mc².
Einstein was right in his intuition that there can be an energy associated with an object at rest. That recognition is the fact that pulled physics into the atomic age. But this is a new kind of energy, or better, a new family of energies and forces that were not suspected before the discovery of radioactivity.
If mass is ‘lost’ in a nuclear reaction or transformation we should not seek to explain it by means of a deformation of space or time - it is not relativity theory that we need but a detailed look at the nuclear fragments and radiation that takes the place of the ‘missing’ mass. If we allow radiation to have mass along with the rest of the nuclear debris than nothing is missing, and then light can carry energy as well.
The bottom line is E = mc² raises more questions than answers. It is derived from the dubious gamma factor. It does not clarify the relation between matter, energy and light. It does not prove the validity of special relativity, but is suggestive of the immense power of atomic energy.
[Note: That mass might change due to speed was under intense discussion in 1904. The Kaufmann experiments indicated that as the speed of electrons increases e/m decreases. Einstein’s results favored an increase in mass instead of a decrease in the effect of charge. That became the prevailing view. Whether Einstein was aware of these experiments is not clear, but highly probable.]
No amount of math can hide the absurdity, or contradiction – on the one hand, mass retains its quantity and its identity in the classical concept of kinetic energy, even as the velocity, v, of its impact increases and gets close to the velocity of light; but on the other hand the mass increases and becomes infinite according to special relativity (not to mention one interpretation of the Kaufmann experiments), but on the other hand (assuming a third hand), on reaching the velocity of light, it vanishes, and turns into energy (of some sort or other) – the meaning of the term ‘energy’ is left to the imagination of the reader.
An example from a different context may help in understanding the concept ‘energy’: If I am making pancakes I take water from one container and flour from another and combine them in a third container that then contains a mixture from which I can make pancakes. But in the case of ‘energy’ we don’t have a container of mass and a container of energy that we can combine to make a third and separate entity called ‘energy’. Kinetic energy IS that combination of mass and velocity – it is not just ‘equal’ to that combination. That is a distinction that the word ‘equals’ does not contain – and it can be the source of confusion when we use the concept ‘energy’. This is not physics, it is metaphysics or linguistics or common sense. But without this distinction much of physics can be very confusing indeed. What the word ‘energy’ can mean apart from ‘mass times the square of the velocity’ has not been clearly defined by Einstein or other physicists of that time.
“Energy cannot exist except in connection with matter.”
(James Clerk Maxwell, 1877, “Matter and Motion” Chapter 6 paragraph 108)