An interesting upshot of Mr. Einstein’s famous equation is that in units of the speed of light, energy and mass are equal. Mr. Poincare once observed that mathematics is the exercise of making different things the same (equal), and sophistry above with the square of the speed of light might be just the sort of trick he was referring to.
Nevertheless, the human concept of squareness derives from carefully drawn figures in the sand, subdivided and counted. A square divided in two each way yields four smaller squares, so it is natural to say that four is two “squared”. In this frame of reference, when you don’t subdivide your square at all, each side remains one, and you have but one square. That one squared equals one is unassailable.
We have noted before that equating energy to matter times the speed of light squared seems peculiar. Here we explore an alternate notion that the huge asymmetry between energy and mass in units smaller than the speed of light, disappearing to unity at the speed of light, is precisely why the equation works.
Mr. Einstein did not know about fermions. Fermions are subatomic particles that take up space, have mass, and constitute the matter in the universe. Electrons, quarks, and the triplets of quarks we call protons and neutrons are fermions.
Mr. Einstein knew about photons. He was instrumental in their discovery. Photons have no mass, take up no space, represent force rather than matter, and travel at the speed of light. In our standard model photons are bosons, and we see the difference between fermions and bosons as a fundamental division of the universe.
Complicating the picture, the divide between fermions and bosons is not about mass. The divide is about the ability (even propensity) to condense into or occupy the same space. Many bosons have mass derived from Mr. Einstein’s equation because they do not travel at the speed of light. Of the bosons, only photons, gluons (carriers of the force holding quarks into protons and neutrons), and the (as yet) hypothetical gravitons have no mass and travel at the speed of light.
This brings us to an important distinction between mass and matter, often loosely interchanged as the “m” in E=mc^2. All matter has mass, but not all mass is matter. While we can write E=f(ermions)c^2 and the expression is true, we cannot write E=mb(osons)^2.
If we use units of the speed of light, in some sense we pose our subject energy, mass, and matter to be travelling at the speed of light. This is not possible, as mass and matter cannot travel at the speed of light. There is still something the matter with our conception.