I have always been fascinated by the famous equation, E=mc2, relating energy, mass, and the speed of light, that Albert Einstein gifted from his insight. Some of his peers criticized it as a circular derivation when he published it in September 1905. I wanted to see if I could arrive at the famous equation, employing other known equations, and demonstrate its circular nature…
To suit those among you interested in typical derivations, a YouTube clip of one such is attached. In this method, the narrator depends upon one of the implications of Einstein’s special theory of relativity, that defines a rest mass and an effective mass that increases as it speeds up. This idea – of mass increasing without limit as velocity increases to approach that of the speed of light – is controversial; I’ve seen a published academic paper that disputes it. Nevertheless, the simple mass-energy relationship, shown to be accurate in nuclear fission experiments, can indeed be derived by this method.
Now for a circular derivation! Energy E=hv=pc, where the first part defines Max Planck’s constant h, where Planck defines energy as proportional to the frequency v (greek nu) of the electromagnetic radiation. The second part of the relation, E=pc, derives from the De Broglie wavelength defined as λ=h/p, where p is the linear momentum (of a photon or “quantum” of electromagnetic radiation, but it is said to be applicable to other particulate matter as well) and c the velocity (of light, or electromagnetic radiation, which is λv or how many wavelengths are traversed per second). If we now define a “mass-equivalent” to this linear momentum p, as p=mv, where v is velocity (or just speed), and circle back to the idea of a photon, which has speed c, we get E=pc=(mc)c = mc2 – all from definitions!
Now for something even more imaginative; hey, was it not Einstein who averred that it was his imagination that led to his theories that changed physics in the twentieth century? Kinetic energy is defined by KE=(1/2)mv2, where v is speed. Knowing the wave-particle duality of components of particles, sub-atomic elements, can we not argue that any mass is simply condensed light of varied energy content? The energy of atoms, comprised of sub-atomic elements that move about as waves of electromagnetic energy, is therefore the kinetic energy of these elements countered by a corresponding amount of other forms of energy that maintain the physical integrity of these atoms. Viewed as condensed light, therefore, the energy content is, at a maximum when released, twice the kinetic energy of elements moving at the speed of light, or E=2(1/2)mv2=mc2, where c is the speed of light.
These are rather specious derivations, and may not hold water with any physicist, but the same energy-mass equation can be derived from a so-called “Gedanken” or thought experiment with a box issuing photons inside it at one end, recoiling from the linear momentum imparted to the photons, and then coming to a stop as these photons strike the other end of the box. Much the same as a cannon inside a rail car at one end, shooting a cannon ball that strikes and stops at the other end of the rail car. These simple derivations (and more refined versions of it) also arrive at the same energy-mass relation, no “Theory of Relativity” required.
That is not to say Einstein’s theory was in any way wrong: many aspects and extensions of it have been conclusively shown to be valid, including time dilation, a most counter-intuitive phenomenon, but widely applied in our GPS (global positioning) satellites today. With many ways to arrive at the energy-mass relation, it has nevertheless also been shown by researchers much later that the relationship isn’t necessarily “absolute” – it does not accurately determine what transpires in all cases, or so I read. Einstein himself marked in his paper that higher order terms were ignored in his original derivation of a slightly different form: change in mass is given by energy radiated divided by the square of the velocity of the radiated energy. Stephen Hawking recently published something to the effect that Relativity and Quantum Mechanics do not work well together when studying event horizon boundaries around black holes, or regions of inescapable gravity, in space.
Hence, to me, the equation is an excellent working model… nothing sacred or earth shattering, just a better understanding of the convertibility of mass and energy. But knowing that mass, energy, and light are all closely related continues to fascinate me…