## måndag 1 juli 2019

### $E=mc^2$: Definition or Physical Fact?

All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta (photons)?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. (Albert Einstein, 1954)

We continue exploring the meaning of the most famous equation of physics $E=mc^2$, which Einstein suggested in 1905 to be a consequence of his special theory of relativity and struggled through all his life to justify theoretically, however without success.

The equation $E=mc^2$ carries the same ambiguity as the basic postulate of Einstein's special theory of relativity, the constancy of the speed $c$ of light, for which it is never clear if it is only a definition true by logic, or a law of physics which may be true or false.

Is then $E=mc^2$ a definition or a physical law as a statement about a physical fact, which may be false?

We start with the following natural question: From where does the factor $c^2$ come, which attributes the energy $mc^2$ to mass of size $m$?

This question can be given an answer for a photon of frequency $\nu$, which can be observed e.g. through the photoelectric effect, to have the energy
• $E=h\nu$,
with a properly specified Planck's constant $h$. We can now, if we want, attribute the mass $m=\frac{h\nu}{c^2}$ to a photon, to get
• $E=h\nu =mc^2$
simply by definition.  We can do this because the physics of a photon is unclear.  We can supplement by naturally attributing the momentum $p=mc$ to a photon of mass $m$ and speed $c$ and so obtain $E=pc$ as an equivalent form of $E=mc^2$ (also discussed in the previous post).

In short, we can argue that for the fictitious concept of a photon (compare with the Einstein quote above), energy and mass indeed can be viewed to be, in the words of Einstein,  different manifestations of the same thing (namely energy).

By claiming that a radiating body loses the mass attributed to emitted photons (although the loss is too small to be measured), we can then give the relation $E=mc^2$ a general meaning beyond photons, still however essentially by definition with eventual physical meaning remaining to be explained.

That mass indeed can be converted to energy in nuclear fission and fusion processes, was a surprise to Einstein and cannot be seen as a consequence of his special theory of relativity, because it has no connection to nuclear physics. The  first quantum field theory Standard Model proof (with quarks and gluons) of $E=mc^2$ was presented only in 2008.

The general idea of a connection between mass and energy is as old as physics, with in particular the kinetic energy of a body of mass $m$ moving with speed $v$ being equal to $\frac{1}{2}mv^2$. Moreover, the factor $c^2$ was suggested prior to Einstein by Poincare and Hasenörhl preceded by Heaviside and Wien, among others. But it was Einstein who got the Prize.

Recall that in Maxwell's equations for electromagnetic waves in vacuum, with the electric field $\hat E$ (in Gaussian units) satisfying an equation of the form
• $\frac{1}{c^2}\frac{\partial^2\hat E}{\partial t^2}-\nabla^2\hat E=0$,
with the factor $\frac{1}{c^2}$ serving the same role as mass $m$ in a mechanical wave equation with connection to energy, thus supporting a relation of the form $m=\frac{E}{c^2}$, which was the original form suggested by Einstein, among others, as an expression for (fictitious) mass rather than (real) energy.

Recall that with the new meter standard as a certain fraction of a light second, the speed of light is by definition equal to exactly 1 (light second per second). For perspective see Many-Minds Relativity. So today, the constancy of the speed of light is a matter of definition. To Einstein in 1905 it was both a definition and a physical fact with truth secured by pure logic as the unique nature of Einstein's physics, which so impressed the world, although there has been many critical views from start which however have been muted since they did not fit the success story of modern physics.

Is it then true that a radiating body loses mass, even if the emitted energy comes from a source of internal heat energy as a form of kinetic vibrational energy measured by temperature (and not mass) according to a new analysis of blackbody radiation? I will seek to return with an answer.

To start with let us recall that the concept of mass $m$ connects to force $f$ and motion/acceleration $\frac{dv}{dt}$ with $v$ velocity through Newton's 2nd law $\, m\frac{dv}{dt}=f$, which can be used to define mass in terms of force and motion, as well as momentum $mv$ as the integral of force, and kinetic energy $m\frac{v^2}{2}$ as the integral of $fv$ as work. Force can be measured by a spring and motion by a meter stick and time, which defines mass threefold in terms of Newton's law, momentum and kinetic energy. The basic relation is Newton's 2nd law, while the integrals of $f$ and $fv$ are computed/collected in physical form as momentum and kinetic energy. Newton's 2nd law is Galilean invariant, while momentum and kinetic energy as integrals depend on initial velocity. Momentum and kinetic energy thus carries information about mass modulo initial velocity: If you travel at the same velocity as a cannon ball, its mass is hidden and you cannot detect it by being hit.

Defining mass by Newton's 2nd law in terms of force and motion/acceleration makes mass = inertial mass, from which equality of inertial mass and gravitational mass follows by definition, since gravitation appears as force. Einstein's Equivalence Principle as the basic assumption of the general theory of relativity is thus empty of physical content,  as the general nature of Einstein's physics jumping freely between definition and physical fact, as exposed in detail  Many-Minds Relativity. Take a look and get enlightened by understanding the confusion between definition and fact, which has corrupted modern physics into a mess of subjective epistemology instead of a science of objective ontology in the spirit of Enlightenment. This was understood by Einstein but he only gave cryptic evidence to this insight like in the above quote and
• If I would be a young man again and had to decide how to make my living, I would not try to become a scientist or scholar or teacher. I would rather choose to be a plumber or a peddler...