Energy Cannot Be Equivalent to Mass

Identifying Mass with Energy is the same as identifying Being with Doing. Not so Clever.

The most famous formula attributed to the most famous physicist all times is Einstein's Law

•  $E=mc^2$      (1)

apparently stating that energy $E$ is equal or "equivalent" to mass $m$ multiplied by the large coefficient $c^2$, where $c$ is the speed of light in vacuum.  

A modern physicist will inform you that (1) is a consequence of Einstein's Special Theory of Relativity SR, even if the details of such a derivation cannot be recalled. To counter further questions you will be informed that in fact (1) is just a special example of of a more "relativistically correct" Einstein Law of the form 

• $E^2=(pc)^2+(mc^2)^2$,     (2)

where $p$ is momentum, which is supposedly easier to prove even if details of proof cannot be recalled.  You will also be informed that both (1) and (2) have been confirmed by the same experiments. And do not forget that atom bombs build on (1) and so show the amazing "power" of this "equivalence".

First, let us seek to understand the meaning of (1). We recall that in the physics of thermodynamics 

• energy is capacity to do (mechanical) work 
• work = force x distance measured in Joule = Newtonmeter.   

Energy comes in forms of large scale ordered kinetic energy and potential energy and heat energy as small scale unordered kinetic energy. Here the kinetic and potential energies associated with a body are extrinsic or relational quantities i.e. depending on the environment of the body. The energy produced by the decent of the bob of a classical pendulum clock depends on bob weight and decent distance. 

The typical expression of kinetic energy of a body of mass $m$ and and speed $v$ is viewed to be $m\frac{v^2}{2}$ as the work required to bring the body from rest to speed $v$, with the rest state as the reference state. This energy/work can be regained letting the body impact with an environment at rest.

Likewise potential energy is created by lifting an object from some reference level, which can be regained by letting the body return to the reference level. Large scale ordered kinetic and potential energy can be transformed to heat energy as small scale unordered kinetic energy in turbulent dissipation, and the 2nd Law of Thermodynamics puts limits to recovery of large scale energy from heat energy, that is limits on production of work from heat energy. 

We conclude that thermodynamical energy is an extrinsic relational quantity which is measured in terms of what it can do depending on the environment. 

What then is the mass $m$ of a body? Is it an extrinsic or intrinsic quantity/quality? Well, mass is inertial mass which is equal to gravitational mass as resistance to motion induced by a force.  This is expressed in Newton's 2nd Law $m=\frac{F}{a}$, where $F$ is force and $a$ acceleration. This is an intrinsic quantity in terms what it is. All bodies independent of quantity of mass react the same way on gravitational force, that is,  carry an intrinsic property of reacting to inertial or gravitational force. We can think of the mass of a body as being equal to he sum of the masses of the pieces of atoms forming the body. All the atoms react the same way on inertial or gravitational force, and this explains why a body is not ripped apart by such forces. Mass is maybe the most intrinsic quality of all. 

Sum up: Energy is extrinsic (what it can do) while mass is intrinsic (what it is). Is it possible that an extrinsic quantity can be equivalent to an intrinsic quantity as expressed by $E=mc^2$? There seems to be no sufficient reason to insist that  "energy is equivalent to mass", so the answer can only be No. 

Let us now turn to (2) as an "improved version" of $E=mc^2$, keeping the first term, that is let us look at the Law, motivated by Many-Minds Relativity MR:

• $E=pc$,     (3)

where $p=mv$ is momentum $v$ velocity. This Law seems to make a bit more sense since $p$ is both intrinsic thorough $m$ and extrinsic through $v$, but the previous post shows that this only an illusion. There is no sufficient reason to insist that "energy is equivalent to momentum".

When confronted with the above arguments a modern physicist will say that "is is all wrong" without showing what is wrong. 

In any case, the bottomline may well be that the by many witnessed crisis of modern physics ultimately depends on (1) as a foundational relation that does not make sense. More detailed arguments in recent previous posts. 

Photons 

Physicists use (2) with $m=0$ to give momentum to the massless photon with energy $E=h\nu$ through the connection $p=\frac{h}{\lambda}=\frac{h\nu}{c}$ with thus $E=pc$. Magic!

Thermodynamics and Atom Physics 

$E=mc2$ is supposed to be a result of SR which does not describe thermodynamics nor atom physics. 

Basic postulates of thermodynamics say that mass and energy are conserved. This means that $E=mc^2$ in the sense of actual transformation of mass into energy cannot happen in thermodynamics. Unless you say that by definition energy and mass are equal and so $E=mc^2$ is a tautology without physical meaning. 

Can $E=mc^2$ or $E=pc$ have a meaning in atom physics, when SR and MR say nothing about atom physics?  Or is also here $E=mc^2$ a tautology without physical meaning? Probably. MR says that (3) is an illusion. Einstein was a master of double play mixing physical fact with definition/logical truth. SR is filled with this ambiguity: Is time dilation and length contraction real or illusion? Ask your physics professor!