Einstein's 7 Erroneous Proofs of E=mc2

The book Einstein's Mistakes by Hans Ohanian gives a chronology of Einstein's many scientific mistakes including 7 erroneous proofs presented by Einstein of the crown jewel of his theory of relativity in the form of $E=mc^2$ stating proportionality between energy $E$ and mass $m$ with $c$ the speed of light. The need of a 7th proof indicates that proofs 1-6 are all incorrect and so it is not far-fetched to expect that also Einstein's 7th proof is incorrect.

To give perspective, let us recall the proof from Many-Minds Relativity chapter 14-15 of a related connection, this time between mass and momentum $P$ of the form $P=mc$. We recall that this relation can be seen as a consequence of a new Many-Minds form of Newton's 2nd Law stating the following connection between a velocity $v(t)$ and acceleration $\frac{dv}{dt}$ of a body of mass $m$ acted upon by a force $F=F(t)$ depending on time $t$:

• $\frac{m}{1+v}\frac{dv}{dt}=F$,   

This form of Newton's 2nd Law results from measuring velocity of a moving object through Doppler shift $\frac{1}{1+v}$ of received signals from the object with the speed of light normalised to 1. It states that objects in approach/recession with respect to an observer, appear to be subject to an increase/decrease of mass connecting to acceleration. Here $v$ is negative in approach and positive in recession and with $v > -1$ in approach, but unlimited in recession allowing far away galaxies to recede faster than the speed of light as observed in large redshift.

Using that for  $\vert v\vert$ much smaller than 1, $\frac{1}{1+v}\approx 1-v$, Newton's 2nd Law takes the form

• $F\approx m\dot v -mv\dot v\approx (m+P)\dot v$ 

with $P=-mv$ momentum. This relation has the form of a classical Newton's 2nd Law with the mass $m$ augmented by $P$, which trades to a connection between momentum $P$ and mass $m$ of 

the stated form $P=mc$ without normalisation to $c=1$. 

We have thus given a proof of the relation $P=mc$, as an alternative the relation $E=mc^2$, which Einstein could not prove and maybe nobody else can.

PS1 In Einstein's special relativity also the recession speed is limited by the speed of light.  This is not what is observed, since galaxies outside the Hubble sphere at a distance of 4300 megaparsecs are by their redshift observed to recede faster than the speed of light. The apparent contradiction with Einstein's special theory of relativity is handled in the usual way: The special theory is correct but it does not apply to receding galaxies, for which instead the general theory of relativity must be used and the general theory is so complicated that contradictions is beyond human

PS2 The suspicion that $E=mc^2$ is just a matter of definition, which is true by defining mass in terms of force and acceleration through Newton’s 2nd Law (thus in terms of energy), and not a physical fact, which could be true or not, is growing stronger and stronger. Einstein is the master of ambiguity between definition and fact, with the constancy of the speed of light as a key example, which by physicists mislead by Einstein is viewed to be both a definition and a physical fact.