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.

Demystifying the New SI Base Units.

In the previous post we observed that Planck's constant $h$ appears as a conversion factor connecting light of frequency $\nu$ with attributed energy $h\nu$ (in eV or Joule) through the photoelectric effect with the release of an electron from a surface exposed to light (of sufficient high frequency). The inner mechanics of the atoms delivering the electrons upon excitation by exposure to light does not enter into the discussion and so Planck's constant can be given a meaning in macroscopic physics, thus without quantum mechanics, as a trade between light and electron energy and then further to mechanical energy. Its role in quantum mechanics then appears as an after construction.

Let us now turn to Boltzmann's constant $k$ to see its connection to Planck's constant and macroscopic physics. Boltzmann's constant appears in Planck's universal law of blackbody radiation law of the form

• $E(\nu ,T) = W(a)\, kT\nu^2$,
• $W(a) = \frac{a}{\exp(a )-1}$ with $a = \frac{h\nu}{kT}$,

where $E(\nu ,T)$ is the (suitably normalised) intensity of radiation of frequency $\nu$ from a blackbody of temperature $T$ and $W(a)$ is a cut-off factor with $W(a)=1 $ for small $a$ and
$W(a)$ small for medium to large $a$, expressing Wien's displacement law stating cut-off of high frequencies. We see that Planck’s constant only appears in the cut-off factor.

Experimental observation of $E(\nu ,T)$ makes it possible to determine $W(a)$ and thus $kT$ in terms of $h\nu$, from which Boltzmann's constant $k$ can be determined with respect to a chosen scale for temperatur $T$, or the other way around as in the new SI units by specifying by definition 

• $k=1.380650\times 10^{-23}\, J/K$,  

thus setting a new standard for Kelvin $K$ as measure of temperature. The connection between the energy measures $h\nu$ and $kT$ then shows to be

• $h\nu_{max} \approx 2.8214391\times kT$,

where $\nu = \nu_{max}$ gives maximum of the spectrum $E(\nu ,T)$.

Again, this can be done without having to invoke quantum mechanics in its standard form with $h$ as a "smallest quantum of action" as exposed in detail on Computational Blackbody Radiation, which presents a derivation of Planck's law using deterministic continuum physics instead of as usual statistics of discrete quanta. In particular, the new derivation captures the universality of blackbody radiation beyond specific inner atomic mechanics.

The universality of Planck's law is expressed by the fact that an ideal blackbody can take the form of a set of oscillators without very specific inner structure. In particular different blackbodies with different inner structure can share the same temperature scale.

To sum up, both Planck's constant and Boltzmann's constant are specified by definition in the new SI units, from which the new units kilogram and Kelvin can be determined by macroscopic experiments without resort to quantum mechanics in its standard form.



Hopefully this helps to demystify both Planck's and Boltzmann's constant, and the new SI units.
 

New Perspective on New Unit of Mass in terms of Planck's Constant

In the 2019 redefinition of the SI base units the kilogram as unit of mass is defined in terms of Planck's constant

• $h$ set by definition to exactly $6.62607015×10^{−34}$ Joule-second ($J\cdot s$), 

where

• $Joule = Newton\times m = M\times\frac{m}{s^2}\times m=M\frac{m^2}{s^2}=Mc^2$

with $m$ meter, $s$ second, $M$ mass in kilogram and $c$ the speed of light.

This defines kilogram in terms of Planck's constant $h$, second $s$ and speed of light $c$ with meter $m$ defined in terms of $c$. The relation $E=Mc^2$, viewed as a profound discovery attributed to Einstein's relativity theory, then appears simply as a definition (of mass).

The connection to quantum mechanics comes by attributing a certain energy $h\nu$ to light of frequency $\nu$ through the law of the photoelectric effect

• $h\nu = eV_0 + \phi = eV_0 + h\nu_0$, 

where $eV_0$ in electronVolts is the energy of a released electron with charge $e$ and $V_0$ a stopping potential in Volt, and $h\nu_0$ is the work to release an electron with $\nu_0$ a threshold frequency. This relation determines $h\approx 4.1357\times 10^{-15}$ in $eV\cdot s$, which fits with the new definition of $h$ in terms of $J\cdot s$ with the conversion $eV= 1.602176634×10^{−19} J$.

The photoelectric effect connects the macroscopic phenomena of light of different frequencies and stopping potential to the microscopic phenomenon of electron charge. In this connection there is nothing that says light of frequency $\nu$ is to be viewed as a stream of discrete photon particles of energy $h\nu$ and that Planck's constant $h$ has the physical meaning of a discrete smallest quantum of action.  Instead Planck's constant has the role of connecting light energy to electron potential energy ultimately to mechanical energy.

For a new continuum physics approach to blackbody radiation and the photoelectric effect with discrete quantum replaced by a threshold condition (as in the photoelectric effect), see Computational BlackBody Radiation.

The new definition of kilogram gives perspective on the very small size of Planck's constant $\sim 6.6\times 10^{-34}\, J\cdot s$ misleading to an idea of an absurdly small Planck length $\sim 1.6\times 10^{-35}\, m$ believed to have a physical meaning, in string theory in particular.  On the other hand, the length scale of atoms (and X-ray light of frequency about $3\times 10^{18}$) is about $10^{-10}\, m$ and that of a proton $10^{-15}\, m$, with the Planck scale 20 orders of magnitude smaller, way beyond any thinkable experimental exploration and thus meaning. Planck time  $\sim 5.3\times 10^{-44}\, s$ is even more absurd. No wonder that modern physics playing with Planck length and time is in a state of deep crisis, with the scientific madness come to full expression in the  Chronology of the Universe starting with the Planck Epoch before $10^{-43}\, s$ after Big Bang.

In Schrödingers equation $h$ multiplies the time derivative of the wave function, which means that
the atomic energy (potential + "kinetic" energy) of an eigenfunction of frequency $\nu$ is equal to $h\nu$, which comes to expression in the photoelectric effect.  Schrödinger's equation is a continuum model without any smallest quantum of action, only discrete eigenvalues representing different energies.

A reformulation of quantum mechanics in the form of a Schrödinger equation as a continuum model in real 3d space plus time without statistics, can be inspected at Real Quantum Mechanics.

To get an idea of the absurdly small Planck length scale $1.6\times 10^{-34}\, m$, one may compare with the estimated size of the observable Universe which is about $10^{27}\, m$ or $10^{33}\, \mu m$ with $\mu m$ mikrometer.



In short, Planck's constant $h$ converts light energy to mechanical energy through electron potential energy, and as such does not ask for a meaning as a "smallest quantum of action" in a mist of "quantisation" into absurdly small "quanta".

The new kilogram standard is specified to high precision using a Kibble balance, where gravitational force is balanced by an electromagnetic force (between two coils), which depends on Planck's constant. Mass is then derived by measuring the local gravitational constant.