E = 3/4 mc² or E = mc²? The forgotten Hassenohrl 1905 work.

In March 1905, six months before Einstein, the Austrian physicist Fritz
Hasenohrl published his third and final paper about the relationship
between mass and radiant energy in the same journal Annalen der Physik
that received and published his papers about relativity.
His final paper, a review of his former two since 1904, was an
elaborated thought experiment to determine if the mass of a perfect
black body radiation increased, from rest, while it was slowly
accelerated (the same hypothesis used by Einstein in his SR paper, to
deal with electrons). The final result was that this relationship:
m = 4/3 E/c² , which can be expressed as  E = 3/4 mc²
which he found to be independent of the velocity of the cavity.
His work received much attention from the physics community, and won the
Haitinger Prize of the Austrian Academy of Sciences. In 1907 he
succeeded Boltzmann as professor of theoretical physics at the
University of Vienna.
This is the translation of his first paper, in 1904, where he derived m
= 8/3 E/c². In the next two papers, he corrected some mistakes,
publishing the last one in March 1905, six months before Einstein's
paper deriving E = mc².
https://en.wikisource.org/wiki/Translation:On_the_Theory_of_Radiation_in_Moving_Bodies#cite_note-21
Prior to Hassenohrl, and since 1881 paper from J.J. Thomson, different
works were published correcting Thomson and relating mass increase and
changes of the electrostatic energy of a moving charged sphere (later
the electron) by Fitzgerald, Heaviside, Larmor, Wien and (finally) by
Abraham in 1903. The work of Hassenohrl was based on Abraham, but with
the fundamental change of using radiant energy from inside a perfect
black body moving. This alone was considered a breakthrough in physics,
and Einstein took note of it and simplified the thought experiment of
Hassenohrl (a closed system) for other in an open system, which has
theoretical deficiencies, which Einstein was never able to solve, giving
up in 1942 (7th. attempt).
The remarkable work of Hassenohrl showed, beyond doubts, that any energy
(electrostatic or radiant) is related to mass increase, when moving, by
the relationship m = 4/3 E/c².
This fact, known for almost a decade since FitzGerald, couldn't be
explained correctly until 1922, when Enrico Fermi focused on the
problem.
All these works are considered today as pre-relativistic, even when
ether is barely mentioned.
Hassenohrl himself used two references (Einstein jargon didn't exist
yet): A fixed reference frame and a co-moving reference (along with the
cavity). The popularization of relativity and the easiness of having a
relationship E = mc² (even with restricted use of velocities) made it
much more appealing to the scientific community than having to deal with
E = 0.75 mc².
Even more, in the next decades, using c = 1 became popular, and so the
direct use E = m, as it was shown in the calculations done by Chadwick
(1932) to justify that he had proven the existence of the neutron. A
different world would exist if E = 0.75 mc² had been adopted, which is a
proof of what I've sustained for years about that such a simple equation
was adopted for convenience and colluded consensus (like many other
constants and formulae. GR?).
Hassenohrl's work proved that his equation is independent of the
velocity, and that mass is an invariant property of matter. On the
contrary, E = mc² has a limited range of applicability, forcing its use
to rations v/c << 1. This is because its derivation comes from using the
first term (the cuadratic one) of an infinite McLauring series used on
the expansion of the Gamma factor minus one:
γ - 1 = 1/√(1 - v ²/c²) - 1 = 1/√(1 - β²) - 1 = 1/2 β²  + 3/4 β⁴ + 15/24
β⁶ + 105/192 β⁸ + ..
Einstein used L (γ - 1) ≈ L/2 β² = 1/2 (L/ c²) v², from where he
extracted m = L/ c² as the mass in the kinetic energy equation. Nor him
neither von Laue (1911) nor Klein (1919) could solve this very limited
approximation for uses on closed systems. Yet, the equation stayed (for
consensus due to its convenience).
The work of Hassenohrl, based on his thought experiment, is very
detailed. Much more than the loose arrangement of Einstein's paper. He
did care to present his closed system with severe restrictions:
- A perfect black body cylindrical cavity, with the walls covered with a
perfectly reflective mirror, exterior temperature of 0"C, and two
perfect black body caps on the ends, tightly fixed and having zero
stress from the forces of radiation and motion.
- A very small acceleration, in order to cause smooth changes in
velocity of the cavity.
- The black body radiation is taken from its intensity i (he never
mentioned Planck), which he described as a "pencil of energy", which
formed an angle θ with the vector of velocity.
In modern terms, it's the Monochromatic Irradiance or Spectral Flux
Density: Radiance of a surface per unit frequency or wavelength per unit
solid angle.
- This directional quantity differs from Planck's Spectral Radiant
Energy formula by (c/4𝜋), which he accepted when integrating along the
volume of the cavity, giving original Planck's density u of radiation
energy.
With the above considerations, and many others, Hassenohrl wrote his
final paper, for which he gained recognition and a prize. But the
problem for him, and for physics, is that it was a pre-relativistic work
where absolute reference at rest was used (as in all the other works
from legions of physicists during the centuries). Relativity
cannibalized all the classic physics, except when it's not convenient to
do so: a blatant hypocrisy (take the merging of reference frames in
particle physics, or just the Sagnac effect).
The problem that Hassenohrl's work poses for physics is his enormous
complexity, which has consumed a lot of manpower since 1905 up to these
days, in order to be understood.
This paper
Fritz Hasenohrl and E = mc²
Stephen Boughn
Haverford College, Haverford PA 19041
March 29, 2013
https://arxiv.org/abs/1303.7162
is one of many modern papers that try to understand Hassenohrl's work by
using relativity and Planck, which simplify the complex work of the
Austrian physicist. Even this paper poses some doubts about the validity
(or not) of Hassenohrl's work in these days, where a notion of absolute
reference frame is gaining momentum within physics. The paper try to
explain (but fails) which were Hassenohrl's mistakes (of course under
the light of relativity), but it serves as a guide to analyze
Hassenohrl's work.
However, the author is highly biased, because he focused on the first
1904 paper and not in the final publication in Annalen der Physik, where
Hassenohrl had changed substantially his first proposal. For instance,
introducing the idea of a slowly accelerated cavity (which is essential
to prove the independence of the gain in mass with respect to the
velocity).
I'm sorry not being able to get the March 1905 paper to cite it here. It
seems that efforts to erase Hassenohrl's work (or Abraham's work with
electrons) from the history have been successful. You have to resort to
find books from the '50s to get some info, like the one cited by Stephen
Boughn.
Now, E = 3/4 mc² or E = mc²?  Which one would the physics community
adopt?
Hmmm....