An argument against the existence
of massless electric charges.
Corrado Massa
Via Fratelli Manfredi 55
42124 Reggio Emilia
Italy
Abstract
A simple gedankenexperiment suggests that
massless particles with electric charge do not
exist.
In the following, I call mass what the old literature
calls “rest mass”. The work done by an electric field
with potential V on a particle with electric charge q
as it traverses a potential difference dV is – q dV .
This must equal dE, the gain in the particle kinetic
energy. For a massless particle the gain equals h df
(h = Planck’s constant, f = frequency) and therefore
d f / f = – q dV / h f ( 1 )
Integrating Eq (1) over a finite path from 1 to 2 we
find
f1 / f 2 = exp ( ʃ ( q / h f ) dV ( 2 )
In stationary condition, from f1 > f 2 ( f1 < f 2 ) it
follows that time itself slows down (quicks up)
near point 1 (to my knowledge the first author to
point out a connection between redshift (blueshift)
and slowing down (quicking up) of time was Einstein (1911).
Since f 1 / f 2 and the time-time component g 00 of
the metric tensor are related by (Rindler 2003)
.
f1 / f 2 = [ g 00 ( 2 ) / g 00 ( 1 ) ] 1 / 2 ( 3 )
where g 00 ( 1 ) and g00 ( 2 ) are respectively g 00 at
point 1 and at point 2, we have from (2) and ( 3 )
g 00(2) / g 00 (1) = exp ( 2 ʃ ( q / h f ) dV ( 4 )
which implies
q = K W ( 5 )
K is a universal constant, and W = h f is the energy
of the charged massless particle. Why is K a universal constant ? K is a universal constant because g 00 is
an intrinsic property of the spacetime geometry and
cannot be influenced neither by q nor by f ; in
other words, the metric is expected to be
influenced only by the electric potential (I neglect
the general relativistic influence on g 00 due to
the energy stored in the electric field) and not by
the test particle travelling the field.
It is easy to see that equation ( 5 ) is at variance
with observations. Indeed, according to eq ( 5 ),
a box with mirroring internal walls, at rest in an
inertial frame S, and filled with massless particles
of total energy W (as measured in S ) has total
electric charge Q = K W, while in another inertial
frame S* has total energy ( γ = the Lorentz factor)
W* = γ W and, for ( 5 ), has total electric charge
Q* = K W* = K γ W . This conclusion conflicts very
strongly with observations; indeed, according to it
the Lorentz – invariance of the equilibrium of a configuration of charges (invariance that cannot
be given up! ) taken together with eq ( 5 ), demands the transformation law
χ * = γ χ ( 6 )
not only for a charged bulk of massless particles, but
also for the charge χ of any charged particle ( such as electron, proton…); needless to say, law ( 6 ) is not
obeyed by electrons, protons…
Conclusion: the gedankenexperiment given above
allows us to the highly probable, if not certain, conclusion that charged particles with zero mass do
not exist.
Interestingly, an analogous gedankenexperiment
suggests that massless fermions cannot exist in an
Einstein – Cartan spacetime with torsion (Massa 2001).
References
A. Einstein: 1911, “ Ueber den Influss der Schwerkraft
auf die Ausbreitung des Lichtes “ Annalen der Physik,
vol. 35.
English translation: “ On the influence of gravitation on the propagation of light “, published in “ The Principle of Relativity “ ( Dover, 1952 ).
W. Rindler: 2003, Relativity, special, general and
cosmological, Oxford, New York; see Ch. 9, Sec 4,
eq. (9.4).
C. Massa: 2001, “ Torsion, neutrinos, and intergalactic magnetic field “ Il Nuovo Cimento B, vol. 116, p. 249.