^{*}

Prevailing and conventional wisdom as drawn from both Professor Albert Einstein’s Special Theory of Relativity (STR) and our palatable experience, holds that photons are massless particles and that, every particle that travels at the speed of light must—accordingly, be massless. Amongst other important but now resolved problems in physics, this assumption led to the Neutrino Mass Problem—namely, “Do neutrinos have mass?” Neutrinos appear very strongly to travel at the speed of light and according to the afore-stated, they must be massless. Massless neutrinos have a problem in that one is unable to explain the phenomenon of neutrino oscillations because this requires massive neutrinos. Experiments appear to strongly suggest that indeed, neutrinos most certainly are massive particles. While this solves the problem of neutrino oscillation, it directly leads to another problem, namely that of “ How can a massive particle travel at the speed of light? Is not this speed a preserve and prerogative of only massless particles?” We argue herein that in principle, it is possible for massive particles to travel at the speed of light. In presenting the present letter, our hope is that this may aid or contribute significantly in solving the said problem of “ How can massive particles travel at the speed of light?”

Despite the death and lack of solid experimental proof [

where

If (1) and (2) are both applicable to the photon with all the identical symbols holding the same meaning, then, it follows directly that

Beginning in Section 2, we herein, place the two dispersion relations (1) and (2) into the dock for some cross- examination, whereafter we come to the interesting and startling conclusion that it must in principle be possible to have massive photons (i.e. non-zero rest mass photons) obeying these two relations simultaneously and concurrently i.e., massive particles that travel at the speed of light

As a first step, we would like to consider what we consider a hidden assumption in our proclamations on the photon mass. This hidden assumption in all the reasoning leading to the fact that for photons

Above, we have argued that if the

In the above Formula (3), the term,

The condition,

The term, ^{1} (as the first order approximation) formula for the energy of a photon, then, what (4) essentially means is that the new extra energy in (4), i.e.

Rightly, according to Dr. Sviratcheva (1999)^{2}, an experiment or observational measurement that fails to conclusively find or positively detect a mass for the photon, such a measurement does not prove that the photon mass is zero as is widely believed (without the direct proof); it merely shows that this mass is less than the limit or accuracy of the experiment in question. From this, we conclude that if the photon has mass as we believe, then, this mass must be very small so much that our measuring precession and accuracy may not have reached those levels where this photon mass can safely be pinned down. However, small it may be, this mass plays an important and pivotal role in the physics of the photon and the very foundations of physics. The currently accepted upper bound on the photon mass,

Clearly, the latter argument―that the mass of the photon maybe small, so small that it most certainly has escaped detection of our finest experiments; this argument is much stronger than the former argument. If the energy, ^{3} where too quick to conclude from experiments that the formula, E = pc, is binding. By this (binding) we mean that the formula,

Before we depart in the present section, it is important that we take note of what the relation (4) implies or means insofar as the group velocity

that is to say, the speed of of light for massive photons now depends on the photon’s momentum

^{1}Actually, there is no reason whatsoever to reject this first order approximation (4). If (3) is accepted as a first order approximation for the regime

^{2}http://www.phys.lsu.edu/students/kristina/PhMass/PhMass.html: Accessed on this day March 14, 2014@12h05GMT+2.

^{3}The “we” here refers to the generality of all physicists since the time the photon was assigned a vanishing mass.

In continuing in the building of our thesis, we shall in this section demonstrate a simple and yet important point, namely that, the rest mass of a particle as defined in Professor Einstein’s Special Theory of Relativity (STR) can be identified with the inertial mass (as defined fundamentally in Newtonian mechanics) of a particle. This is not a new idea, it is as old as the STR itself. What is new perhaps is that this idea can be used to identify and distinguish gravitational mass from inertial mass. As is well known, there is at least two distinct and important kinds of mass that enter Newtonian mechanics, these are the gravitational and inertial mass. Most notably, due to Professor Einstein (in 1907 cf. [

The first fundamental mass is the inertial mass

By motion, Newton meant the momentum

The vector quantity

The second kind of fundamental mass enters Newtonian mechanics in Newton’s law of universal gravitation is the gravitational mass (

where

Pertaining the gravitational and inertial mass, we have the Weak Equivalence Principle (WEP) due to Galileo which states that test bodies fall with the same acceleration independent of their internal structure or com- position: in other words, the gravitational mass appearing in (8) and inertial mass appearing (7) are the same i.e.

As has been done in [

Now, taking a step further toward our desired end, we know from Einstein’s STR that the total energy of a particle

where

The term

inertial mass as defined in Newton second law, namely

From the forgoing, it means we can write (11) as:

Now, the energy E is equal to

further deliberations. This leaves us with no choice but to identify the

In this case where

At this point, we have attained our desired objective i.e., we have shown that in principle, one can define or identify the rest mass as the inertial mass. Further, we have defined the gravitational mass as-well.

Having somewhat defined the gravitational and inertial mass, let us revisit the idea presented in Section 2 where we stated that there is a hitherto hidden assumption in all the reasoning leading to the fact that for photons

Is not the energy “

If these two energies are different, that is, say the energy

where generally

In the present section, we shall―accordingly; derive an exact expression for the group velocity of massive photons of spin-

This equation i.e.

Since the photon is a spin-1 particle, according to this new equation,

Now, we come to the main issue of the present section―that of calculating the speed massive photons. The dispersion relation corresponding to the equation,

where

Setting,

Now, if the mass,

where

so that (19) becomes:

From (20), we have:

Solving this equation, i.e. (22), gives:

Writing (23) in-terms of the momentum, that is, with

It is interesting to note that

Now, we have to find a way to deduce the value for

and for

As one can verify for themselves, the expression (26) does reduce to (5) when evaluated to first order approxi- mation. We take (26) is the exact expression giving the group velocity of spin-

In this section, we shall ask and subsequently make the endeavour to answer the question “What distinguishes matter from radiation or radiation from matter?” We find that the answer to this question is within reach of the present ideas. One distinguishing feature of radiation (photons) is it appears to always propagate at the constant speed

If one where to go by the thesis presented in Section 6, then, they may very well draw the following conclusion:

1) For matter, we must have,

2) For radiation, we must have,

While the above define the energy-momentum regimes satisfied by both matter and energy respectively, we believe these conditions do not in general define matter and radiation. For example, it would mean that if a material particle were accelerated to high enough energy-momentum regimes where

Let us consider the scenario where

If the momentum of the particle is defined as it is defined in the STR i.e., as,

substituting this into (27), one can show that

1) For matter to be distinctly defined as matter, we must have,

2) For radiation to be be distinctly defined as radiation, we must have,

We shall take the above as distinctly defining matter and radiation.

We here calculate or deduce the

From the equation

In [

We put forward a proposal to test the hypothesis here set-forth that

where

In the above,

If a seasonal variation in ^{11} m,

For example, let us take a simple pendulum of length

In this time interval, the most important figures are the last three numbers in bold and with an over-bar. These are the figures that will determine the correctness of our assertion because over the course of a year (preferably from January to December), these three figures are expected to vary in a way conforming to our assertion set- forth here-above. So, the proposed experiment must have the capacity to measure time durations to an accuracy of at least

From (4), i.e.

What the above means is that deflection of light by the Sun should take at least the Newtonian deflection of

0.87". This means that deflections in the range 0.87" - 1.75" are permitted: this deepens of the value of

Without shying away from the truth, theories without any grounds on which to test them are worthless. They stand there with their miniature beauty only to be marvelled at for their shear beauty and nothing more and beyond. A true scientist needs to find solid ground on which their theories may be falsified―only this way will our science take great forward leaps into new territory. As such, we propose grounds on which the idea of photon mass as proposed herein can be investigated.

For this effort to test the proposed theory, we make an appeal to the measurement of time-delays in the arrival times of gamma-rays. These gamma rays of different energies which exhibit an arrival time-delay are supposedly emanating from the same event, that is to say, these flashes of gamma-rays where produced at the same time and left the same place at the same time. If the speed of light is to be dependent on the wavelength of light (energy), then, it is expected that there must be an observed time delay in the arrival times of these energy bundles.

If correct, then, in no uncertain terms, Equation (19) spells it out clearly that if the mass of a photon is a function of its wavelength (energy), then, photons of different energies will travel at different speeds. Clearly, an energy dependent speed of light has the potential to explain the puzzling time delays observed in Gamma-ray Bursts (GRBs) which where first reported by in 1973 by Klebesadel et al. [

GRBs are the most intense gamma-ray events in the known Universe, originating from any random directions from the sky in a manner that is without precedent in all history of astronomy. When GRBs are observed, the more energetic emission arrives well after the low energetic emission. If these gamma-rays are emanating from the same event (source), they must leave this event at the same time and the observed delay must be due to a difference in speed of the low and high energy electromagnetic waves.

To demonstrate this, let

From this equation, it is clear that if

where

Therefore, from (36), if for the data for which

As already said―in the reading [

What this essentially means for the Solar gravitational bending of starlight where the deflection angle has been measured to be very close to that predicted by the GTR i.e. δ = 1.75"; is that for such photons, we must have

Now, given that

Since,

What is interesting here about the Formula (38) is that is naturally fits the gravitational deflection of star light whose spin in 1 because for spin-1 photons, the value of

If what we have presented herein is proven or shown to hold, then, the implications thereof have serious foresee- able repercussions and ramifications across the broad spectrum of physics as contemporary physics hitherto assumes that the mass of a photon is identically equal to zero, especially the embellished Standard Model of Particle Physics (SMPP). We have argued that this assumption may not be correct as it is based on a hitherto hidden and clandestine assumption that is not necessary; this assumption can be gotten reed off. It appears highly likely that is photons have mass, this mass must be extremely small behold the direct.

More than this, we have argued that, in principle, physics (i.e. Einstein’s STR and the theory of waves) has no problem with a massive photon that travels at the speed of light. If the ideas herein are accepted or acceptable, then, this places physics on a sure pedestal to consider massive photons as plausible physical objects of the Universe.

For example, in Quantum Electrodynamics (QED) and Quantum Field Theory (QFT) which are one of the two critical foundational pillars of the SMPP, a massive photon is not consistent with gauge invariance or renormalizability. However, via Proca Electrodynamics, one can devise a theory of massive photons [

On the more realistic side of things, if one can obtain results which are in satisfactory agreement with experience using a massless photon, why bother with an additional unessential? Simple let the sleeping dogs lay. In any case, Occam’s “all-powerful and very sharp” Razor forbids the unnecessary addition of non-essentials, so, there are very many good reasons to ignore massive photons. Perhaps physicist will consider them when they can longer avoid them at all. For now, there strongly appears to be no real need for them.

However, the notion that every particle that travels at the speed of light must―accordingly, be massless leads to an important but now solved problem in physics, this assumption has led to the Neutrino Mass Problem― namely, “Do neutrinos have mass?” According to Einstein’s STR, if neutrinos are massless, they must travel at the speed of light and conversely, if neutrinos travel at the speed of light, they must be massless. According of a recent CERN press release^{4} refuting the claim of faster-than light speed for neutrinos supposedly detected by the OPERA Collaboration in September of 2011, neutrinos strongly appear to travel at the speed of light.

Further, in order to explain neutrino oscillation, that is, the change of neutrinos from one state to the other^{5}, neutrinos must have mass. Endowing neutrinos with mass helped solve one of the outstanding problems in Solar physics known as the Solar neutrino problem. The Standard Solar Model (SSM) which is a theory detailing how the Sun produces its energy predicted that the Sun must produce a specific amount of neutrinos. Prior to the 1988 Sudbury Neutrino Observatory (SNO) Experiment, all neutrino detectors that had ever attempted to measure the rate at which the Sun made neutrinos were getting values between 1/3 and 2/3 of what was expected from theory. This meant that there was a dearth in neutrinos produced by the Sun if the SSM was right―this deficiency came to be known as the Solar Neutrino Problem (SNP). Neutrino oscillations resolved the SNP: the electron neutrinos produced in the Sun partly change into other flavors which prior experiments could not detect. Other than this, measurements indicate that neutrinos most certainly have mass. Additionally―as alreadly stated above, neutrinos most certainly travel at the speed of light. We seem to have a apparent contradiction here!

Ceteris paribus: in the light of popular contemporary physics, how can a massive particle travel at the speed of light? Is not this speed a preserve and prerogative of only massless particles? If observations are to take their rightful place in Science, which is that they take precedence over all our theories, then, we have but no choice except to accept that massive particles most certainly can travel at the speed of light as strongly appears to be the case with neutrinos.

The only way is to amend our theories to conform with observations and experiments. To that end, we have demonstrated herein that in principle, it is possible for massive particles to travel at the speed of light. In our supposition that only massless particles must travel at the speed of light and conversely, that, if a particle does happen to travel at the speed of light, it must accordingly be massless, we have argued that this assumption may not be correct as it hinges on a hitherto hidden and clandestine assumption that is surely not necessary; this assumption can swiftly be gotten reed off. Further, a small mass photon has here been shown to be a possible explanation as to the failure by both observations and experiments to pin-down the photon mass.

If they exist as we propose herein, massive photons pause a problem namely that they are according to Maxwell-Proca Electrodynamics supposed be a short range phenomenon. As far as observations and experi- ments reveal and can tell, the electromagnetic phenomenon is a long range phenomenon. This raises the question: “If photons are massive, how do they come to be a long range phenomenon?. This question is desively dealt with in [

Assuming the correctness (or acceptability) of the ideas presented herein, we hereby make the following conclusions:

^{4}http://press.web.cern.ch/Press/PressReleases/Releases2011/PR19.11E.html

^{5}Neutrinos come in three states known as flavors and these are the electron neutrino

1) As is typically assumed, massive photons do not necessarily lead to a violation of gauge invariance as we have shown that an appropriate choice of the gauge leads to a gauge invariant electrodynamics of massive photons.

2) The current belief or position that for a particle to travel at the speed of light it must be massless may not entirely be correct as this is based on a hitherto hidden and clandestine assumption that the energies

3) Neutrinos may very well be good candidates to be described by the present ideas of massive particles travelling at the speed of light. These neutrinos must have a non-zero

4) Gravitational deflection of monochromatic electromagnetic waves (such as gamma and radio waves) by the Sun presents the best way to measure the mass of photons by measuring^{6}.

5) It has been shown herein that it is possible for a photon to be massive. What is left to be answered is how massive photons come about to mediate as a long range phenomenon. This question is decisively dealt with in [

^{6}The theory on how to interpret

We are heftily grateful to the National University of Science & Technology (NUST)’s Research & Innovation Department and Research Board for their unremitting support rendered toward our research endeavours; of particular mention, Prof. Dr. P. Mundy, Dr. P. Makoni, Dr. D. J. Hlatywayo and Prof. Dr. Y. S. Naik’s unwavering support. We are grateful to the insights of Dr. William Straub―his insights have helped refine some portions of the present theory. This reading is dedicated to my mother Setmore Nyambuya and to the memory of my dearly missed departed father Nicholas Nyambuya (27.10.1947-23.09.1999).