Photons Must Have a Token Mass.

May 30, 2022
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Photons must have a token mass since we need to be able to form p out of m and v. Using E = hf and E = pc we get pc = mvc.. Since m and c are fixed (m is fixed at 5*10^(-19)) we have only v to encode photons with capability of having any energy and this v > c. We can call v the encoded velocity. We can't encode p directly because then mass is undefined, which means p (= mv) is also undefined - a contradiction.
 
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Apr 24, 2023
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This is a topic of some interest to me. You may want to look at my publication on the topic of photon reflection dynamics. It's hypothetical of course, but I have always believed that photons have some trivial mass. In my own model of physics, photons are simply electrons with reduced mass and should not be considered to be in the distinct category of "leptons." The mass reduction is the result of the transference of angular momentum into spin momentum upon the redirection of orbiting electrons in an electron shell and the expulsion of mass carriers during this brief window of time required to invert direction (not a truly instantaneous process.)

As light speed is consistent regardless of frequency (also sometimes referred to as energy level,) it stands to reason, as I explained in one of my recent publications, that spin speed (and spin direction oscillation) varies between photons of different energy levels. Something I've hinted at in my own work is that given a physical model in which neutrinos (the cream filling of electrons) may be mass-inverted by the discrete magnetism emitted by an electron/photon (or other neighboring electrons/photons) and that this mass-inversion results in the temporal inversion of direction and the eventual depletion of energy in a photon, limiting the distance it may travel through space, even in a vacuum.

The rate of this structural inversion of mass carriers (Higgs Bosons) determines the velocity of the photon. As the inter-relationship between phase height and wavelength is a constant (presumably universal,) neutrinos within photons are "spent" over time and distance traveled at a remarkably consistent rate. This rate of expenditure of neutrinos dictates the light speed constant.

That is my take on the matter.
 
Apr 24, 2023
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Phasing is the tendency of photons to oscillate "up and down" as they travel through space in a roughly linear fashion. Photons do not move in perfectly straight lines, as I would hope you are aware, but they move up and down in a sine wave pattern as they move in a general direction. The distance they move up and down in each oscillation is greater the greater the wavelength. Phase height is generally about 2% of wavelength.

The electric charge inherent in an electron must be continually replenished in order for it to continue to exist over the long-term. In the case of electrons (proper) orbiting atoms, neutrinos fluxing toward positively charged protons provide this charge (like a cell phone in use, but plugged into a wall outlet.) A photon is more like a cell phone operating on reserve power, but which is not plugged in. Eventually, its charge is depleted. This is the result of the interaction of quantum magnetism with quantum charge, an interaction that results in structural inversion of Higgs Bosons. The charge-carriers (neutrinos) are kicked into "the past" and the light wave is moved by a single neutrino width forward through space for each inversion that takes place.

I am rather glad you brought up the topic. It gave me another idea. Please see my thread for my latest posting.
 
Jan 15, 2024
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Photons must have a token mass since we need to be able to form p out of m and v. Using E = hf and E = pc we get pc = mvc.. Since m and c are fixed (m is fixed at 5*10^(-19)) we have only v to encode photons with capability of having any energy and this v > c. We can call v the encoded velocity. We can't encode p directly because then mass is undefined, which means p (= mv) is also undefined - a contradiction.
Then where does the mass go when the lights go out?
 
Jan 15, 2024
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The mass gets absorbed by matter when the photons are absorbed. Photons stop being created when the lights go out.
Easily proved or disproved as all one would need to do, to prove this would be to pulse a laser at a known mass that would have to increase its weight from the pulses, which does not happen.
 
May 30, 2022
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Light's path gets bent by a gravitational field. Its just not measurable (the weight). I postulate it as having mass smaller than the theoretical upper bound ( 10^(-18) kg or so).
 
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Light's path gets bent by a gravitational field. Its just not measurable (the weight). I postulate it as having mass smaller than the theoretical upper bound ( 10^(-18) kg or so).
If gravity bent light, you could never shine a flashlight onto a wall as the Earths gravity would suck it into the floor. As for astronomical observations no one has a clue as to whether light is bending or is it space itself that is bending and the light merely following. And before you say that you know, you do not know, no human does, we might someday, but not now.
 
Recall E=mc2, which equates mass and energy.

The logic can be constructed in many ways, and the following is one such. Take an isolated system (called a "particle") and accelerate it to some velocity v (a vector). Newton defined the "momentum" p of this particle (also a vector), such that p behaves in a simple way when the particle is accelerated, or when it's involved in a collision. For this simple behaviour to hold, it turns out that p must be proportional to v. The proportionality constant is called the particle's "mass" m, so that p = mv.

In special relativity, it turns out that we are still able to define a particle's momentum p such that it behaves in well-defined ways that are an extension of the newtonian case. Although p and v still point in the same direction, it turns out that they are no longer proportional; the best we can do is relate them via the particle's "relativistic mass" mrel. Thus

p = mrelv .

When the particle is at rest, its relativistic mass has a minimum value called the "rest mass" mrest. The rest mass is always the same for the same type of particle. For example, all protons have identical rest masses, and so do all electrons, and so do all neutrons; these masses can be looked up in a table. As the particle is accelerated to ever higher speeds, its relativistic mass increases without limit.

It also turns out that in special relativity, we are able to define the concept of "energy" E, such that E has simple and well-defined properties just like those it has in newtonian mechanics. When a particle has been accelerated so that it has some momentum p (the length of the vector p) and relativistic mass mrel, then its energy E turns out to be given by

E = mrelc2 , and also E2 = p2c2 + m2restc4 . (1)

There are two interesting cases of this last equation:
  1. If the particle is at rest, then p = 0, and E = mrestc2.
  2. If we set the rest mass equal to zero (regardless of whether or not that's a reasonable thing to do), then E = pc.
In classical electromagnetic theory, light turns out to have energy E and momentum p, and these happen to be related by E = pc.

Quantum mechanics introduces the idea that light can be viewed as a collection of "particles": photons. Even though these photons cannot be brought to rest, and so the idea of rest mass doesn't really apply to them, we can certainly bring these "particles" of light into the fold of equation (1) by just considering them to have no rest mass. That way, equation (1) gives the correct expression for light, E = pc, and no harm has been done. Equation (1) is now able to be applied to particles of matter and "particles" of light. It can now be used as a fully general equation, and that makes it very useful.

Alternative theories of the photon include a term that behaves like a mass, and this gives rise to the very advanced idea of a "massive photon". If the rest mass of the photon were non-zero, the theory of quantum electrodynamics would be "in trouble" primarily through loss of gauge invariance, which would make it non-renormalisable; also, charge conservation would no longer be absolutely guaranteed, as it is if photons have zero rest mass. But regardless of what any theory might predict, it is still necessary to check this prediction by doing an experiment.

It is almost certainly impossible to do any experiment that would establish the photon rest mass to be exactly zero. The best we can hope to do is place limits on it. A non-zero rest mass would introduce a small damping factor in the inverse square Coulomb law of electrostatic forces. That means the electrostatic force would be weaker over very large distances.

Likewise, the behavior of static magnetic fields would be modified. An upper limit to the photon mass can be inferred through satellite measurements of planetary magnetic fields. The Charge Composition Explorer spacecraft was used to derive an upper limit of 6 × 10−16 eV with high certainty. This was slightly improved in 1998 by Roderic Lakes in a laboratory experiment that looked for anomalous forces on a Cavendish balance. The new limit is 7 × 10−17 eV. Studies of galactic magnetic fields suggest a much better limit of less than 3 × 10−27 eV, but there is some doubt about the validity of this method.

See:


E. Fischbach et al., Physical Review Letters 73, 514–517 25 July 1994.

Chibisov et al., Sov. Ph. Usp. 19, 624 (1976).

See also the Review of Particle Properties at http://pdg.lbl.gov

You can get a mass metric from the energy of a photon, everything else is a matter of definition of the English words. It is a universal statement to say that photons do not have rest mass.

Coulomb's inverse square law implies that the photon does not have a mass. So if you wish find the lower limit for the photon mass, you should look at deviations from the Coulomb's law at large distances. If photon does indeed have a mass then the force between charged particles will go as the Yukawa potential*.

Renormalization of the mass of an electron is studied within the framework of the Extended Holstein model at strong coupling regime and nonadiabatic limit. In order to take into account an effect of screening of an electron-phonon interaction on a polaron it is assumed that the electron- phonon interaction potential has the Yukawa form and screening of the electron-phonon interaction is due to the presence of other electrons in a lattice. The forces are derived from the Yukawa type electron-phonon interaction potential. It is emphasized that the early considered screened force is a particular case of the force deduced from the Yukawa potential and is approximately valid at large screening radiuses compared to the distances under consid- eration. The Extended Holstein polaron with the Yukawa type potential is found to be a more mobile than polaron studied in early works at the same screening regime.

See: https://arxiv.org/pdf/1308.2197.pdf

*Yukawa potential is the effective non-relativistic description of the interaction of two particles due to the exchange of a massive particle of mass ⁠. It was proposed by H. Yukawa as a low-energy description of the strong interactions between nucleons, due to the exchange of massive particles, now known as pions.

See: Dark Universe phenomenology from the Yukawa potential at

The electron has mass:
eimg270.gif
eimg22.gif
eimg271.gif
eimg22.gif
eimg272.gif

The particle of electromagnetic radiation is often assumed to be massless, but the laws of physics do not require that assumption. If the photon has a mass, however, it must be exceedingly small.
Hartmann352
 
Apr 24, 2023
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"If photons have mass"

Imagine paying that much money for a college education only to have phrases like that in your coursework. This is why we're falling behind. People are actually still suggesting that there is a chance that photons have zero mass. In fact, until recently, textbooks stated that "photons are certainly massless."

They have even been able to estimate the mass of the neutrino. If the neutrino has mass, a photon certain does. How is this even up for debate?
 
Jan 15, 2024
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Recall E=mc2, which equates mass and energy.

The logic can be constructed in many ways, and the following is one such. Take an isolated system (called a "particle") and accelerate it to some velocity v (a vector). Newton defined the "momentum" p of this particle (also a vector), such that p behaves in a simple way when the particle is accelerated, or when it's involved in a collision. For this simple behaviour to hold, it turns out that p must be proportional to v. The proportionality constant is called the particle's "mass" m, so that p = mv.

In special relativity, it turns out that we are still able to define a particle's momentum p such that it behaves in well-defined ways that are an extension of the newtonian case. Although p and v still point in the same direction, it turns out that they are no longer proportional; the best we can do is relate them via the particle's "relativistic mass" mrel. Thus

p = mrelv .

When the particle is at rest, its relativistic mass has a minimum value called the "rest mass" mrest. The rest mass is always the same for the same type of particle. For example, all protons have identical rest masses, and so do all electrons, and so do all neutrons; these masses can be looked up in a table. As the particle is accelerated to ever higher speeds, its relativistic mass increases without limit.

It also turns out that in special relativity, we are able to define the concept of "energy" E, such that E has simple and well-defined properties just like those it has in newtonian mechanics. When a particle has been accelerated so that it has some momentum p (the length of the vector p) and relativistic mass mrel, then its energy E turns out to be given by

E = mrelc2 , and also E2 = p2c2 + m2restc4 . (1)

There are two interesting cases of this last equation:
  1. If the particle is at rest, then p = 0, and E = mrestc2.
  2. If we set the rest mass equal to zero (regardless of whether or not that's a reasonable thing to do), then E = pc.
In classical electromagnetic theory, light turns out to have energy E and momentum p, and these happen to be related by E = pc.

Quantum mechanics introduces the idea that light can be viewed as a collection of "particles": photons. Even though these photons cannot be brought to rest, and so the idea of rest mass doesn't really apply to them, we can certainly bring these "particles" of light into the fold of equation (1) by just considering them to have no rest mass. That way, equation (1) gives the correct expression for light, E = pc, and no harm has been done. Equation (1) is now able to be applied to particles of matter and "particles" of light. It can now be used as a fully general equation, and that makes it very useful.

Alternative theories of the photon include a term that behaves like a mass, and this gives rise to the very advanced idea of a "massive photon". If the rest mass of the photon were non-zero, the theory of quantum electrodynamics would be "in trouble" primarily through loss of gauge invariance, which would make it non-renormalisable; also, charge conservation would no longer be absolutely guaranteed, as it is if photons have zero rest mass. But regardless of what any theory might predict, it is still necessary to check this prediction by doing an experiment.

It is almost certainly impossible to do any experiment that would establish the photon rest mass to be exactly zero. The best we can hope to do is place limits on it. A non-zero rest mass would introduce a small damping factor in the inverse square Coulomb law of electrostatic forces. That means the electrostatic force would be weaker over very large distances.

Likewise, the behavior of static magnetic fields would be modified. An upper limit to the photon mass can be inferred through satellite measurements of planetary magnetic fields. The Charge Composition Explorer spacecraft was used to derive an upper limit of 6 × 10−16 eV with high certainty. This was slightly improved in 1998 by Roderic Lakes in a laboratory experiment that looked for anomalous forces on a Cavendish balance. The new limit is 7 × 10−17 eV. Studies of galactic magnetic fields suggest a much better limit of less than 3 × 10−27 eV, but there is some doubt about the validity of this method.

See:


E. Fischbach et al., Physical Review Letters 73, 514–517 25 July 1994.

Chibisov et al., Sov. Ph. Usp. 19, 624 (1976).

See also the Review of Particle Properties at http://pdg.lbl.gov

You can get a mass metric from the energy of a photon, everything else is a matter of definition of the English words. It is a universal statement to say that photons do not have rest mass.

Coulomb's inverse square law implies that the photon does not have a mass. So if you wish find the lower limit for the photon mass, you should look at deviations from the Coulomb's law at large distances. If photon does indeed have a mass then the force between charged particles will go as the Yukawa potential*.

Renormalization of the mass of an electron is studied within the framework of the Extended Holstein model at strong coupling regime and nonadiabatic limit. In order to take into account an effect of screening of an electron-phonon interaction on a polaron it is assumed that the electron- phonon interaction potential has the Yukawa form and screening of the electron-phonon interaction is due to the presence of other electrons in a lattice. The forces are derived from the Yukawa type electron-phonon interaction potential. It is emphasized that the early considered screened force is a particular case of the force deduced from the Yukawa potential and is approximately valid at large screening radiuses compared to the distances under consid- eration. The Extended Holstein polaron with the Yukawa type potential is found to be a more mobile than polaron studied in early works at the same screening regime.

See: https://arxiv.org/pdf/1308.2197.pdf

*Yukawa potential is the effective non-relativistic description of the interaction of two particles due to the exchange of a massive particle of mass ⁠. It was proposed by H. Yukawa as a low-energy description of the strong interactions between nucleons, due to the exchange of massive particles, now known as pions.

See: Dark Universe phenomenology from the Yukawa potential at

The electron has mass:

eimg270.gif
eimg22.gif
eimg271.gif
eimg22.gif
eimg272.gif


The particle of electromagnetic radiation is often assumed to be massless, but the laws of physics do not require that assumption. If the photon has a mass, however, it must be exceedingly small.
Hartmann352
LOL Einstein claimed that the universe was not moving, so get over it already, he was wrong
 
Jan 15, 2024
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"If photons have mass"

Imagine paying that much money for a college education only to have phrases like that in your coursework. This is why we're falling behind. People are actually still suggesting that there is a chance that photons have zero mass. In fact, until recently, textbooks stated that "photons are certainly massless."

They have even been able to estimate the mass of the neutrino. If the neutrino has mass, a photon certain does. How is this even up for debate?
LOL, how come a neutrino's mass can be measured and not a photon's. LOL you are actually claiming that a flashlight generates mass.
 
LOL Einstein claimed that the universe was not moving, so get over it already, he was wrong

The cosmological constant, alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equations of general relativity. He later removed it. Much later it was revived and reinterpreted as the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy.

Albert Einstein admitted that his insertion of a cosmological constant, to insure a steady state universe, was what he termed as his biggest mistake. He later calculated it properly.

You should understand the expansion of the universe, which is not “moving” as you indicate. The actual space, between the galaxies, is expanding. Hence, the farther you look into the universe means that you are looking back into time.

Astronomers have known that the universe is expanding for a century. Space-time is stretching itself out over billions of light-years, carrying the galaxies within it apart, like raisins embedded within a rising loaf of bread. This steady expansion, pitted against the cosmos’ urge to collapse under its own gravity, means there are two main scenarios for how the universe will eventually end. These scenarios are dubbed the Big Crunch — where gravity overcomes expansion and the Big Bang occurs in reverse — and the Big Freeze — where gravity loses out to the expansion and all matter is isolated by unfathomable distances.

For a while, researchers believed the universe’s fate was leaning toward the final scenario. But, in the late 1990s, Saul Perlmutter, Martin Riese and Brian Schmidt discovered something unexpected that changed our understanding of the future of the universe: They found that the most distant galaxies weren’t just moving away from us. They were accelerating away from our vantage point. These three physicists shared the 2011 Nobel Prize for this discovery.

Perlmutter et al found that large percent of the universe is made up of something previously undiscovered and unexpected, though hinted at by Veras Rubin in her study of galactic stars and their rotation around the center bulge. And this so-called dark energy is overpowering gravity and driving space-time apart from within.

dark energy.png

The Dark Energy Survey conducted by the Dark Energy Camera mounted on the Víctor M. Blanco 4-meter Telescope at Cerro Tololo Inter-American Observatory in Northern Chile shows that observations of supernovas remain integral to solving the mystery that such investigations triggered 25 years ago.

The new Dark Energy Survey results were just presented at the 243rd meeting of the American Astronomical Society on Jan. 8, 2024, with the team behind them adding they are consistent with the standard model of cosmology, the so-called "Lambda cold dark matter" model (ΛCDM), that features a universe with accelerating expansion.

These results place the tightest constraints on the history of expansion throughout the 13.8 billion-year history of the cosmos, but they also leave breathing room for more complex models of the universe.
Hartmann352
 
Jan 15, 2024
21
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30
The cosmological constant, alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equations of general relativity. He later removed it. Much later it was revived and reinterpreted as the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy.

Albert Einstein admitted that his insertion of a cosmological constant, to insure a steady state universe, was what he termed as his biggest mistake. He later calculated it properly.

You should understand the expansion of the universe, which is not “moving” as you indicate. The actual space, between the galaxies, is expanding. Hence, the farther you look into the universe means that you are looking back into time.

Astronomers have known that the universe is expanding for a century. Space-time is stretching itself out over billions of light-years, carrying the galaxies within it apart, like raisins embedded within a rising loaf of bread. This steady expansion, pitted against the cosmos’ urge to collapse under its own gravity, means there are two main scenarios for how the universe will eventually end. These scenarios are dubbed the Big Crunch — where gravity overcomes expansion and the Big Bang occurs in reverse — and the Big Freeze — where gravity loses out to the expansion and all matter is isolated by unfathomable distances.

For a while, researchers believed the universe’s fate was leaning toward the final scenario. But, in the late 1990s, Saul Perlmutter, Martin Riese and Brian Schmidt discovered something unexpected that changed our understanding of the future of the universe: They found that the most distant galaxies weren’t just moving away from us. They were accelerating away from our vantage point. These three physicists shared the 2011 Nobel Prize for this discovery.

Perlmutter et al found that large percent of the universe is made up of something previously undiscovered and unexpected, though hinted at by Veras Rubin in her study of galactic stars and their rotation around the center bulge. And this so-called dark energy is overpowering gravity and driving space-time apart from within.

View attachment 3462

The Dark Energy Survey conducted by the Dark Energy Camera mounted on the Víctor M. Blanco 4-meter Telescope at Cerro Tololo Inter-American Observatory in Northern Chile shows that observations of supernovas remain integral to solving the mystery that such investigations triggered 25 years ago.

The new Dark Energy Survey results were just presented at the 243rd meeting of the American Astronomical Society on Jan. 8, 2024, with the team behind them adding they are consistent with the standard model of cosmology, the so-called "Lambda cold dark matter" model (ΛCDM), that features a universe with accelerating expansion.

These results place the tightest constraints on the history of expansion throughout the 13.8 billion-year history of the cosmos, but they also leave breathing room for more complex models of the universe.
Hartmann352
It was wrong, why do you celebrate a buffoon
 
Apr 24, 2023
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Is there theoretical or observational reason(s) to set photon rest mass to exactly zero?
I am not clear on what you mean by "rest mass" but of course, it is difficult to estimate the mass of photons because its magnetic and electrical effects are much more pronounced than their mass-related effects. I have reached the conclusion that photons have extremely marginal, but non-zero mass.

One experiment we might try in order to estimate the mass of a single photon would be to suspend a photon positionally in an ion trap and using a Coulomb Force Line of known intensity in order to exert electroweak force against the photon. Proximity of the trapped photon to the walls of an ion trap could then be measured using an exciton-based induction mechanism comparable to those now being investigated as potentially useful in advanced LED displays. The amplitude of current flow through excitons in the walls of the ion trap could be used to estimate the mass of the individual photon.
 
Apr 24, 2023
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Photons never rest, thus they have no rest mass. Its impossible
Regardless of the rectitude of your statement, this got me to thinking about the topic of how it is that a photon goes from having little mass to having full mass as an electron once again during a photoelectric conversion process. I just published something to explain this, which complements another recent publication of mine on the topic of how electrons are converted into photons and what causes them to shed their mass.

I have concluded that charge and mass are restored through two separate processes and that the shedding of mass is linked to the emission of discrete magnetism. I would therefore submit that photons have variable mass depending upon factors like distance traveled.

To measure mass in a photon, it would not only need to be trapped, but it would need to be trapped in such a way that it is permitted to continue moving in a circular fashion within a large scale trap as motion is in the nature of a photon. As I stated in the previous post, Coulomb Force Lines of known strength and proximity could be used to nudge the photon in this specialized trap toward excitons, which would drain varying levels of current from the photons depending upon proximity. The rate at which a CFL will nudge something like a photon into a wider orbit in such a trap would be dictated by the inertia of the photon, which would be determined by its mass. In such a scheme, a CFL would have a uniform effect on photons according to charge in terms of the extent to which orbit is influenced, but the rate at which that orbit is altered would have to be dictated by inertia which, as I said, can be used as a proxy for identifying mass.

Importantly, you would need to keep the photons in a vacuum and away from any protons which would generate a Higgs Field that would have the effect of restoring mass to the photon and turning it either into an electron or into a heavy photon somewhere between the mass of an electron and the mass of a photon.

I think that if we performed this experiment, we would find that the mass is quite low, indeed, but varies within a range depending upon the energy level and flight time of a photon. I would predict that high-energy photons would have less mass than low-energy ones and I would, furthermore, predict that the greater the distance traveled, the further mass is reduced.
 
May 30, 2022
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Their mass needs to be non zero so that E = pc = mcc can be any size. Its the only variable in this equation's right side.

For E = 1 J this means it's mass should be 1/6* 10^(-16) kg which is larger than the maximum mass. So I don't know what to do.
 
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Their mass needs to be non zero so that E = pc = mcc can be any size. Its the only variable in this equation's right side.

For E = 1 J this means it's mass should be 1/6* 10^(-16) kg which is larger than the maximum mass. So I don't know what to do.


Answering your question over in my thread now. You're welcome.
 
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The maximum value is 10^(-18) eV/c^2 so this mass is much too large. Converting this to kg gives: 10^(-18-36) = 10^(-54) kg !

Poe tells me 10^(-36) is the conversion factor, but it uses E = mc^2 and in this formula energy must be in Joules! According to me the conversion factor is 10^(-19-16) = 10^(-35) giving the maximum mass = 10^(-18-35) = 10^(-53) kg !

I can't find your thread S_Edwards!
 
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All of my postings are in .png images below the posts, which are simply the titles. #308 is a concept for an experiment for establishing photon masses at different energy levels. #309 will be for a prism which enables a technology I came up with back in June which enables neutrino wave detection using polarity measurement.

Actually I changed my mind, the 25 June 2023 publication actually describes something more advanced than simple polarity measurement over the 2 1/2 mile distance since it is direction-specific and is less noisy and doesn't require a 2 1/2 mile tunnel.

Now working on concept for a prism designed to aid in measuring polarity with unprecedented precision via conversion of polarity shifts into changes in angular momentum which can be more easily measured in support of a novel SSD voltage cell read mechanism I came up with 20 September 2023.

For measuring neutrino waves associated with EM, the best way is to fire the beam at the signal source and measure the slowing of the arrival of light over short distances but for assessing the precise charge level of a voltage cell, one needs to measure magnetism precisely using an indirect observation method. Good times.
 
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Photons do not have mass.
You can say it has relativistic mass, due to m^2 = E^2/c^4 - p^2/c^2, but it has no mass.