In response to my column the other day about whether or not neutrinos could travel faster than light, several of you raised the possibility that:

It could simply mean that neutrinos have less mass than photons, so that the true c should be the speed of neutrinos, not photons.

This was also brought up in the comments section at io9. There’s an element of truth to this observation, since we don’t know for certain that photons have zero mass. From an observational perspective, there’s no functional difference between “zero” and “too tiny to matter.”

But how small could the photon mass be and still be consistent with observations? Short answer: waaaay smaller than is required to explain the neutrino results. Long answer is a bit technical, so if you’re averse to equations, you may want to quit here.

Let’s play along and assume:

1. Neutrinos are massless (which they clearly aren’t, since they oscillate) and travel at $c$ (which I will cease to call “the speed of light” for this discussion).
2. Photons have some (as yet unknown) mass, $m$.
3. The disparity between the speed of light and neutrinos is:

$1-\frac{v}{c}=2\times 10^{-5}$

where $v$ is the speed of the photon. This is the size of the difference we’re talking about, if we take the OPERA result literally.

4. For our initial purposes, we’ll suppose that all calibration is done using light at $\lambda=700nm$ (red).
5. Special relativity is correct.

So, under these assumptions, how massive is the photon? Well, we can compute the $\gamma$ factor easily enough:

$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{1}{\sqrt{1-\left(1-2\times 10^{-5}\right)^2}}\simeq 158$

Problem 1: This is not a big $\gamma$ factor.

Protons in the LHC have a factor of something like 3500, which means that we would regularly produce particles traveling faster than light. We would notice that.

But, moving on…

Supposing this is the case, we can now compute the mass of the photon. The energy of a relativistic particle can be computed from its wavelength:

$E=\frac{hc}{\lambda}=\frac{(6.626\times 10^{-34}J\cdot s)(3\times 10^8 m/s)}{7\times 10^{-7}m}=2.84\times 10^{-19}J$

And since
$E=mc^2\gamma$

we can compute:
$m=\frac{E}{c^2\gamma}=2\times 10^{-38}kg$

I should point out that this wouldn’t account for dark energy or anything like that. The mass-energy for a photon would be only about 0.01 eV, and even in aggregate would constitute less total mass than protons and neutrons.

But even at this tiny mass…

Problem 2: Photons would be non-relativistic, even in the Infrared.

To put things in perspective, thermal emissions given off by people (approximately) are about 0.02eV. If half of that is mass energy, then the photons would move at much less than c. We’d notice. Even worse, photons from the cosmic microwave background (T=3K), are a hundred times smaller. The spectrum wouldn’t even look like what we normally consider thermal, but observationally it does and to insane precision.

But even stranger, photons are the carrier particle for electromagnetism. If you have a massive carrier particle, it drops off after a finite range.

Problem 3: Electromagnetism would be a short-range force.

The range of a force with a mediator particle, $m$ is:

$r\simeq \frac{l_p}{m/m_p}$

where $l_p$ is the Planck length (around $1.6\times 10^{-35 m}$) and $m_p$ is the Planck mass (around $2\times 10^{-8}kg$).

You’ll notice that if the mass of the mediator goes to zero, the force becomes universally long range. For the massive photon, this would yield:

$r\simeq 16\mu m$

Magnets and capacitors and so forth wouldn’t work on even remotely macroscopic scales.

Even worse, we know that magnetic fields operate on the earth (and really, on scales of the solar system and Galaxy, but since your compass works, let’s just focus on the earth), meaning that the maximum possible photon mass consistent with earth-scale magnetic fields is a factor of $10^{11}$ times smaller than what we’re discussing here.

Finally, there’s the fact that if the photon were massive, different frequencies would travel at different speeds.

Problem 4: Blue light would travel much faster than red.

I’ve done the equations already, but using this mass, we’d find:

$v_{blue}-v_{red}=2081 m/s$

But since the uncertainty in the speed of light is much less than 1 m/s (or really, since the uncertainty in the definition of the meter is that small fractionally), we’d notice this effect, even using just visible light.

And remember, all of this is based on assuming the minimum possible photon mass consistent with the explanation provided. If neutrinos also have mass (which they do) photons would have to be even more massive.

In summary: The OPERA results probably don’t tell us anything new about how photons work.

-Dave

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What about the possibility that the photon has a mass, and the mass varies depending on the wavelength of the photon?

How long do you figure, ’till we start hearing possible causes, or that the experiment has been replicated?

And I havn’t quite been able to find what the experiment they were doing was meant to do in the first place. They didn’t specifically build the experiment around an idea that neutrinos might travel faster than light did they?

/P

Does this mean that c is actually a number higher than the rate of a neutrino, since as you have stated, a neutrino does have mass, however small? If a smaller yet particle is found, then would it not have a higher rate of speed and c ultimately would be only truly found by extrapolating the rates of speed of a zero mass particle, which probably doesn’t exist, making c only a theoretical value.

Not quite. Some readers had asked if it was possible that the explanation of the OPERA results could have been due to a massive photon. I argue it can’t be. Not only is c not a theoretical value, it’s a VERY well-measured value.

While neutrinos have mass, they have very little of it, compared to their energies (though exactly how little, we still can’t say). Thus, the expectation is that they are traveling at 99.9999999999% the speed of light (or something like that), which functionally would be indistinguishable from c.

I was thinking along the lines of: if a photon’s speed is somewhat limited by the fact that it has a non-zero mass, and perhaps a neutrino has even less mass, then that may explain the results of the Cern experiment, and imply that less massive particles have their own intrinsic speed. Which would also imply that c is based on the mass, however small, of the particle, and thus as the mass truly approaches zero, the measurable speed increases.

This is exactly what Dave addresses in the article and disproves handily. Even if the neutrino is massless, then the velocity difference from OPERA implies that the photon, to satisfy the eqn for the velocity difference, would require a mass completely inconsistent with our universe. There must be another explanation. I too at first thought of this angle but was too dim to do the math that Dave has done (thanks!).

Do photons have mass? Hell I didn’t even think they were religious.

Thanks, I kind of got lost on some of the math and the point he was making, I read it again and can sort of understand it….

Photons have energy which is directly equivalent to mass. This makes sense if you consider that photons don’t interact with the Higgs so they won’t accumulate mass relative to their high velocities.