RE: Dark Matter

Hi Matthew,

Your question is an interesting one because the theory of relativity predicts that particles that travel at the speed of light have zero mass. Actually, it's probably better to say that relativity relies on there being one, limiting universal velocity (the speed of light) and particles that have mass can approach this velocity but never equal or exceed it. Particles with zero mass, on the other hand, must travel at the speed of light. It's therefore an interesting question to measure whether photons travel at exactly the speed of light.

If we assume that photons have mass, then we can use the equations of relativity to construct an equation that relates the mass of a photon to its speed and its wavelength. This equation predicts that the speed of a photon in empty space is related to its wavelength, unless the photon has zero mass. Basically, if the photon has mass, then short wavelength photons should travel at much lower speeds than long wavelength photons.

So, we can try to measure the mass of the photon, or at least put limits on how large its mass can be, by measuring the speed of light for photons of different wavelengths. One of the most accurate ways to do this is by using observations of pulsars. Pulsars are the remnants of dead stars which emit reqular pulses of radiation over a range of wavelengths. If we assume that all of the photons in a single pulse are emitted by a pulsar at exactly the same time then, if photons have mass, long wavelength photons in the pulse will arrive at the Earth slightly ahead of short wavelength photons. Therefore, we can estimate the mass of a photon by attempting to measure this time delay.

This experiment has been carried out using observations of a pulsar in the Crab nebula. In visible light, there is no observed time difference between the arrival times of photons of different wavelengths. This puts a limit on the mass of the photon (from the measurement sensitivity of the experiment) of 10 to the power -40 kg (about 10 to the power 13 times smaller than the mass of a proton.) With radio observations, which are much more sensitive than observations of visible light, this limit has been pushed down to 10 to the power -47 kg.

This mass is too small to explain the dark matter problem. Physicists are currently looking to the neutrino, another massless particle, as a possible answer to the problem. However, a similar experiment to measure the mass of the neutrino, which measured the arrival times at the Earth of neutrinos from supernova 1987a, seems to indicate that neutrinos, like protons, have a negligible, probably zero, mass.

For more discussion of this, including a derivation of the equations involved and a brief discussion of how the mass of the photon is measured, I recommend looking in "An Introduction to Mechanics" by Kleppner and Kolenkow, which has an excellent discussion in Section 13.4

Jim O'Donnell

jim@astroscu.unam.mx

Addendum:
Checking my math for the answer to this question, I found a mistake in my answer. I said that if photons had a rest mass, then short wavelength photons would travel more slowly than long wavelength photons. This should be the other way around; the speeds of photons would approach the speed of light as their wavelengths got shorter. So, for example, if a pulsar emits a burst of red and blue photons then the blue photons would arrive at the Earth slightly ahead of the red photons, and a measurement of the delay between the arrival times would allow us to measure (or at least place a limit on) the rest mass of the photon.

Added by the mad scientist astronomy moderator:

Jim's comments on the constraints on a potential mass for the photon are entirely accurate.

In response to the original question I would like to add that the energy of a photon (E=h*nu=Planck's Constant times frequency) can indeed be related to a "mass" by Einstein's famous formula E=m*c^2(=mass times speed of light squared). This gives an equivalent gravitational mass for the photon even though as Jim said, the rest mass of the photon is most probably zero. This gravitational mass determines how much gravitational attraction is caused by a photon. Such gravitational attraction generally turns out to be negligibly small, and all the light observed in the universe does not make a significant contribution to dark matter. marc herant