Updated 1998 by PEG.
Updated 1996 by PEG.
Updated 1992 by SIC.
Original by Matt Austern.

What is the Mass of a Photon?

This question falls into two parts:

Does the photon have mass, after all it has energy and energy is equivalent to mass?

This question comes up in the context of wondering whether photons are really "massless," since, after all, they have nonzero energy and energy is equivalent to mass according to Einstein's equation E=mc². The problem is simply that people are using two different definitions of mass. The overwhelming consensus among physicists today is to say that photons are massless. However, it is possible to assign a "relativistic mass" to a photon which depends upon its wavelength. This is based upon an old usage of the word "mass" which, though not strictly wrong, is not used much today. See also the Faq article Does mass change with velocity?.

The old definition of mass, called "relativistic mass," assigns a mass to a particle proportional to its total energy E, and involved the speed of light, c, in the proportionality constant:

                m = E / c².                                    (1)

This definition gives every object a velocity-dependent mass.

The modern definition assigns every object just one mass, an invariant quantity that does not depend on velocity. This is given by

                m = E₀ / c²,                                   (2)

where E₀ is the total energy of that object at rest.

The first definition is often used in popularizations, and in some elementary textbooks. It was once used by practicing physicists, but for the last few decades, the vast majority of physicists have instead used the second definition. Sometimes people will use the phrase "rest mass," or "invariant mass," but this is just for emphasis: mass is mass. The "relativistic mass" is never used at all. (If you see "relativistic mass" in your first-year physics textbook, complain! There is no reason for books to teach obsolete terminology.)

Note, by the way, that using the standard definition of mass, the one given by eqn (2), the equation "E = m c²" is not correct. Using the standard definition, the relation between the mass and energy of an object can be written as

                E  = m c² / sqrt(1 - v²/c²),                   (3)

or as

                E² = m² c⁴  +  p² c²,                          (4)

where v is the object's velocity, and p is its momentum.

In one sense, any definition is just a matter of convention. In practice, though, physicists now use this definition because it is much more convenient. The "relativistic mass" of an object is really just the same as its energy, and there isn't any reason to have another word for energy: "energy" is a perfectly good word. The mass of an object, though, is a fundamental and invariant property, and one for which we do need a word.

The "relativistic mass" is also sometimes confusing because it mistakenly leads people to think that they can just use it in the Newtonian relations

                F = m a                                       (5)

and

                F = G m₁ m₂ / r².                             (6)

In fact, though, there is no definition of mass for which these equations are true relativistically: they must be generalized. The generalizations are more straightforward using the standard definition of mass than using "relativistic mass."

Oh, and back to photons: people sometimes wonder whether it makes sense to talk about the "rest mass" of a particle that can never be at rest. The answer, again, is that "rest mass" is really a misnomer, and it is not necessary for a particle to be at rest for the concept of mass to make sense. Technically, it is the invariant length of the particle's four-momentum. (You can see this from eqn (4).) For all photons this is zero. On the other hand, the "relativistic mass" of photons is frequency dependent. UV photons are more energetic than visible photons, and so are more "massive" in this sense, a statement which obscures more than it elucidates.

Reference: Lev Okun wrote a nice article on this subject in the June 1989 issue of Physics Today, which includes a historical discussion of the concept of mass in relativistic physics.

Is there any experimental evidence that the photon has zero rest mass?

If the rest mass of the photon was non-zero, the theory of quantum electrodynamics would be "in trouble" primarily through loss of gauge invariance, which would make it non-renormalizable; also, charge-conservation would no longer be absolutely guaranteed, as it is if photons have vanishing rest-mass. However, whatever theory says, it is still necessary to check theory against experiment.

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

The behavior of static magnetic fields is likewise modified. A limit on the photon mass can be obtained through satellite measurements of planetary magnetic fields. The Charge Composition Explorer spacecraft was used to derive a limit of 6x10^-16 eV with high certainty. This was slightly improved in 1998 by Roderic Lakes in a laborartory experiment which looked for anomalous forces on a Cavendish balance. The new limit is 7x10^-17 eV. Studies of galactic magnetic fields suggest a much better limit of less than 3x10^-27 eV but there is some doubt about the validity of this method.

References:

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

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

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