What is the Mass of a Photon?
updated 24-JUL-1992 by SIC
original by Matt Austern
Or, "Does the mass of an object depend on its velocity?"
This question usually comes up in the context of wondering whether photons are really "massless," since, after all, they have nonzero energy. 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.
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 / c^2. (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 = E_0 / c^2, (2)
where E_0 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 Eq. (2), the equation "E = m c^2" is not correct. Using the standard definition, the relation between the mass and energy of an object can be written as
E = m c^2 / sqrt(1 -v^2/c^2), (3)
E^2 = m^2 c^4 + p^2 c^2, (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)
F = G m1 m2 / r^2. (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 Eq. (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.
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