Updated 2007 by Don Koks.
Updated 1998 by Phil Gibbs.
Updated 1992 by Scott Chase.
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.

Photons are traditionally said to be massless.  This is a figure of speech that physicists use to describe something about how a photon's particle-like properties are described by the language of special relativity.

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 such that it behaves in well-defined ways that are an extension of the newtonian case.  The vector p is no longer proportional to the vector v (although they do both grow or shrink together), but these two vectors do still lie in the same direction; so we can define the ratio of the length of p to the length of v to be 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, electrons, and neutrons have the same rest mass; it's something that 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.

Because the energy of a particle just equals its relativistic mass times c2, physicists have learned to economise the language by only ever referring to a particle's energy.  When they use the term "mass", they mean rest mass.  This is purely a linguistic convention.  When the two sorts of mass are referred to together, relativistic mass is usually written m and rest mass is written m0.  But when only rest mass is being used, then the word "mass" is assumed to mean rest mass, and it tends to be written simply as m.

One reason why the term relativistic mass is sometimes avoided is because there is actually a directional dependence to a particle's resistance to being accelerated; it's much easier to push a fast-moving particle sideways than it is to alter its speed in its direction of motion.  Another reason is that physicists do not want anyone thinking that relativistic mass can be used in the newtonian relations F = ma and F = G m1 m2 /r2.  There is no definition of mass for which these equations are true relativistically; they must be generalised.  The generalisations are more straightforward using the idea of rest mass, as opposed to using relativistic mass.  For more discussion of the term "relativistic mass", see the FAQ entry Does mass change with speed?

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

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.  However, whatever theory says, it is still necessary to check theory against 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 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.

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.

References:

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.

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/