I was recently asked on facebook:

Professor, I have a question about photons. Photons are packets of energy that have no mass, correct? Also sunlight has been calculated to have mass and Einstein's theory of general relativity relates energy and mass. How is then possible that photons have energy and no mass?

That is an interesting question, and has to do with the full expression of energy in special relativity. We are likely all familiar with the equation

$$E = mc^{2}.$$

However, this is a simplification of the more general equation

$$E^{2} = (mc^{2})^{2} + (pc)^{2},$$

where \(E\) is the energy of a particle, \(m\) is the rest mass of the particle, \(p\) is the momentum of the particle, and \(c\) is the speed of light. In this argument, the photon is a massless particle, meaning its energy would be given by the expression

$$E = pc.$$

So, does a photon have momentum if it has no mass? The answer is yes. In explaining the shape of the thermal emission spectrum Max Planck found that the energies emitted by a thermal source were quantized and that these energies (\(E\)) were proportional to the frequency (\(\nu\)) of the emitted photons. The proportionality constant, (\(h\)), is now known as Planck’s constant. So a photon’s energy is given by the relation


Einstein combined his theory of relativistic energy with Planck’s theory of thermal radiation and one of the relationships derived by combining the above equations is that the momentum of a photon is

$$p = \frac{h\nu}{c},$$

and its kinetic energy is

$$K = h\nu.$$

This turns out to be crucial in explaining the photoelectric effect in which photons are able to dislodge electrons from a metal surface. It turns out the frequency of the photons is important, because unless the frequency is high enough the kinetic energy of the photons will be unable to overcome the binding energy of the electrons to their atoms. Classical physics could not explain why frequency would matter, but these concepts in relativity explained the effect and are the reason why even though a photon is massless it carries kinetic energy and momentum.