special relativity - Do light and sound waves have mass

I have been reading Hawking's 'A Brief History of Time' and it has gotten me thinking about Einstein's theory of relativity, in that it assumes that an object must have infinite mass if it is to be traveling at the speed of light (please correct if I'm wrong in my beginner's knowledge of physics).

But, do light waves have any sort of measurable mass? Or in that same vein, do sound waves? Is it somehow possible?

Answer

The particles of light waves - the photons - have the rest mass $m_0$ equal to zero. However, at the speed of light, $v=c$, the total mass $$ m= \frac{m_0}{\sqrt{1-v^2/c^2}} $$ is increased to an indeterminate form, $0/0$, which should be evaluated as a finite number. The photons - and everything else - carry the total mass that is proportional to the total energy via the famous $E=mc^2$ relation.

Yes, this mass may be measured. For example, uranium nuclear power plants burn the uranium and reduce its mass by 0.1 percent or so because the waste products (the nuclei) are actually a little bit lighter. This energy may be completely transformed to the radiation coming from light bulbs - and the light from these light bulbs carry 0.1 percent of the uranium mass away. This mass is a source of gravitational field and adds inertia to boxes with this light etc.

Sound is different. The speed of sound is much smaller than the speed of light.

While "phonons" in low-temperature condensed matter physics - particles of sound - are analogous to photons in many respects, and $E=mc^2$ still applies, the same is not true for sound waves in the air etc. Because the temperature of the air is nonzero, the "ground state" - the lowest-energy state at fixed conditions, with the minimum number of "sound quanta" or "phonons" - is not really unique. Instead, there are many states of the air "without any sound" which correspond to chaotic configurations of the air molecules. So one can't consistently divide the energy of the air to the energy of its ground state and the energy of the phonons.

But of course, if you produce some loud sounds, they will carry lots of energy in the air and the mass of the air will inevitably increase by $m=E/c^2$ which is, well, not too high because $c^2$ is a large number.