## Saturday, August 18, 2018

### Phonons and negative mass

There has been quite a bit of media attention given to this paper, which looks at whether sound waves involve the transport of mass (and therefore whether they should interact with gravitational fields and produce gravitational fields of their own).

The authors conclude that, under certain circumstances, sound wavepackets (phonons, in the limit where we really think about quantized excitations) rise in a downward-directed gravitational field.  Considered as a distinct object, such a wavepacket has some property, the amount of "invariant mass" that it transports as it propagates along, that turns out to be negative.

Now, most people familiar with the physics of conventional sound would say, hang on, how do sound waves in some medium transport any mass at all?  That is, we think of ordinary sound in a gas like air as pressure waves, with compressions and rarefactions, regions of alternating enhanced and decreased density (and pressure).  In the limit of small amplitudes (the "linear regime"), we can consider the density variations in the wave to be mathematically small, meaning that we can use the parameter $\delta \rho/rho_{0}$ as a small perturbation, where $\rho_{0}$ is the average density and $\delta \rho$ is the change.  Linear regime sound usually doesn't transport mass.  The same is true for sound in the linear regime in a conventional liquid or a solid.

In the paper, the authors do an analysis where they find that the mass transported by sound is proportional with a negative sign to $dc_{\mathrm{s}}/dP$, how the speed of sound $c_{\mathrm{s}}$ changes with pressure for that medium.  (Note that for an ideal gas, $c_{\mathrm{s}} = \sqrt{\gamma k_{\mathrm{B}}T/m}$, where $\gamma$ is the ratio of heat capacities at constant pressure and volume, $m$ is the mass of a gas molecule, and $T$ is the temperature.  There is no explicit pressure dependence, and sound is "massless" in that case.)

I admit that I don't follow all the details, but it seems to me that the authors have found that for a nonlinear medium such that $dc_{\mathrm{s}}/dP > 0$, sound wavepackets have a bit less mass than the average density of the surrounding medium.  That means that they experience buoyancy (they "fall up" in a downward-directed gravitational field), and exert an effectively negative gravitational potential compared to their background medium.  It's a neat result, and I can see where there could be circumstances where it might be important (e.g. sound waves in neutron stars, where the density is very high and you could imagine astrophysical consequences).  That being said, perhaps someone in the comments can explain why this is being portrayed as so surprising - I may be missing something. Anonymous said...

Astro folks are trying to have fun again... Anonymous said...

Douglas Natelson said...

Anon@7:35, love it. When I was a kid, like 10, I saw some of these when they would broadcast his segments on PBS in Pittsburgh. They looked really old even then, but they were still fun. Josh W said...

Coming to this a little later, I think it's probably because visualising the mass of waves is something that tends to go along with quantum intuition, and sound is something that everyone already understands, so you know, it's a cool overlap of those two ideas.

Also, I was looking at phonons in crystals under the usual approach using sin functions, and they also seem to have negative effective masses for non-zero energy; based on finding the inverse of the second derivative matrix of the dispersion relation.

I did a back of envelope calculation, and it came out something like (planck's constant)/((Debey Velocity)/(2*k) *(lattice constant)*(delta_i,j - *k_i*k_j /K^2))

for small deviations from zero,

but the more important thing is that looking at the dispersion relation curve, it has no minima, only maxima, so with the exception of 0 energy phonons, it will always have a negative curvature and a slightly negative mass.

I'm still not sure what status effective mass has as a physical thing, but if it does, that would match up at least.