^{1,2,3}

^{1}

^{1}

^{3}.

We show that the commonly accepted statement that sound waves do not transport mass is only true at linear order. Using effective field theory techniques, we confirm the result found by Nicolis and Penco [

It is usually said that sound waves do not transport mass. They carry momentum and energy, but it is an accepted fact that the net mass transported by a sound wave vanishes. Here, we question this “fact.”

A first indication that sound waves can carry a nonzero net mass is contained in the results of Ref.

Now this effect is completely equivalent to standard refraction: in the presence of gravity, the pressure of the superfluid depends on depth, and so does the speed of sound. As a result, in the geometric acoustics limit, sound waves do not propagate along straight lines. Because of this, one might be tempted to dismiss any interpretation of this phenomenon in terms of “gravitational mass.” However, since, in the formalism of Ref.

Thus, in a very physical sense, the phonon carries (negative) mass. Moreover, this is not due to the usual equivalence of mass and energy in relativity: the effect survives in the nonrelativistic limit. And, finally, it is not a quantum effect, because the formalism of Ref.

In this Letter, we confirm this result by computing explicitly the mass carried by a classical sound wave packet. As we will see, from the wave mechanics standpoint, the fact that such a mass is nonzero is a nonlinear effect, and that is why, from a linearized analysis, we usually infer that sound waves do not carry mass. We also generalize the result to sound waves in ordinary fluids and to longitudinal and transverse sound waves in solids. We find that, in the nonrelativistic limit, the mass carried by a sound wave traveling in these media is its energy

A final qualification: for excitations propagating in a Poincaré-invariant vacuum, the invariant mass is a fundamental, completely unambiguous quantity, whose value also directly determines the gap. Not so in a medium that breaks boost invariance: there, the dispersion relation

For technical convenience, we use gravity to probe how much mass our excitations carry, since in the nonrelativistic limit gravity only couples to mass.

Conventions: We set

As a general framework, we use the recently developed effective field theories (EFT) for the gapless excitations of generic media. These can be characterized in terms of spontaneous symmetry breaking. In particular, all media break at least part of the fundamental spacetime symmetries of nature, which are spacetime translations, spatial rotations, and Lorentz boosts (the Poincaré group)

Here we compute the mass carried by a sound wave packet. For each of the media considered, we use the corresponding effective action for gapless excitations to derive the equations of motion by varying the action with respect to the fields, and the energy-momentum tensor

Finally, we keep a relativistic notation and take the nonrelativistic limit only when needed. It should be noted that the same results can also be derived imposing Galilean invariance from the beginning. However, the relativistic analysis is not any more demanding to carry out, and it is more general. Once powers of

The low-energy dynamics of a zero-temperature superfluid can be characterized by a single scalar field

The superfluid phonon field

We can solve this equation to nonlinear order by writing

To find the energy-momentum tensor, we vary the effective action [Eq.

To find the mass transported by the sound wave, we integrate

As far as the quadratic terms in

The above result is fully relativistic. In the nonrelativistic limit, the mass carried by our wave packet is

As mentioned in the Introduction, the bending of a sound wave’s trajectory in the presence of gravity is equivalent to Snell’s law. In light of this, given the generality of Snell’s law, we expect sound waves to interact with gravity in fluids and solids much in the same way as they do in superfluids. We will again analyze phonons in a solid from an effective field theory viewpoint. Moreover, since in the EFT language a fluid is simply a solid with an enhanced symmetry (see, e.g., Refs.

The EFT for a three-dimensional solid can be built out of three scalar fields,

The low energy effective action is a generic function of the combination

A perfect fluid can be viewed as an infinitely symmetric solid

Sound waves once again correspond to the fluctuations of the fields about their backgrounds,

Let us now find the gravitational mass carried by sound waves in a solid (or fluid). It is convenient to take the nonrelativistic limit right away. The only

The volume integral reduces to a boundary term and, for large enough volumes, one might be tempted to discard it. This is certainly allowed for a linearized solution, since we can take it to be as localized as we wish. However, in general we must be careful about nonlinear corrections. Splitting as before the fluctuation field into

To compute such a contribution, we first take the divergence of Eq.

The right-hand side of Eq.

Putting together Eqs.

Longitudinal and transverse sound waves in general have different propagation speeds—in fact, one generally has

This result holds equally well for the sound waves in a perfect fluid, for which

We showed that contrary to common belief, sound waves carry gravitational mass in a standard Newtonian sense: they are affected by gravity, but they also

The mass transported is in general quite small, of order

One possibility is to employ ultracold atomic or molecular gases. In these systems, not only might one be able to achieve very small sound speeds and enhance the effect, but one could also use suitable traps to simulate strong gravitational potentials

Another possibility might be to consider seismic phenomena. The wave generated by an earthquake of Richter magnitude

The effects we are considering may also be of relevance to neutron star dynamics, since gravity would affect phonon-mediated transport properties in the superfluid stellar interior

Finally, since sound waves both source gravity and are affected by it, they can interact through gravity. In particular, for ordinary equations of state (higher sound speeds at higher pressures), their gravitational mass is negative. Still, since gravity is attractive for like charges, two sound wave packets running parallel to each other should start converging. Very, very slowly, of course.

We are grateful to N. Bigagli, G. Chiriacò, S. Garcia-Saenz, L. Hui, A. Lam, R. Mcnally, R. Penco, P. Richards, I. Z. Rothstein, C. Warner, and S. Will for illuminating discussions. This work has been partially supported by the US Department of Energy Grant No. de-sc0011941. The work done by A. E. is also supported by the Swiss National Science Foundation under Contract No. 200020-169696 and through the National Center of Competence in Research SwissMAP.