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A fun calculation of the maximum speed of sound

Let us spend some time by temporarily abandoning fears that Trump won't decisively beat the loons and domestic terrorists on November 3rd, and from our peaking Covid-19 infection in Czechia (which is almost reaching the level of epidemics in the first district, so for the first time, I have at least a "moderate" understanding for the online schools and some other changes, hopefully a very short-lived policy; I think it's great that Czechia avoided excessive preventive policies and only regulates the virus when it's arguably defensible). I could write lots of stuff about various news reported on my Twitter account but at the end, I think that most of these topics aren't deep enough to deserve a full-blown, extensive, penetrating TRF blog post. Or I am rather exhausted by the news about insanity from so many directions that is overwhelming us.

Willie sent me a fun article that just appeared Science Advances,

Speed of sound from fundamental physical constants
by Trachenko et al. The preprint is from April 2020 but I obviously don't systematically follow all cond-mat preprints.

What is the maximum speed of sound you may have? Well, a relativist or fundamental physicist will immediately say that the maximum speed of anything is \(v=c\), the speed of light in the vacuum. That's the ultimate limit. In principle, a generic enough vacuum of string theory (or at least an effective field theory, an extra swampland constraint could exist) may contain "materials" where the speed of sound approaches the speed of light very closely. At least that's what we normally think; I will throw some doubts on this statement at the end.

Note that in gases, the speed of sound \(v\) obeys\[ v^2 = \frac{\partial p}{\partial \rho}, \] it is the square root of the partial derivative of the pressure with respect to the density. For \(p = \rho c^2 \), which is an extreme equation state that is very far from the "dust" that we can normally hold in our hands but that is allowed by relativity (and the "energy conditions"), you get \(v=c\). However, that's a hypothetical limit that can't be realized with common materials, of course. These condensed matter authors use the Debye model some extreme scenario and bounds on the phonon energy and binding energy and conclude that the maximum speed of sound in a condensed material is given by\[ \frac{v_{\rm max}^2}{c^2} = \alpha \cdot \sqrt{\frac{m_e}{2 m_p}}. \] That is equal to \(1/137.036\times \sqrt{1/(2\cdot 1836.15)}\approx 0.00012\) which is, in normal units, 36 km/s or so. Note that e.g. the speed of sound in iron is 5.12 km/s so their bound is still a factor of 7 higher than iron. But it's not "uselessly" far. Their bound is supposed to be saturated for some condensed hydrogen, whatever it is.

Condensed matter physics is often analogous to fundamental physics, both find the Renormalization Group helpful, the effective quantum field theories may be similar, and so on. But by definition, condensed matter physics is always derived or emergent while by definition, the fundamental physics should be fundamental and non-emergent in the same sense. Some condensed matter physicists who find the truth (about the non-fundamental character of condensed matter physics) inconvenient have been trying to obscure these basic defining facts. They only highlighted that when overstated, their field becomes the squalid state physics, as Murray Gell-Mann called it.

These comments are partly a lore of high-energy physics, something that is not (and cannot be) proven to be totally accurate and relevant in all situations, however. So I am often afraid of (or enthusiastic about?) the scenario that lots of condensed matter physics may be needed to understand the fundamental laws, too. In particular, as I suggested above, there may exist a swampland-like principle that places a nontrivial upper bound (much lower than the speed of light in the vacuum which follows from special relativity) on the speed of sound in condensed phases. Using the result above, such a swampland criterion could be an inequality on a function of the mass ratios and fine-structure constants that exist in every stringy vacuum. Maybe in all string vacua the speed of sound in a condensed phase must be smaller than some particular constant that is strictly smaller than one? Can you disprove this hypothesis?

Note that the particular formula above assumes some "atomic" composition of the condensed matter. You may arguably surpass the speed of sound above in degenerate matter such as white dwarfs and neutron stars but there may still exist a similar, very universal and nontrivial limit that applies those materials, too. Of course, this whole research project could be ill-defined because in the most general environment, it could be impossible to strictly and invariantly separate phonons (and therefore sound) from any boson (a particle; or the corresponding general signal that may spread in a material).

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