Acoustic metric and Planck constants

Based on Akama-Diakonov (AK) theory of emergent tetrads, it was suggested\cite{Volovik2023b} that one can introduce two Planck constants, which are the parameters of the corresponding components of Minkowski metric. In the AK theory, the interval $ds$ is dimensionless, as a result the metric elements and thus the Planck constants have nonzero dimensions. The Planck constant $\hbar$ has dimension of time, and the second Planck constant $\hslash$ has dimension of length. It is natural to compare $\hslash$ with the Planck length $l_{\rm P}$, which is related to the Newton constant as $l_{\rm P}^2= \hslash G$. However, this connection remains an open question, because the microscopic (trans-Planckian) physics of the quantum vacuum is not known. Here we study this question using the effective gravity emerging for sound wave quanta (phonons) in superfluid Bose liquid, such as $^4$He, where the microscopic physics is known: it is atomic physics. The elements of the effective acoustic metric are determined by the parameters of this Bose liquid, and as in the AK theory, the interval $ds$ is dimensionless. One may introduce the effective "acoustic Planck constants" as elements of acoustic metric, $g^{\mu\nu}_{\rm ac}= {\rm diag}(-\hbar_{\rm ac}^2,\hslash_{\rm ac}^2,\hslash_{\rm ac}^2,\hslash_{\rm ac}^2)$. Then one obtains that the acoustic Planck constant $\hslash_{\rm ac}$ has dimension of length, and in liquid helium it is on the order of the interatomic distance in this liquid, $\hslash_{\rm ac} \sim a$. This supports the scenario in which the Planck constant in the relativistic quantum vacuum is on the order of the Planck length, $\hslash\sim l_{\rm P} \sim G$. We also use the acoustic metric for consideration of the possible dependence of the Planck constant on the Hubble parameter in expanding Universe.