Acoustic metric and Planck constants

Abstract

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 $\hslash$ has dimension of time, and the second Planck constant $\hslash$ has dimension of length. It is natural to compare $\hslash$ with the Planck length $l_{\mathrm{P}}$, which is related to the Newton constant as $l_{\mathrm{P}}^2=\hslash G$. 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, $g_{\mathrm{ac}}^{\mu \nu }=\mathrm{diag}(-{\hslash }_{\mathrm{ac}}^2,{\hslash }_{\mathrm{ac}}^2,{\hslash }_{\mathrm{ac}}^2,{\hslash }_{\mathrm{ac}}^2)$. Then one obtains that the acoustic Planck constant ${\hslash }_{\mathrm{ac}}$ has dimension of length, and in liquid helium it is on the order of the interatomic distance in this liquid, ${\hslash }_{\mathrm{ac}}\sim a$. This supports the scenario in which the Planck constant in the relativistic quantum vacuum is on the order of the Planck length, $\hslash \sim l_{\mathrm{P}}\sim G$. We also use the acoustic metric for consideration of the possible dependence of the Planck constant on the Hubble parameter in expanding Universe.

arXiv comment10 pages, no figures, version accepted in JETP Letters

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