Acoustic Metric and Planck Constants

Abstract

Based on Akama–Diakonov (AD) theory of emergent tetrads, it was suggested that one can introduce two Planck constants, $$\hbar $$ and $$\not h $$, which are the parameters of the corresponding components of Minkowski metric, $$g_{\text{Mink}}^{\mu \nu }=\text{diag}(-{{\hbar }^{2}},{{\not h }^{2}},{{\not h }^{2}},{{\not h }^{2}})$$. In the Akama–Diakonov 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 Planck constant $$\not h $$ has dimension of length. It is natural to compare $$\not h $$ with the Planck length $${{l}_{{\text{P}}}}$$. 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, where the microscopic physics is known, and the elements of the effective acoustic metric are determined by the parameters of the Bose liquid. Since the acoustic interval is dimensionless, one may introduce the effective “acoustic Planck constants.” The acoustic Planck constant $${{\not h }_{\text{ac}}}$$ has dimension of length and is on the order of the interatomic distance. This supports the scenario in which $$\not h \sim {{l}_{\text{P}}}$$. We also use the acoustic metric for consideration of dependence of $$\hbar $$ on the Hubble parameter in expanding Universe.