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We discuss the effective metric experienced by the Nambu-Goldstone modepropagating in the broken symmetry spin-superfluid state of coherent precessionof magnetization. This collective mode represents the phonon in the RF drivenor pulsed out-of-equilibrium Bose-Einstein condensate (BEC) of optical magnons.We derive the effective BEC free energy and consider the phonon spectrum whenthe spin superfluid BEC is formed in the anisotropic polar phase of superfluid3He, experimentally observed in uniaxial aerogel 3He-samples. The coherentprecession of magnetization experiences an instability at a critical value ofthe tilting angle of external magnetic field with respect to the anisotropyaxis. From the action of quadratic deviations around equilibrium, thisinstability is interpreted as a Minkowski-to-Euclidean signature change of theeffective phonon metric. We also note the similarity between the magnon BEC inthe unstable region and an effective vacuum scalar "ghost" condensate.

Effective Minkowski to Euclidean signature change of the magnon BEC pseudo-Goldstone mode in polar 3He