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Topologically protected superfluid phases of $^3$He allow one to simulatemany important aspects of relativistic quantum field theories and quantumgravity in condensed matter. Here we discuss a topological Lifshitz transitionof the effective quantum vacuum in which the determinant of the tetrad fieldchanges sign through a crossing to a vacuum state with a degenerate fermionicmetric. Such a transition is realized in polar distorted superfluid $^3$He-A interms of the effective tetrad fields emerging in the vicinity of the superfluidgap nodes: the tetrads of the Weyl points in the chiral A-phase of $^3$He andthe degenerate tetrad in the vicinity of a Dirac nodal line in the polar phaseof $^3$He. The continuous phase transition from the $A$-phase to the polarphase, i.e. in the transition from the Weyl nodes to the Dirac nodal line andback, allows one to follow the behavior of the fermionic and bosonic effectiveactions when the sign of the tetrad determinant changes, and the effectivechiral space-time transforms to anti-chiral "anti-spacetime". This condensedmatter realization demonstrates that while the original fermionic action isanalytic across the transition, the effective action for the orbital degrees offreedom (pseudo-EM) fields and gravity have non-analytic behavior. Inparticular, the action for the pseudo-EM field in the vacuum with Weyl fermions(A-phase) contains the modulus of the tetrad determinant. In the vacuum withthe degenerate metric (polar phase) the nodal line is effectively a family of$2+1$d Dirac fermion patches, which leads to a non-analytic $(B^2-E^2)^{3/4}$QED action in the vicinity of the Dirac line.

Dimensional crossover of effective orbital dynamics in polar distorted 3He-A: Transitions to anti-spacetime