The Hybrid Euclidean–Lorentzian Universe: Stability Considerations and the Point of Transition

The Lorentzian structure of space-time limits velocities, preventing the universe from achieving the thermal equilibrium suggested by the cosmic microwave background (CMB). However, if one follows Hawking’s idea of an initial Euclidean metric—where no such velocity limit exists—thermalization become possible without invoking an arbitrary new field. A mathematical framework describing how today’s Lorentzian space-time could evolve from an early Euclidean phase has been proposed, resolving issues like the initial singularity and CMB uniformity. Still, the model’s stability has not yet been examined, nor has a mathematical condition for the Euclidean–Lorentzian transition been established; these aspects are partially addressed in this work. We show that the collapse of the Euclidean sector is an inevitable result of matter moving into thermodynamical equilibrium.