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The classical Einstein--Standard Model system with conformally invariantcoupling of the Higgs field to gravity is investigated. We show that theenergy-momentum tensor is not polynomial in the Higgs field, and hence it mayhave two singularities: In cosmological spacetimes the usual Big Bang typesingularity with diverging matter field variables, and a second, less violentone (Small Bang), in which it is only the geometry that is singular but thematter field variables remain finite. In generic spacetimes, the latterprovides a finite, universal upper bound for the pointwise norm of the Higgsfield in terms of Newton's gravitational constant.  We also show that, in the presence of Friedman--Robertson--Walker orKantowski--Sachs symmetries, the energy density can have finite local minimumonly if the transitivity hypersurfaces of the spacetime symmetries are locallyhyperboloidal and their mean curvature is less than a finite critical value. Inparticular, in the very early era of an expanding universe or in a nearlyspherically symmetric black hole near the central singularity, the Higgs sectordoes not have any instantaneous vacuum state, and hence its rest mass is notdefined, and, via the Brout--Englert--Higgs (BEH) mechanism, the gauge andspinor fields do not get non-zero rest mass. For smaller mean curvatureinstantaneous symmetry breaking vacuum states of the Higgs sector emerge,yielding non-zero rest mass and electric charge for some of the gauge andspinor fields via the BEH mechanism. These rest masses are decreasing withdecreasing mean curvature, but the charge remains constant. It is also shownthat globally defined instantaneous vacuum states that are invariant withrespect to the spacetime symmetries do not exist at all in the $k=1,0$cosmological models and in Kantowski--Sachs spacetimes (e.g. inside sphericallysymmetric black holes).

Gravity, as a classical regulator for the Higgs field, and the origin of rest masses and electric charge