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