  There does not exist a notion of time which could be transferred
straightforwardly from classical to quantum gravity. For this reason, a method
of time quantification which would be appropriate for gravity quantization is
being sought. One of the existing proposals is using the evolving matter as an
intrinsic `clock' while investigating the dynamics of gravitational systems.
The objective of our research was to check whether scalar fields can serve as
time variables during a dynamical evolution of a coupled multi-component
matter-geometry system. We concentrated on a neutral case, which means that the
elaborated system was not charged electrically nor magnetically. For this
purpose, we investigated a gravitational collapse of a self-interacting complex
and real scalar fields in the Brans-Dicke theory using the 2+2 spacetime
foliation. We focused mainly on the region of high curvature appearing nearby
the emerging singularity, which is essential from the perspective of quantum
gravity. We investigated several formulations of the theory for various values
of the Brans-Dicke coupling constant and the coupling between the Brans-Dicke
field and the matter sector of the theory. The obtained results indicated that
the evolving scalar fields can be treated as time variables in close proximity
of the singularity due to the following reasons. The constancy hypersurfaces of
the Brans-Dicke field are spacelike in the vicinity of the singularity apart
from the case, in which the equation of motion of the field reduces to the wave
equation due to a specific choice of free evolution parameters. The
hypersurfaces of constant complex and real scalar fields are spacelike in the
regions nearby the singularities formed during the examined process. The values
of the field functions change monotonically in the areas, in which the
constancy hypersurfaces are spacelike.
