  We develop an extension of Bohmian mechanics by defining Bohm-like
trajectories for quantum particles in a curved background space-time containing
a spacelike singularity. As an example of such a metric we use the
Schwarzschild metric, which contains two spacelike singularities, one in the
past and one in the future. Since the particle world lines are everywhere
timelike or lightlike, particles can be annihilated but not created at a future
spacelike singularity, and created but not annihilated at a past spacelike
singularity. It is argued that in the presence of future (past) spacelike
singularities, there is a unique natural Bohm-like evolution law directed to
the future (past). This law differs from the one in non-singular space-times
mainly in two ways: it involves Fock space since the particle number is not
conserved, and the wave function is replaced by a density matrix. In
particular, we determine the evolution equation for the density matrix, a
pure-to-mixed evolution equation of a quasi-Lindblad form. We have to leave
open whether a curvature cut-off needs to be introduced for this equation to be
well defined.
