Bohmian Mechanics at Space-time Singularities. I. Timelike Singularities

We develop an extension of Bohmian mechanics by defining Bohm-like tra-jectories for (one or more) quantum particles in a curved background space-time containing a singularity. Part one, the present paper, focuses on timelike singu-larities, part two will be devoted to spacelike singularities. We use the timelike singularity of the (super-critical) Reissner–Nordström geometry as an example. While one could impose boundary conditions at the singularity that would prevent the particles from falling into the singularity, in the case we are interested in here particles have positive probability to hit the singularity and get annihilated. The wish for reversibility, equivariance and the Markov property then dictate that particles must also be created by the singularity, and indeed dictate the rate at which this must occur. That is, a stochastic law prescribes what comes out of the singularity. We specify explicit model equations, involving a boundary condition on the wave function at the singularity, which is applicable also to other versions of quantum theory besides Bohmian mechanics.