The free boundary problem in general relativity

We study the action principle for space-times whose boundary is singular. We suggest that it is natural to treat the singularity as a {\it free} boundary, where the variation is unconstrained. Demanding that the action is stationary under such free variations then implies certain (on-shell) boundary conditions at the singularity. We derive these boundary conditions for the case of Einstein gravity coupled to matter and show that, when applied to an initial spacelike singularity, they exclude Kasner-like or BKL space-times, but admit conformally regular space-times (including FLRW models) sourced by fluids satisfying $0\le P<\rho$. For standard hot big bang FLRW cosmologies, the admissible linear (scalar, vector, tensor) perturbations satisfy reflecting boundary conditions at the bang, in agreement with large-scale cosmological observations.