On the fate of spacetime singularities

TL;DR

The paper tackles spacetime singularities in quantum cosmology by promoting the observer's proper time to a clock within the Wheeler-DeWitt framework, yielding a Schrödinger equation along the worldline and a mini-superspace reduction near the singularity. The near-singularity dynamics reduce to a radial problem with an effective potential $V_{ m eff}(r) = - Q/r^2 - b/r^{2 w}$, where the anisotropy yields a negative inverse-square term controlled by the conserved anisotropy charge $Q$. Unitarity of evolution imposes a critical threshold $Q_{ m crit} = α^2/8$ (or equivalently γ < 1/4 in dimensionless units); surpassing this threshold leads to nonunique self-adjoint extensions and potential breakdown of unitary evolution, implying that a quantum bounce would require near-isotropy. The work also notes that including additional scalar fields generalizes the problem to higher effective dimensions and may motivate holographic perspectives on the interior dynamics of spacetime singularities.

Abstract

I investigate spacetime singularities from the point of view of the wavefunction of the universe. In order to extend the classical notion of geodesic incompleteness one has to include the proper time of an observer as a degree of freedom in the Wheeler DeWitt equation. This leads to a Schrödinger equation along the observer worldline. Near the singularity, as in the classical BLK treatment, I ignore spatial gradients and effectively describe the spacetime around the worldline in the mini-superspace approximation. Then the problem proves identical to a spherically symmetric scattering of a quantum particle off a central potential and singularity avoidance is tantamount to unitary evolution for this system. Standard types of matter (dust, radiation) correspond to regular potentials and thus lead to a bounce. The most singular component, spatial anisotropy, is associated to a conserved charge and yields a negative inverse-square potential -- like standard angular momentum, but with opposite sign. This potential is critical, in that the unitarity of the evolution depends on the actual numerical factor in front of it, i.e., on the anisotropy charge.