Directional Quantum Singularities in Curzon Spacetime

TL;DR

The paper investigates directional naked singularities in the Curzon spacetime through the Horowitz–Marolf quantum singularity framework using a massless Klein–Gordon field. By analyzing the self-adjointness of the spatial operator $A$ in both charged and uncharged Curzon backgrounds across asymptotic and near-singularity regimes, it finds that the uncharged case ($p=1$) yields quantum regularity at the classical singularity $r=0$, whereas the charged case with an outer singular surface at $r_\star$ remains quantum mechanically singular due to a square-integrable mode near $r_\star$ that prevents a unique self-adjoint extension. These results highlight that quantum effects can smooth directional singularities in some spacetimes but not in others, and motivate future work on higher-spin probes, non-minimal couplings, backreaction, and comparisons with other axisymmetric solutions.

Abstract

The scalar quantum probe method developed by Horowitz and Marolf is applied to the cylindrically symmetric Curzon solution. The main cause for choosing the Curzon solution is that it is the best known example that exhibits directional singularity. Interestingly the singularity at $r=0$, for the uncharged Curzon spacetime, which is classically very strong with a divergence rate of the order $\frac{1}{r^{10}}$ becomes regular when examined using scalar quantum field. The charged Curzon spacetime, however, due to the emergence of a second singularity off the $r=0$ singularity does not regularize quantum mechanically. All three different charged versions, i.e. electric, magnetic and dyonic share the same feature.