Conical singularity in spacetimes with NUT is observer dependent

We discuss the issue of defining and measuring conical deficits (conicity) in spacetimes with the torsion singularity such as the Misner string in Taub-NUT spacetime. We propose a geometric definition that generalizes the standard notion of conicity to stationary axially symmetric spacetimes with torsion singularity, where the conical deficit becomes observer dependent---it depends on the choice of a timelike Killing vector. This implies the existence of observers who perceive no conical singularity along the symmetry axis. As a result, in any spacetime with a nonvanishing NUT parameter, there are observers for whom the conicity has the same value on both semiaxes. This challenges the usual interpretation of conicity differences as indicators of string/rod-induced acceleration along the axis. We illustrate our definition across the full Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski--Demia\ifmmode \acute{n}\else \'{n}\fi{}ski class, including the recently identified accelerated Taub-NUT solution. Our attempts in determining a canonical observer lead to even less desirable definitions of conicity.