Conical singularity in spacetimes with NUT is observer-dependent

TL;DR

The paper introduces an observer-dependent geometric definition of conical deficits in spacetimes with NUT charge, revealing that conicity perception varies with the observer's choice of timelike Killing vector.

Contribution

It generalizes the notion of conicity to stationary axially symmetric spacetimes with torsion singularities, showing observer dependence and implications for interpreting conical deficits.

Findings

Conicity depends on the observer's choice of timelike Killing vector.

Some observers perceive no conical singularity along the axis.

The definition applies to the full Plebański–Demiański class, including accelerated Taub–NUT solutions.

Abstract

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 non-vanishing NUT parameter, there are observers for whom the conicity has the same value on both semi-axes. 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\'nski--Demia\'nski class, including the recently identified accelerated Taub--NUT solution. Our attempts in determining a canonical observer lead to even less desirable definitions of conicity.