Conical singularity in spacetimes with NUT is observer-dependent

TL;DR

The paper addresses how to define conical deficits in spacetimes with torsion singularities, where standard conicity fails due to Misner strings. It introduces a geometric, observer-dependent definition of conicity grounded in the relation between axial and cyclic Killing vectors, and shows that conicity and a associated time-shift depend on the chosen timelike Killing vector. Through the Plebański–Demiański class, including Taub–NUT and accelerated Taub–NUT solutions, it demonstrates that nonzero NUT parameter generally yields an ineliminable, observer-dependent conicity, with explicit expressions for the two semi-axes and for several physically relevant spacetimes. The work concludes that there is no canonical observer fixing conicity in general, challenging interpretations of conicity differences as indicators of axis-accelerating sources and suggesting alternative characterizations of acceleration in these spacetimes.

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 Plebański--Demiański class, including the recently identified accelerated Taub--NUT solution. Our attempts in determining a canonical observer lead to even less desirable definitions of conicity.

Figures (3)

• Figure 1: The cyclic $\boldsymbol{c}$ (blue), axial $\boldsymbol{a}$ (green), and timelike $\boldsymbol{t}$ (red) Killing vectors are shown on the orbit of $\Gamma$. The orbit of the Killing vectors $\Gamma$ has the topology of a cylinder and the induced flat Minkowski geometry. The left picture corresponds to ${\mathcal{T} = 0}$, while the middle and right pictures depict two equivalent representations of the case ${\mathcal{T} \neq 0}$. The cylinder in the right picture has been 'untwisted' in angular direction and oriented horizontally to highlight the spacelike and timelike character of the Killing vectors.
• Figure 2: The picture shows two timelike $\boldsymbol{t}_{\textrm{I,II}}$ (magenta) Killing vectors measuring trivial conicity, which are proportional to the sum and difference of the cyclic $\boldsymbol{c}$ (blue) and axial $\boldsymbol{a}$ (green) Killing vectors via \ref{['eq:unitcon']}. Both pictures are equivalent representation of the case ${\mathcal{T} \neq 0}$. The cylinder in the right picture has been untwisted and oriented horizontally (relative to the left one) to highlight the spacelike and timelike character of the Killing vectors.
• Figure 3: The orbit of $\Gamma$ covered by coordinates $\tau$ (blue) and $\varphi$ (red). The left picture corresponds to ${\mathcal{T} = 0}$, while the middle and right pictures depict two equivalent representations of the case ${\mathcal{T} \neq 0}$. The left picture shows the circle of circumference $L_{\circ}$ for ${\mathcal{T}=0}$, while the middle and right pictures show the segment of length $L$ for ${\mathcal{T} \neq 0}$ (cyan); these are used in the conicity formulas \ref{['eq:conicity-standard']} and \ref{['eq:extcon']}, respectively. Going from ${\mathcal{T} = 0}$ to ${\mathcal{T} \neq 0}$, we clearly see the necessity of introducing an observer ${\boldsymbol{t}}$, corresponding to the lines of constant ${\varphi}$, which defines the notion of "same spatial position". This replaces the integration along the circle with integration along the spiral segment (straightened in the representation on the right).