Asymptotic charges of a quadrupolar naked singularity
Edgar Gasperin, Mariem Magdy
TL;DR
This work computes the asymptotic charges of the Zipoy–Voorhees ($q$-metric) vacuum spacetime by constructing its Newman–Penrose gauge and NP coordinates and deriving the required asymptotic expansions of the tetrad and Weyl scalars. It then evaluates the Bondi–Sachs energy–momentum, the NP constants, and the BMS charges, finding a nonzero NP constant despite the spacetime being asymptotically algebraically special. The results demonstrate that the algebraically special condition in the Wu–Shao theorem is essential, since the Zipoy–Voorhees metric provides a counterexample to relaxing this condition, and they illuminate how quadrupole content and naked singularities influence asymptotic charges. The methods and findings have implications for the study of asymptotic symmetries and higher-spin charges in spacetimes beyond the classic Kerr–type solutions.
Abstract
The purpose of this article is to compute the asymptotic charges of a vacuum solution to the Einstein field equations describing a naked singularity with a non-vanishing quadrupole moment, known in the literature as the Zipoy-Voorhees spacetime (q-metric). In addition to the well-known asymptotic quantities such as the Bondi-Sachs energy-momentum, the BMS charges and NP constants of this spacetime are computed. Explicit calculations of the latter are relatively scarce in the literature. Moreover, it has been proven that the NP constants of asymptotically flat, stationary, vacuum, and algebraically special spacetimes vanish (for instance, those of the Kerr spacetime). A by-product of the present analysis is to show that the algebraically special condition in the aforementioned result appears to be crucial, since the q-metric provides a counterexample to the conjecture that all asymptotically flat, stationary, vacuum, and asymptotically algebraically special spacetimes (a weaker version of the algebraically special condition) have vanishing NP constants.