Asymptotic charges of a quadrupolar naked singularity

TL;DR

This paper computes various asymptotic charges, including BMS and NP constants, for the Zipoy-Voorhees spacetime, revealing that algebraically special conditions are crucial for NP constants to vanish.

Contribution

It provides explicit calculations of NP constants for a quadrupolar naked singularity, challenging previous assumptions about their vanishing in certain spacetimes.

Findings

NP constants of the q-metric do not vanish, contrary to previous conjectures.

The algebraically special condition is essential for NP constants to be zero.

The q-metric serves as a counterexample to the conjecture about vanishing NP constants.

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.