Abstract
We investigate the properties of static and axisymmetric vacuum solutions of Einstein equations which generalize the Schwarzschild spherically symmetric solution to include a quadrupole parameter. We test all the solutions with respect to elementary and asymptotic flatness and curvature regularity. Analysing their multipole structure, according to the relativistic invariant Geroch definition, we show that all of them are equivalent up to the level of the quadrupole. We conclude that the q-metric, a variant of the Zipoy–Voorhees metric, is the simplest generalization of the Schwarzschild metric, containing a quadrupole parameter.
Highlights
Most applications of Einstein’s gravity theory follow from the investigation of exact solutions of the corresponding field equations
We analysed all the exact solutions of Einstein’s vacuum field equations which contain the Schwarzschild solution as a particular case and, in addition, possess an arbitrary parameter which determines the quadrupole of the gravitational source
We found that in general there are three types of naked singularities which are located at the origin of coordinates r = 0, between the origin and the Schwarzschild horizon r = m(1 ± cos θ) and on the horizon r = 2m, where m is the mass of the gravitational source
Summary
Most applications of Einstein’s gravity theory follow from the investigation of exact solutions of the corresponding field equations. In the case of relativistic astrophysics, asymptotically flat solutions in empty space are of particular importance in order to describe the physical properties of the exterior field of compact objects [1]. An elegant method to derive explicit expressions for the multipole 2 moments of a given stationary and axially symmetric space–time with asymptotic flatness was found by Fodor et al [7] using the Ernst formalism. This FHP method was generalized by Hoenselaers & Perjés [8].
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