The charged Zipoy–Voorhees metric with astrophysical applications

Abstract

Starting from an integral of the interaction region of colliding Einstein–Maxwell waves and by applying a coordinate transformation, we obtain the charged version of the static Zipoy–Voorhees (ZV) metric valid for all values of the distortion parameter $$\gamma $$ γ . In Schwarzschild coordinates, we investigate the effect of the charge in the newly found spacetime, stress the analogy with Reissner–Nordstrom metric and discuss some of its features. It is shown that from the expression of Weyl curvature, directional singularities become manifest. For astrophysical importance, we find lensing of null geodesics from the Gauss–Bonnet theorem in such non-spherically charged objects. To prepare the ground for our null, circular geodesics we consider the angular equation linearized about the symmetry plane $$\theta =\pi /2$$ θ = π / 2 . This, in turn, suggests the distortion parameter (the ZV parameter) must be in the interval $$1/2<\gamma <1$$ 1 / 2 < γ < 1 . It is found that the lensing angle is highly dependent on the distortion parameter, and becomes decisive on the effect of the charge. For a class of charged compact stars, we plot the deflection angle versus the ratio of impact parameter to the radius of the star. Plots have revealed that for perfectly spherical compact stars, it is hard to identify the effect of electric/magnetic charge, but for non-spherical compact stars the effect of electric charge becomes apparent. For comparison, the same lensing angle has also been found for the stationary ZV metric in the equatorial plane. Our analysis indicates that depending on the value of $$\gamma $$ γ , the stationary state induces counter effect on the bending angle and thus, when compared with the uncharged static ZV case, the bending angle decreases. The influence of the parameter $$\gamma $$ γ on the gravitational redshift is also displayed.

Geodesics and gravitational waves in chaotic extreme-mass-ratio inspirals: the curious case of Zipoy-Voorhees black-hole mimickers

“…In this study, we will investigate the simplest deformation of Schwarzschild geometry, the Zipoy-Voorhees (ZV) metric [ 102 , 103 ] (also known as the -metric, that has also been generalized in [ 104 , 105 ] to include electric charge), that describes a vacuum, static, and spheroidal solution of Einstein equations which is continuously connected to Schwarzschild by a deformation parameter (in our convention). The ZV metric, presenting a spheroidal deformation of an otherwise static geometry can pose as a good model for a static compact object surrounded by a compact environment or an accretion disk, such as those residing in galactic centers.…”

Quasi–Hyperbolically Symmetric γ-Metric

“…In other words, the γ-metric is an appealing candidate to describe spacetimes close to Schwarzschild, by means of exact analytical solutions to Einstein vacuum equations. This of course is of utmost relevance and explains why it has been so extensively studied in the past (see [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] and the references therein).…”

Quasi-Hyperbolically Symmetric γ-Metric

“…In other words, the -metric is an appealing candidate to describe spacetimes close to Schwarzschild, by means of exact analytical solutions to Einstein vacuum equations. This of course is of utmost relevance and explains why it has been so extensively studied in the past (see [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 ] and the references therein).…”