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 $γ$. 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 $θ=π/2$. This, in turn, suggests the distortion parameter (the $ZV$ parameter) must be in the interval $1/2<γ<1$. 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 $γ$, 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 $γ$ on the gravitational redshift is also displayed.