A stationary vacuum solution of Einstein’s field equations

We investigate a stationary generalization of the static metric. The static -metric is a variant of the Zipoy-Voorhees metric and simplest generalization of the Schwarzschild metric, containing a quadrupole parameter. In the present work, we introduce the stationary version of the -metric, and this stationary metric find by using the complex Ernst potential . The metric function determined by two first-order differential equations that can be integrated by quadratures once  is known. To obtain an explicit form of the new Ernst potential, we use the solution generation techniques that allows us to generate stationary solutions from a static solution. It possesses three independent parameters related to the mass, quadrupole moment and angular momentum. We investigate the geometric and physical properties of this exact stationary solution of Einstein’s vacuum equations and show that it can be used to describe the exterior gravitational field of rotating, axially symmetric, compact objects. According to the relativistic invariant Geroch definition, we analyze multipole structure using the corresponding Ernst function and we compute the lowest ten relativistic multipole moments for the static quadrupole metric. The particular choice of parameters we obtain the known solutions. i.e., the exterior Schwarzschild Solution find with the vanishing quadrupole and rotating parameter, Corresponding static -metric find with the vanishing rotating parameter and non-zero quadrupole parameter. The multipole moments of the well-known Kerr solution are given by the vanishing quadrupole parameter and non-zero rotating parameter.