S ep 2 00 1 A stationary vacuum solution dual to the Kerr solution

We present a stationary axially symmetric two parameter vacuum solution which could be considered as " dual " to the Kerr solution. It is obtained by removing the mass parameter from the function of the radial coordinate and introducing a dimensionless parameter in the function of the angle coordinate in the metric functions. It turns out that it is in fact the massless limit of the Kerr-NUT solution. The axially symmetric stationary spacetime hosts the most interesting two parameter Kerr family of vacuum black hole solution of the Einstein equation. It is well known that the family is unique under the assumptions of asymptotic flatness and of existence of regular smooth horizon. The Kerr solution has turned out to be the most interesting solution of the Einstein equation for the astrophysical applications, particularly in the context of high energy sources like quasars, pulsars, gamma ray bursters and active galactic nuclei. The two parameters in the solution represent mass and specific angular momentum of the hole. When rotation is switched off, the solution reduces to the static Schwarzschild black hole. In the limit of vanishing mass it reduces to flat space. That is, the rotation by itself cannot be a source of gravity. This is however quite understandable in the Newtonian framework. The question is, should that always be the case in general relativity as well? In this paper, we wish to address this question and would like to present a solution in which mass parameter is replaced by a dimensionless parameter appropriately. The solution so obtained would in a particular sense be " dual " to the Kerr solution. The form of the axially symmetric stationary metric we choose is motivated by the physical considerations of separability of the Hamilton-Jacobi equation for particle motion and the Klein-Gordon equation. That is, we seek the solution of the Einstein vacuum equation in the metric form in which these equations characterizing the physical features of the solution are solvable. For this, we follow the method employed by one of