Relativistic discs by Appell-ring convolutions
We propose a novel approach to generate the gravitational field of thin discs within the Weyl class of static and axially symmetric spacetimes. Similar to the Newtonian potential theory, such a gravitational field is described by a solution to the Laplace equation. In general relativity, however, there is also another metric function present which may deviate from the Newtonian picture. We show, how to obtain various analytic thin disc solutions of the Laplace equation by convolving a certain weight function – an Abel transformation of the physical surface density profile – with the Appell-ring potential. Due to the rather simple form of the Appell potential, we also explicitly derive the second metric function in some cases. Moreover, we focus on discs empty in their central region and "superpose" them with the Schwarzchild black hole – resulting in an elementary, but fully general relativistic model of an accretion disc around a compact body. While the superposition problem is simple (linear) for the potential, the nonlinearity of Einstein equations is manifested in the second metric function. Again, in particular cases, we derive the nonlinear part, thus obtaining the complete metric of the entire superposition in closed form.