New sources for Kerr and other metrics: rotating relativistic disks with pressure support

Abstract

Complete sequences of new analytic solutions of Einstein's equations which describe thin super massive disks are constructed. These solutions are derived geometrically. The identification of points across two symmetrical cuts through a vacuum solution of Einstein's equations defines the gradient discontinuity from which the properties of the disk can be deduced. The subset of possible cuts which lead to physical solutions is presented. At large distances, all these disks become Newtonian, but in their central regions they exhibit relativistic features such as velocities close that of light, and large redshifts. Sections with zero extrinsic curvature yield cold disks. Curved sections may induce disks which are stable against radial instability. The general counter rotating flat disk with planar pressure tensor is found. Owing to gravomagnetic forces, there is no systematic method of constructing vacuum stationary fields for which the non-diagonal component of the metric is a free parameter. However, all static vacuum solutions may be extended to fully stationary fields via simple algebraic transformations. Such disks can generate a great variety of different metrics including Kerr's metric with any ratio of a to m. A simple inversion formula is given which yields all distribution functions compatible with the characteristics of the flow, providing formally a complete description of the stellar dynamics of flattened relativistic disks. It is illustrated for the Kerr disk.