Rotating disk of dust

5.9 Rotating disk of dust

The stationary axisymmetric spacetime around a rigidly rotating disk of dust was first studied in terms of a numerical solution to Einstein’s field equation by Bardeen and Wagoner [18 , 19 ]. The exact solution was found much later by Neugebauer and Meinel [237 ]. It is discussed, e.g., in [236 ]. The metric cannot be written in terms of elementary functions because it involves the solution to an ultraelliptic integral equation. It depends on a parameter $μ$ which varies between zero and $μc = 4.62966 ...$ . For small $μ$ one gets the Newtonian approximation, for $μ → μc$ the extreme Kerr metric ( $a = m$ ) is approached. The lightlike geodesics in this spacetime have been studied numerically and the appearance of the disk to a distant observer has been visualized [345 ]. It would be desirable to support these numerical results with exact statements. From the known properties of the metric, only a few qualitative lensing features of the disk can be deduced. As Minkowski spacetime is approached for $μ → 0$ , the spacetime must be asymptotically simple and empty as long as $μ$ is sufficiently small. (This is true, of course, only if the disk is treated as transparent.) The general results of Section 3.4 imply that in this case the gravitational field of the disk produces finitely many images of each light source, and that the number of images is odd, provided that the worldline of the light source is past-inextendible and does not go out to past lightlike infinity. For larger values of $μ$ , this is no longer true. For $μ > 0.5$ there are two counter-rotating circular lightlike geodesics in the equatorial plane, a stable one at a radius $&tidle;ρ1$ inside the disk and an unstable one at a radius $ρ&tidle;2$ outside the disk. (This follows from [10 ] where it is shown that for $μ > 0.5$ timelike counter-rotating circular geodesics do not exist in a radius interval $[ρ&tidle;1,ρ&tidle;2]$ . The boundary values of this interval give the radii of lightlike circular geodesics.) The existence of circular light rays has the consequence that the number of images must be infinite; this is obviously true if light source and observer are exactly on the spatial track of such a circular light ray and, by continuity, also in a neighborhood. For a better understanding of lensing by the disk of dust it is desirable to investigate, for each value of $μ$ and each event $pO$ : Which past-oriented lightlike geodesics that issue from $pO$ go out to infinity and which are trapped? Also, it is desirable to study the light cones and their caustics.