Exact Relativistic Gravitational Field of a Stationary Counterrotating Dust Disk

Abstract

We present a solution to the Ernst equation which represents an infinitesimally thin dust disk consisting of two streams of particles circulating with constant angular velocity in opposite directions. These streams have the same density distribution but their relative density may vary continuously. In the limit of only one component of dust, we get the solution for the rigidly rotating dust disk of Neugebauer and Meinel; in the limit of identical densities, the static disk of Morgan and Morgan is obtained. We discuss the Newtonian and the ultrarelativistic limit, the occurrence of ergospheres, and the regularity of the solution.PACS numbers: 04.20.JbIn astrophysics thin disks of collisionless matter, socalled dust, are discussed as models for certain galaxies (see, e.g., [1]) or for accretion disks. A fully relativistic treatment of these models is necessary if a black hole is present since black holes are genuinely relativistic objects. But the exact treatment of dust disks without a central object would provide deep insight both in the mathematical structure of the field equations and in the physics of rapidly rotating relativistic bodies, since dust disks can be viewed as a limiting case for extended matter sources. Corresponding exact solutions hold globallyin the vacuum and in the matter region-and can thus provide physically realistic test beds for numerical codes. Since Newtonian dust disks are known to be unstable and since there are hints by numerical work (see, e.g., [2]) that the same holds in the relativistic case, such solutions could be taken as exact initial data for numerical collapse calculations. Whereas the Newtonian theory of such disks is well established (see [1], and references therein), the same holds in the relativistic case only for static disks which can be interpreted as consisting of two counterrotating streams of matter with vanishing total angular momentum. The first disk of this type was considered by Morgan and Morgan [3]. Infinitely extended dust disks with finite mass were studied by Bičák, and Katz [4] in the static case and by Bičák and Ledvinka [5] in the stationary case. The first to construct the exact solution for a finite stationary dust disk were Neugebauer and Meinel [6] who gave the solution for the rigidly rotating dust disk which was first treated numerically by Bardeen and Wagoner [2]. They solved the corresponding boundary value problem for the Einstein equations with the help of a corotating coordinate system.In this Letter we present a class of new disk solutions to the Ernst equation where such a coordinate system cannot be used. The disks of finite radius r 0 consist of two counterrotating components of dust with respective density s 6 ͑r͒. The angular velocity of both streams of particles is of the same constant absolute value V but of different sign. The relative density g ͑s 1 2 s 2 ͒͑͞s 1 1 s 2 ͒ is a constant with respect to r and z which varies between one, the rigidly rotating dust disk [6], and zero, the static Morgan and Morgan disk [3]. Interesting...

Electrovacuum static counterrotating relativistic dust disks

“…The physical properties of the matter distribution are then studied by an analysis of the surface energy-momentum tensor so obtained. On the other hand, a "direct problem" approach, called by Synge the "Tmethod", is also used by other authors [23,24,25,26,27,28,29] by taking a given surface energy-momentum tensor and solving the Einstein equations in the matter region. The inner solution is then used to obtain boundary data for the vacuum field equations in the outer region.…”

“…[23] to obtain the solution to the problem of a (one-component) uniformly rotating disk of dust, and in Refs. [24,25,26,27,28,29] to generate counterrotating dust disks, but no condition is imposed there about the (electro-) geodesic motion of the two counterrotating streams.…”

Ernst equation, Fay identities and variational formulas on hyperelliptic curves

“…The complete metric and the Ernst potential can be given explicitly in terms of theta functions. This explicit form free of derivatives of the metric made it possible in [12] to solve a boundary value problem for a relativistic dust disc in terms of a thetafunctional Ernst potential. The description of the dust discs requires partial degeneration of the curve L and subsequent "condensation" of the double points, as was done in [14].…”

Time-Independent Gravitational Fields

“…(This would also follow from (3.33) and the standard formula for the behaviour of R ij under conformal rescalings.) As for the potential, rewrite (3.26) as 35) and observe that the operator in (3.35) obeys…”