Exact relativistic treatment of stationary counter-rotating dust disks III. Physical Properties

This is the third in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the physical properties of a class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. It is parameterized by two physical parameters, the central redshift and the relative density of the two counter-rotating streams in the disk. We discuss the dependence of the metric on these parameters using a combination of analytical and numerical methods. Interesting limiting cases are the Maclaurin disk in the Newtonian limit, the static limit which gives a solution of the Morgan and Morgan class and the limit of a disk without counter-rotation. We study the mass and the angular momentum of the spacetime. At the disk we discuss the energy-momentum tensor, i.e. the angular velocities of the dust streams and the energy density of the disk. The solutions have ergospheres in strongly relativistic situations. The ultrarelativistic limit of the solution in which the central redshift diverges is discussed in detail: In the case of two counter-rotating dust components in the disk, the solutions describe a disk with diverging central density but finite mass. In the case of a disk made up of one component, the exterior of the disks can be interpreted as the extreme Kerr solution.Comment: 30 pages, 20 figures; to appear in Phys. Rev.

Self-dual SU(2) invariant Einstein metrics and modular dependence of theta-functions

We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.Comment: A wrong sign in the formula for W_1 is corrected; we thank Owen Dearricott who pointed out this mistake in the original version of the pape

Szeg\"o kernel and symplectic aspects of spectral transform for extended spaces of rational matrices

We revisit the sympectic aspects of the spectral transform for matrix-valued rational functions with simple poles. We construct eigenvectors of such matrices in terms of the Szeg\"o kernel on the spectral curve. Using variational formulas for the Szeg\"o kernel we construct a new system of action-angle variables for the canonical symplectic form on the space of such functions. Comparison with previously known action-angle variables shows that the vector of Riemann constants is the gradient of some function on the moduli space of spectral curves; this function is found in the case of matrix dimension 2, when the spectral curve is hyperelliptic.Comment: 19 page

Exact relativistic treatment of stationary counter-rotating dust disks I: Boundary value problems and solutions

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin's finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter.Comment: 32 pages, 1 figure, to appear in Phys. Rev.

Differentially rotating disks of dust

We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer-Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Baecklund transformations to appropriate seed solutions of the Weyl class.Comment: 14 pages, 3 figures. To appear in "General Relativity and Gravitation". Second version with minor correction

Dirichlet Boundary Value Problems of the Ernst Equation

We demonstrate how the solution to an exterior Dirichlet boundary value problem of the axisymmetric, stationary Einstein equations can be found in terms of generalized solutions of the Backlund type. The proof that this generalization procedure is valid is given, which also proves conjectures about earlier representations of the gravitational field corresponding to rotating disks of dust in terms of Backlund type solutions.Comment: 22 pages, to appear in Phys. Rev. D, Correction of a misprint in equation (4

Minimal Models of CFT on Z_N-Surfaces

The conformal field theory on a Z_N-surface is studied by mapping it on the branched sphere. Using a coulomb gas formalism we construct the minimal models of the theory.Comment: 16 pages, latex, no figures; two important early references on the coset construction have been included; to appear in Mod. Phys. Let

Differentially rotating disks of dust: Arbitrary rotation law

In this paper, solutions to the Ernst equation are investigated that depend on two real analytic functions defined on the interval [0,1]. These solutions are introduced by a suitable limiting process of Backlund transformations applied to seed solutions of the Weyl class. It turns out that this class of solutions contains the general relativistic gravitational field of an arbitrary differentially rotating disk of dust, for which a continuous transition to some Newtonian disk exists. It will be shown how for given boundary conditions (i. e. proper surface mass density or angular velocity of the disk) the gravitational field can be approximated in terms of the above solutions. Furthermore, particular examples will be discussed, including disks with a realistic profile for the angular velocity and more exotic disks possessing two spatially separated ergoregions.Comment: 23 pages, 3 figures, submitted to 'General Relativity and Gravitation