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

TL;DR

This paper develops an exact relativistic model for stationary counter-rotating dust disks using boundary value problems and Riemann surface methods, extending Newtonian techniques to Einstein's equations for astrophysical disk structures.

Contribution

It introduces explicit solutions for relativistic counter-rotating dust disks using hyperelliptic Riemann surfaces, advancing the mathematical modeling of astrophysical disks in general relativity.

Findings

Constructed explicit solutions for disks with constant angular velocity and energy density.

Analyzed genus 1 and 2 Riemann surfaces to solve boundary value problems.

Linked metric functions and multipoles to Riemann surface properties.

Abstract

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.