Physical Properties

Abstract

In Chap. 5 we studied solutions describing stationary counter-rotating dust disks in terms of hyperelliptic functions. As an example of this approach we gave an explicit solution on a Riemann surface of genus 2 in Theorem 5.16 where the two counter-rotating dust streams have constant angular velocity and constant relative density. In the present chapter we discuss the physical features of the class of hyperelliptic solutions (4.19) which are a subclass of Korotkin‚s .nite gap solutions [52, 94] for the example of this disk. We demonstrate how one can extract physically interesting quantities from the hyperelliptic functions in terms of which the metric is given. The metric depends on two physical parameters: ϶ = zR/(1 + zR) is related to the redshift zR of photons emitted from the center of the disk and detected at infinity; γ is the relative density of the counter-rotating streams in the disk. In the Newtonian limit e is approximately 0 whereas it tends to 1 inthe ultrarelativistic limit where the central redshift diverges. The limit of a single component disk is reached for γ = 1 (we will only consider positive values of γ), the static limit for γ = 0.