Relativistic Static Thin Disks: The Counter-Rotating Model

TL;DR

This paper analyzes the Counter-Rotating Model for static thin disks with radial pressure, deriving conditions for counter-rotating velocities, and exploring stability and energy conditions in specific disk configurations.

Contribution

It provides a general constraint on counter-rotating velocities and examines the physical viability of CRM in static thin disks with nonzero radial pressure.

Findings

Counter-rotating velocities generally do not follow geodesics.

Some CRM configurations are stable against radial perturbations.

Disks satisfy strong energy condition but violate weak energy condition in some regions.

Abstract

A detailed study of the Counter-Rotating Model (CRM) for generic finite static axially symmetric thin disks with nonzero radial pressure is presented. We find a general constraint over the counter-rotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counter-rotating perfect fluids. We also found expressions for the energy density and pressure of the counter-rotating fluids. Then we shown that, in general, there is not possible to take the two counter-rotating fluids as circulating along geodesics neither take the two counter-rotating tangential velocities as equal and opposite. An specific example is studied where we obtain some CRM with well defined counter-rotating tangential velocities and stable against radial perturbations. The CRM obtained are in agree with the strong energy condition, but there are regions of the disks with negative energy density, in violation of the weak energy condition.