Exact General Relativistic Thick Disks

TL;DR

This paper introduces a new method for constructing exact general relativistic thick disks by extending the classical 'displace, cut and reflect' approach, applicable to various metrics, ensuring physically acceptable properties.

Contribution

It generalizes the 'displace, cut and reflect' method to include a 'fill' step, enabling the construction of physically consistent relativistic thick disks from known solutions.

Findings

Constructed thick disks using Weyl gamma, Chazy-Curzon, and Schwarzschild metrics.

All models satisfy energy conditions within certain parameter ranges.

Method produces physically acceptable thick disk solutions in general relativity.

Abstract

A method to construct exact general relativistic thick disks that is a simple generalization of the ``displace, cut and reflect'' method commonly used in Newtonian, as well as, in Einstein theory of gravitation is presented. This generalization consists in the addition of a new step in the above mentioned method. The new method can be pictured as a ``displace, cut, {\it fill} and reflect'' method. In the Newtonian case, the method is illustrated in some detail with the Kuzmin-Toomre disk. We obtain a thick disk with acceptable physical properties. In the relativistic case two solutions of the Weyl equations, the Weyl gamma metric (also known as Zipoy-Voorhees metric) and the Chazy-Curzon metric are used to construct thick disks. Also the Schwarzschild metric in isotropic coordinates is employed to construct another family of thick disks. In all the considered cases we have non trivial ranges of the involved parameter that yield thick disks in which all the energy conditions are satisfied.