Invariant construction of solutions to Einstein's field equations - LRS perfect fluids II

TL;DR

This paper develops a geometric method to construct solutions to Einstein's field equations for LRS class II perfect fluids, including new non-stationary, inhomogeneous, and self-similar solutions, extending previous spherical symmetry results.

Contribution

It introduces an invariant geometric approach to derive and analyze solutions for LRS class II perfect fluids, including cases beyond spherical symmetry.

Findings

Derived new non-stationary, inhomogeneous solutions with shear, expansion, and acceleration.

Presented self-similar solutions with specific equations of state.

Demonstrated the limitations of Riemann tensor and Ricci coefficients alone in describing certain geometries.

Abstract

The properties of LRS class II perfect fluid space-times are analyzed using the description of geometries in terms of the Riemann tensor and a finite number of its covariant derivatives. In this manner it is straightforward to obtain the plane and hyperbolic analogues to the spherical symmetric case. For spherically symmetric static models the set of equations is reduced to the Tolman-Oppenheimer-Volkoff equation only. Some new non-stationary and inhomogeneous solutions with shear, expansion, and acceleration of the fluid are presented. Among these are a class of temporally self-similar solutions with equation of state given by $p=(\gamma-1)\mu, 1<\gamma<2$, and a class of solutions characterized by $\sigma=-\Theta/6$. We give an example of geometry where the Riemann tensor and the Ricci rotation coefficients are not sufficient to give a complete description of the geometry. Using an extension of the method, we find the full metric in terms of curvature quantities.