Invariant construction of solutions to Einstein`s field equations - LRS perfect fluids I

TL;DR

This paper introduces a new method for constructing solutions to Einstein's field equations for LRS perfect fluid space-times using curvature tensors, enabling coordinate choices at any calculation stage.

Contribution

A novel method utilizing Riemann tensor derivatives for solving Einstein's equations in LRS perfect fluids, with explicit metric constructions for certain classes.

Findings

No LRS space-times depend solely on one null coordinate.

Full metrics derived for classes with fluid rotation and spatial twist.

Method simplifies solution process by flexible coordinate selection.

Abstract

The properties of some locally rotationally symmetric (LRS) perfect fluid space-times are examined in order to demonstrate the usage of the description of geometries in terms of the Riemann tensor and a finite number of its covariant derivatives for finding solutions to Einstein's field equations. A new method is introduced, which makes it possible to choose the coordinates at any stage of the calculations. Three classes are examined, one with fluid rotation, one with spatial twist in the preferred direction and the space-time homogeneous models. It is also shown that there are no LRS space-times with dependence on one null coordinate. Using an extension of the method, we find the full metric in terms of curvature quantities for the first two classes.