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

We use the description of geometries in terms of the Riemann tensor and a finite number of its covariant derivatives in order to find locally rotationally symmetric (LRS) perfect-fluid solutions to Einstein's 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 (LRS class I), one with twist in the preferred spacelike direction (LRS class III), and the spacetime homogeneous models. It is also shown that there are no LRS spacetimes with dependence on one null coordinate. Using an extension of the method, we find the full metric in terms of curvature quantities for LRS class I and LRS class III.