B\"{a}cklund Transformations of Einstein's Field Equations for the Interior of a Uniformly Rotating Stationary Axisymmetric Perfect Fluid

TL;DR

This paper investigates the application of Clairin's method to Einstein's field equations for rotating perfect fluids, revealing it only produces non-trivial transformations under specific conditions related to pressure and density.

Contribution

It demonstrates that Clairin's method yields non-trivial Backlund transformations only when the fluid's pressure and density satisfy a particular relation, linking to Ehlers' transformation.

Findings

No non-trivial Backlund transformations for arbitrary pressure and density.

Non-trivial transformations occur when  + 3p = 0, corresponding to Ehlers' transformation.

Clarifies conditions under which Clairin's method applies to Einstein's equations.

Abstract

Clairin's method of obtaining B\"{a}cklund transformations is applied to Einstein's field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid. It is shown that for arbitrary pressure $p$ and mass density $\mu$ the method does not give non-trivial B\"{a}cklund transformations, while if $\mu + 3p =0$ it gives the transformation of Ehlers.