Uniqueness of Petrov type D spatially inhomogeneous irrotational silent models

TL;DR

This paper proves that spatially inhomogeneous irrotational silent models of Petrov type D are uniquely the Bianchi type I models, and explores conditions under which more general silent solutions can exist.

Contribution

It provides a direct proof that SIIS models of Petrov type D are spatially homogeneous Bianchi type I models, and discusses how relaxing certain conditions can lead to broader solutions.

Findings

SIIS Petrov type D models are spatially homogeneous Bianchi type I.

The Szekeres solutions are the most general SIIS models.

Non-zero pressure does not affect the uniqueness of Petrov type D models.

Abstract

The consistency of the constraint with the evolution equations for spatially inhomogeneous and irrotational silent (SIIS) models of Petrov type I, demands that the former are preserved along the timelike congruence represented by the velocity of the dust fluid, leading to \emph{new} non-trivial constraints. This fact has been used to conjecture that the resulting models correspond to the spatially homogeneous (SH) models of Bianchi type I, at least for the case where the cosmological constant vanish. By exploiting the full set of the constraint equations as expressed in the 1+3 covariant formalism and using elements from the theory of the spacelike congruences, we provide a direct and simple proof of this conjecture for vacuum and dust fluid models, which shows that the Szekeres family of solutions represents the most general class of SIIS models. The suggested procedure also shows that, the uniqueness of the SIIS of the Petrov type D is not, in general, affected by the presence of a non-zero pressure fluid. Therefore, in order to allow a broader class of Petrov type I solutions apart from the SH models of Bianchi type I, one should consider more general ``silent'' configurations by relaxing the vanishing of the vorticity and the magnetic part of the Weyl tensor but maintaining their ``silence'' properties i.e. the vanishing of the curls of $E_{ab},H_{ab}$ and the pressure $p$.