3+1 description of silent universes: a uniqueness result for the Petrov type I vacuum case

TL;DR

This paper proves that vacuum silent universes of Petrov type I are necessarily spatially homogeneous Bianchi I models, using a 3+1 decomposition and algebraic computing, and characterizes the Kasner metric invariantly.

Contribution

It provides a proof of the uniqueness conjecture for vacuum silent universes of Petrov type I using a 3+1 approach and algebraic computation, confirming the models are Bianchi I.

Findings

The initial data for silent universes is non-contorted and embeddable in conformally flat spacetime.

The conjecture holds in the vacuum case, supporting its validity in the general case.

The Kasner metric is characterized invariantly via the Weyl tensor.

Abstract

Silent universes are studied using a ``3+1'' decomposition of the field equations in order to make progress in proving a recent conjecture that the only silent universes of Petrov type I are spatially homogeneous Bianchi I models. The infinite set of constraints are written in a geometrically clear form as an infinite set of Codacci tensors on the initial hypersurface. In particular, we show that the initial data set for silent universes is ``non-contorted'' and therefore (Beig and Szabados, 1997) isometrically embeddable in a conformally flat spacetime. We prove, by making use of algebraic computing programs, that the conjecture holds in the simpler case when the spacetime is vacuum. This result points to confirming the validity of the conjecture in the general case. Moreover, it provides an invariant characterization of the Kasner metric directly in terms of the Weyl tensor. A physical interpretation of this uniqueness result is briefly discussed.