Anti-Newtonian universes do not exist

TL;DR

This paper proves that irrotational dust solutions with vanishing electric Weyl tensor are only FLRW spacetimes, extending previous results to include non-zero cosmological constant and highlighting differences with magnetic Weyl cases.

Contribution

It rigorously confirms the conjecture that only FLRW solutions exist under these conditions, regardless of the cosmological constant.

Findings

Irrotational dust solutions with zero electric Weyl tensor are FLRW.

The proof holds for both zero and non-zero cosmological constant.

Differences are identified between electric and magnetic Weyl curvature cases.

Abstract

In a paper by Maartens, Lesame and Ellis (Class. Quant. Grav. 15, 1005) it was shown that irrotational dust solutions with vanishing electric part of the Weyl tensor are subject to severe integrability conditions and it was conjectured that the only such solutions are FLRW spacetimes. In their analysis the possibility of a cosmological constant Lambda was omitted. The conjecture is proved, irrespective as to whether Lambda is zero or not, and qualitative differences with the case of vanishing magnetic Weyl curvature are pointed out.