The Chevreton Tensor and Einstein-Maxwell Spacetimes Conformal to Einstein Spaces

TL;DR

This paper characterizes Einstein-Maxwell spacetimes with a trace-free Chevreton tensor, exploring their conformal properties, and provides new explicit examples, linking tensor properties to spacetime conformality and Petrov types.

Contribution

It establishes the equivalence between a trace-free Chevreton tensor and pure-radiation type, and identifies conditions for conformality to Einstein spaces, including new explicit solutions.

Findings

Chevreton tensor is of pure-radiation type in these spacetimes

Spacetimes are restricted to Petrov types N or O

New explicit Einstein-Maxwell solutions conformal to Einstein spaces

Abstract

In this paper we characterize the source-free Einstein-Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure-radiation type and that it restricts the spacetimes to Petrov types \textbf{N} or \textbf{O}. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein-Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic field and the Weyl tensor must be aligned, and in the case that the electromagnetic field is null, the spacetime must be conformally Ricci-flat and all such solutions are known. In the non-null case, since the general solution is not known on closed form, we settle with giving the integrability conditions in the general case, but we do give new explicit examples of Einstein-Maxwell spacetimes that are conformal to Einstein spaces, and we also find examples where the vanishing of the Bach tensor does not imply that the spacetime is conformal to a $C$-space. The non-aligned Einstein-Maxwell spacetimes with vanishing Bach tensor are conformally $C$-spaces, but none of them are conformal to Einstein spaces.