Aligned electric and magnetic Weyl fields (Q1768428)

In this paper, the authors discuss the concepts of electric and magnetic Weyl fields; denoted by \(E(v)\) and \(B(v)\), respectively, and defined in terms of a unit (timelike or spacelike) vector field \(v\) and the Weyl tensor \(C\) (the authors use \(W\)) by \[ E(v)_{ab}= C_{acbd}v^cv^d,\quad B(v)_{ab}=C_{acbd}^*v^cv^d, \] where \(C^*\) is the dual to \(C\). The paper discusses the situation when \(E\) and \(B\) are proportional (the aligned case) and seeks the possibility of alignment for each of the Petrov types for \(C\), and how \(v\) would then be related to the principal null directions of \(C\) and the members of the canonical Petrov tetrad(s) for \(C\). In particular, they show every algebraically special space-time possesses such an ``alignment'' vector field \(v\). The Petrov type I case (i.e. the algebraically general case) is discussed in some detail and a condition under which such an alignment vector field is admitted, is found. In the final section of the paper the authors introduce the concept of ``homothetic'' electric and magnetic fields (where the alignment is rather more special).