An example of non-Weyl preserving complex transformation

Abstract

It has been observed on a number of occasions that complex transformations, of real solutions of the field equations to other real solutions, often preserve certain properties of the Weyl tensor. That is, the Petrov type and/or gravito-electromagnetic (GEM) properties of the Weyl tensor are preserved. In this context, we present an outstanding example of a complex windmill transformation of a static (non-physical) anisotropic fluid spacetime of Petrov type $$I(M^+)$$ that maps to a purely magnetic (PM) spacetime of Petrov type $$I(M^{\infty })$$ . The PM spacetime is analyzed and compared to the Arianrhod–Lun–McIntosh–Perjes spacetime. It is shown that these spacetimes, although similar in some aspects, are distinct solutions. The main distinction is that the generated PM spacetime satisfies all the standard energy-conditions. This intriguing but purely mathematical scenario may have implications in the area of GEM duality.