Newman-Penrose formalism and exact vacuum solutions to conformal Weyl gravity

We study the exterior solution for a static, spherically symmetric source in Weyl conformal gravity in terms of the Newman–Penrose formalism. We first show that both the static, uncharged black hole solution of Mannheim and Kazanas and the static, charged Reissner–Nordström-like solution can be found more easily in this formalism than in the traditional coordinate-basis approach, where the metric tensor components are taken as the basic variables. Second, we show that the Newman-Penrose formalism offers a particularly convenient framework that is well suited for the discussion of conformal gravity solutions corresponding to Petrov “type-D” spacetimes. This is illustrated with a two-parameter class of wormhole solutions that includes the Ellis–Bronnikov wormhole solution of Einstein’s gravity as a limiting case. Other salient issues, such as the gauge equivalence of solutions and the inclusion of the electromagnetic field are also discussed.