Algebraically special Einstein–Maxwell fields

TL;DR

The paper re-analyses algebraically special non-null Einstein–Maxwell fields using the Geroch–Held–Penrose formalism, covering aligned and non-aligned cases with a possible non-vanishing cosmological constant. It gives new invariant characterizations of the García–Plebański, Plebański–Hacyan, and Griffiths metrics and corrects/generalizes a no-go claim of Kozarzewski.

Key findings

• In the aligned shearing case, the paper proves Kozarzewski’s no-go claim for \(\Lambda=0\) and generalizes it: from \(R=4\Lambda=-4|\Phi_1|^2\), no solutions occur for \(\Lambda\ge 0\), while for \(\Lambda<0\) the only allowed solutions are the doubly aligned García–Plebański metrics, with only non-zero Weyl spinor component \(\Psi_2=R/6\).
• In the aligned non-geodesic case, the Plebański–Hacyan metrics exhaust the non-diverging subfamily \(\rho=0\) and have \(\Lambda<0\); for \(\Lambda\ge 0\) no solutions exist.
• For non-aligned fields with a shear-free and geodesic multiple Weyl-PND not parallel to a PND of \(F\), the paper shows the cosmological constant must vanish; under the additional restriction \(\pi=0\), the unique solutions are the Griffiths metrics.
• As a corollary, for non-vanishing \(\Lambda\), the double alignment of Debever–McLenaghan “class D” metrics follows from their multiple Weyl-PNDs being geodesic and shear-free.

Methods

• Uses the Geroch–Held–Penrose (GHP) formalism, with tetrad \(k,\ell,m,\bar m\), sign conventions and notation of Kramer §7.4, and primed variables written as NP equivalents \(-\nu,-\lambda,-\mu,-\pi\).
• Re-derives and applies GHP, Bianchi, and Maxwell equations (presented in an Appendix) to classify algebraically special non-null Einstein–Maxwell fields, aligned and non-aligned, with possible non-zero cosmological constant.
• All calculations were done using the Maple symbolic algebra system.

Limitations

• The paper states that the general solution in the aligned non-geodesic family is not known: “So far it has not been possible to complete the analysis of this case.”
• It states that whether the class D double-alignment result persists when \(\Lambda=0\) is still open, with only the Kundt case (\(\rho=0\)) dealt with so far.
• It notes that the \(\pi=0\) class may not admit a larger set of solutions, but “no explicit examples appear to be known”; the proof for the Kundt case is postponed to a possible later publication.