Four-function generalization and separable structures of the Plebanski spacetime with sources

TL;DR

This work presents a four-function generalization of the Plebański spacetime, introducing three radial functions $Q(q)$, $\Xi(q)$, $\Φ(q)$ and one angular function $P(p)$, while recovering the Plebański metric in the appropriate limit. It establishes Hamilton–Jacobi and Klein–Gordon separability for the generalized metric, derives uncharged and charged test-particle trajectories via conserved quantities, and identifies Killing horizons with explicit surface-gravity expressions. A conformal factor is shown to generically break separability, unless restrictive conditions are met, and conformal Killing tensors replace Killing tensors under conformal rescaling. The paper also proposes scalar and nonlinear-electromagnetic sources that can source static generalized Plebański metrics and satisfies the dominant energy condition in several examples, illustrating a wide class of physically viable spacetimes and highlighting avenues for further coupling between matter fields and geometry.

Abstract

We determine a four-function generalization of the Plebanski spacetime, depending on three arbitrary functions of the radial coordinate, and one function on the angular coordinate. For the generalized Plebanski spacetime, we analyze the separability of the Hamilton--Jacobi equations, and the trajectories of charged test particles are derived from the motion constants. The Klein-Gordon equation separability is established and the Killing horizons are presented as well. Then we introduce a conformal factor to the Plebanski metric and discuss the conditions that preserve the separability. Finally we show a possible stress--energy tensor that may be the source of some of the generalized metrics.