Pleban\'ski-Demia\'nski \`a la Ehlers-Harrison: Exact Rotating and Accelerating Type I Black Holes

TL;DR

This paper constructs a new exact solution describing a rotating, accelerating black hole with NUT charge and electric charge, extending the Plebański-Demiański family through Ehlers and Harrison transformations, and clarifies the Ernst formalism for such solutions.

Contribution

It introduces the 'Enhanced Plebański-Demiański' spacetime, a novel exact solution obtained via combined transformations, and provides a detailed pedagogical guide to the Ernst formalism.

Findings

Derived the exact form of the new black hole solution.

Explicitly constructed a rotating, accelerating black hole with NUT and electric charges.

Clarified the Ernst formalism with a step-by-step guide.

Abstract

Recently, it was shown that type D black holes, encompassed in the large Pleban\'ski--Demia\'nski (PD) family, exhibit a wide class of algebraically general generalizations via the application of Ehlers and Harrison transformations. In this work, we first discuss some mathematical details behind the composition of such transformations, and next, we introduce a qualitative picture of the most general type I generalization of the PD family, dubbed ``Enhanced Pleban\'ski--Demia\'nski'' spacetime. We provide the exact form of the solution in the original PD coordinates, obtained via the simultaneous action of an Ehlers and a Harrison transformation on the vacuum PD geometry. In order to make the physics more transparent, we explicitly construct a rotating and accelerating black hole which further has NUT parameter and electric charges, both of them entering, not only the event horizon, but the Rindler horizon as well. This solution is directly obtained in the ``physical'' coordinates recently proposed by Podolsk\'y and Vr\'atny. Finally, a pedagogical appendix is thoughtfully included, providing readers with a user-friendly step-by-step guide to the Ernst formalism, in an attempt to address and resolve various minor inconsistencies frequently appearing in the relevant literature.