Is the type-D NUT C-metric really "missing" from the most general Pleba\'nski-Demia\'nski solution?

Abstract

It remains a long-standing problem, unsettled for almost two decades in the general relativity community, ever since Griffiths and Podolsky demonstrated in Ref. [J.B. Griffiths and J. Podolsky, Class. Quant. Grav. 22, 3467 (2005)] that the type-D NUT C-metric seems to be absent from the most general family of the type-D Pleba\'nski-Demia\'nski (P-D) solution. However, Astorino [Phys. Rev. D 109, 084038 (2024)] presented a different form of rotating and accelerating black holes and showed that all known four-dimensional type-D accelerating black holes (without the NUT charge) can be recovered via various different limits in a definitive fashion. In particular, he provided, for the first time, the correct expressions for the type-D static accelerating black holes with a nonzero NUT charge, which was previously impossible using the traditional parametrization of the familiar P-D solution. Nevertheless, it still remains elusive how these two different forms of the four-dimensional rotating and accelerating solutions are related. In this paper, we aim to fill this gap by finding the obvious coordinate transformations and parameter identifications between the vacuum metrics after two different parametrizations of the generated solution via the inverse scattering method from the seed metric -- the Rindler vacuum background. We then resolve this ``missing" puzzle by providing another M\"{o}bius transformation and linear combinations of the Killing coordinates, which clearly cast the type-D NUT C-metric into the familiar form of the P-D solution. Additionally, we propose an alternative new routine for the normalization of the obtained metric derived via the inverse scattering method from the vacuum seed solution, which could be potentially useful for the construction of higher-dimensional solutions using the trivial vacuum background as the seed metric.