Extremal rotating BTZ black holes cannot be dressed in (anti-)self-dual Maxwell field

Under the (anti-)self-dual condition for orthonormal components of the Faraday tensor, the 3D Einstein-Maxwell system with a negative cosmological constant $\mathrm{\Lambda }$ admits a solution obtained by Kamata and Koikawa and later by Cataldo and Salgado in the most general form. Actually, Clement first obtained this solution and interpreted it as a regular particle-like solution without horizon. Nevertheless, it has been erroneously stated in some literature that this Clement-Cataldo-Salgado (CCS) solution, locally characterized by a single parameter, describes a black hole even in the charged case as it reduces to the extremal rotating Banados-Teitelboim-Zanelli (BTZ) solution in the vacuum limit and its curvature invariants are constant. In this paper, we supplement Clement’s interpretation by showing that there appears a parallelly propagated curvature singularity corresponding to an infinite affine parameter along spacelike geodesics at the location of the Killing horizon in the extremal rotating BTZ solution when the (anti-)self-dual Maxwell field is added. If the spatial coordinate $\theta$ is periodic, closed timelike curves exist near the singularity. It is also shown that the CCS solution is of the Cotton type N (in contrast to charged rotating BTZ black holes which are of type I away from the horizon), and the energy-momentum tensor of the Maxwell field is of the Hawking-Ellis type II. The metric solves the Einstein-$\mathrm{\Lambda }$ equations also with a massless scalar field or a null dust fluid. We demonstrate that it belongs to the Kundt shear-free, non-twisting, non-expanding class of geometries, whereas extremal rotating BTZ black holes have expanding principal null directions. It means that the CCS metric represents the specific null (“radiative”) Maxwell field generated by a singular source, rather than an extremal rotating BTZ black hole dressed in an (anti-)self-dual Maxwell field.