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

TL;DR

This paper clarifies that extremal rotating BTZ black holes cannot be dressed with (anti-)self-dual Maxwell fields, showing the solutions are actually radiative fields with singular sources rather than true black holes.

Contribution

It demonstrates that solutions previously thought to be extremal rotating BTZ black holes with Maxwell fields are actually radiative spacetimes with singularities, not genuine black holes.

Findings

Presence of a curvature singularity at the horizon in extremal solutions.

CCS solutions are of Cotton type N, not black hole type I.

Solutions describe radiative Maxwell fields, not dressed black holes.

Abstract

Under the (anti-)self-dual condition for orthonormal components of the Faraday tensor, the 3D Einstein-Maxwell system with a negative cosmological constant $\Lambda$ 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-$\Lambda$ 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.