Charged rotating BTZ solution revisited: New coordinates and algebraic classifications

Abstract

We revisit the charged rotating Ba\~nados-Teitelboim-Zanelli (BTZ) solution in the three-dimensional Einstein-Maxwell-$\Lambda$ system. After the erroneous announcement of its discovery at the end of the original BTZ paper in 1992, the solution was first obtained by Cl\'ement in the paper published in 1996 by coordinate transformations from the charged non-rotating BTZ solution. While Cl\'ement's form of the solution is valid only for ${\Lambda<0}$, we present a new form for a wider range of $\Lambda$ by uniform scaling transformations and a reparametrization. We also introduce new coordinates corresponding to the Doran coordinates in the Kerr spacetime, in which the metric and also its inverse are regular at the Killing horizon, and described by elementary functions. Lastly, we show that (i) the algebraic Cotton type of the spacetime is type III on the Killing horizon and type I away from the horizon, and (ii) the energy-momentum tensor for the Maxwell field is of the Hawking-Ellis type I everywhere.