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General Relativity and Quantum Cosmology, gr-qc
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.
