We revisit the charged rotating Bañados–Teitelboim–Zanelli (BTZ) solution in the three-dimensional Einstein–Maxwell-Λ system. After the erroneous announcement of its discovery at the end of the original BTZ paper in 1992, the solution was first obtained by Clément in the paper published in 1996 by coordinate transformations from the charged non-rotating BTZ solution. While Clément’s form of the solution is valid only for Λ<0 , we present a new form for a wider range of Λ 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.

中文翻译：

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重新审视带电旋转 BTZ 解决方案：新坐标和代数分类

我们重新审视三维 Einstein-Maxwell-Λ 系统中的带电旋转 Bañados-Teitelboim-Zanelli (BTZ) 解。在 1992 年原始 BTZ 论文末尾错误地宣布其发现后，Clément 在 1996 年发表的论文中首次通过带电非旋转 BTZ 解的坐标变换获得了该解。虽然克莱门特的解决方案形式仅适用于 Λ<0 ，我们通过统一缩放变换和重新参数化提出了一种更广泛的 Λ 的新形式。我们还引入了与克尔时空中的多兰坐标相对应的新坐标，其中度量及其逆在杀伤视界上是规则的，并由初等函数描述。最后，我们证明 (i) 时空的代数 Cotton 类型在杀戮视界上为 III 型，远离视界为 I 型，并且 (ii) 麦克斯韦场的能量动量张量为霍金-埃利斯型我无处不在。