Three dimensional extreme black hole with self (anti-self) dual Maxwell field

The most general charged version of the spinning BTZ black hole solution assuming an (anti-) self dual condition imposed on the orthonormal basis components of the electric and magnetic fields is obtained. It is found that the one-parameter Kamata-Koikawa field is a particular solution of our general solution. A spinning magnetic solution is also found. It is shown that a duality mapping exists among spinning solutions obtained from electrostatic and magnetostatic fields with the help of the local transformation t→t−ωθ, θ→θ−ωt.