THE ELECTRICALLY CHARGED BTZ BLACK-HOLE WITH SELF (ANTI-SELF) DUAL MAXWELL FIELD

The Einstein-Maxwell equations with a negative cosmological constant Lambda in 2 + 1 spacetime dimensions discussed by Banados, Teitelboim and Zanelli are solved by assuming a self (anti-self) dual equation E(rho) = +/-B, which is imposed on the orthonormal basis components of the electric field Ep and the magnetic field B---. This solution describes an electrically charged extreme black hole with mass M = 8 pi GQ(e)(2), angular momentum J = +/-8 pi GQ(e)(2)/\Lambda\(1/2) and electric charge Q(e)(2). Although the coordinate components of the electric field E(tau) and the magnetic field B have singularities on the horizon at r = (4 pi GQ(e)(2)/\Lambda\)(1/2), the spacetime has the same value of constant negative curvature R = 6 Lambda as that of Banados et al.