Static Einstein-Maxwell solutions in 2+1 dimensions.

We obtain the Einstein-Maxwell equations for (2+1)-dimensional static space-time, which are invariant under the transformation q0 = i q2, q2 = i q0, α ⇀↽ γ. It is shown that the magnetic solution obtained with the help of the procedure used in Ref. [6], can be obtained from the static BTZ solution using an appropriate transformation. Superpositions of a perfect fluid and an electric or a magnetic field are separately studied and their corresponding solutions found. It is well known that in (2+1)-dimensional space-time, the metric around a point mass is given by ds = dt − dr2 (1− kM 2π ) 2 − rdφ. (1) This space-time is analogous to the Schwarzschild metric around a point mass in (3+1)-dimensions. The metric (1) corresponds to a flat space-time [3]. Another situation in (2+1) dimensions is to consider the cosmological constant Λ. In this case, outside sources, the exterior gravitational fields are spaces of constant curvature: De Sitter for Λ > 0 and anti-De Sitter for Λ < 0 [1]. The (2+1) local electromagnetic field is given by (“cutting out” one of the spatial dimensions ) F = 1 2 Fab dx a ∧ dx = E1 dx 1 ∧ dx + (2) E2 dx 2 ∧ dx + B dx ∧ dx. Thus the electromagnetic tensor has only three independent components [2, 3], two for the vector electric field and one for the scalar magnetic field ( ~ B to be a vector needs the missing dimension). gr-qc/9605049 e-mail: [email protected] e-mail: [email protected]