STATIONARY STRINGS AND PRINCIPAL KILLING TRIADS IN 2-GRAVITY(1)

A new tool for the investigation of (2 + 1)-dimensional gravity is pro posed. It is shown that in a stationary (2 + 1)-dimensional space-time , the eigenvectors of the covariant derivative of the timelike Killing vector form a rigid structure, the principal Killing triad. Two of th e triad vectors are null, and in many respects they play the role simi lar to the principal null directions in the algebraically special 4D s pace-times. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary 2 + 1 spa ce-times. One of the most interesting applications is a study of minim al surfaces in a stationary space-time. A principal Killing surface is defined as a surface formed by Killing trajectories passing through a null ray, which is tangent to one of the null vectors of the principa l Killing triad. We prove that a principal Killing surface is minimal if and only if the corresponding null vector is geodesic. Furthermore, we prove that if the (2 + 1)-dimensional space-time contains a static limit, then the only regular stationary timelike minimal 2-surfaces t hat cross the static limit, are the minimal principal Killing surfaces . A timelike minimal surface is a solution to the Nambu-Goto equations of motion and hence it describes a cosmic string configuration. A sta tionary string interacting with a (2 + 1)-dimensional rotating black h ole is discussed in detail.