Stationary strings and principal Killing triads in 2 + 1 gravity

Abstract

A new tool for the investigation of (2 + 1)-dimensional gravity is proposed. 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 the triad vectors are null, and in many respects they play the role similar to the principal null directions in the algebraically special 4D space-times. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary 2 + 1 space-times. One of the most interesting applications is a study of minimal 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 principal 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 that 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 stationary string interacting with a (2 + 1)-dimensional rotating black hole is discussed in detail.