Stationary Strings and Principal Killing Triads in 2+1 Gravity

TL;DR

This paper introduces the principal Killing triad as a new tool for classifying stationary 2+1 dimensional spacetimes, and explores its applications to minimal surfaces and cosmic string configurations.

Contribution

It defines the principal Killing triad in stationary 2+1 spacetimes and demonstrates its utility for classification, minimal surface analysis, and cosmic string modeling.

Findings

Principal Killing triad forms a rigid structure in stationary 2+1 spacetimes.

A principal Killing surface is minimal iff the null vector is geodesic.

Regular stationary minimal surfaces crossing static limits are principal Killing surfaces.

Abstract

A new tool for the investigation of 2+1 dimensional gravity is proposed. It is shown that in a stationary 2+1 dimensional spacetime, the eigenvectors of the covariant derivative of the timelike Killing vector form a rigid structure, the {\it 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 4-D spacetimes. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary 2+1 spacetimes. One of the most interesting applications is a study of minimal surfaces in a stationary spacetime. A {\it 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 spacetime 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.