Does a black hole rotate in Chern-Simons modified gravity?

Abstract

Rotating black hole solutions in the $(3+1)$-dimensional Chern-Simons modified gravity theory are discussed by taking account of perturbation around the Schwarzschild solution. The zenith-angle dependence of a metric function related to the frame-dragging effect is determined from a constraint equation independently of a choice of the embedding coordinate. We find that at least within the framework of the first-order perturbation method, the black hole cannot rotate for finite black hole mass if the embedding coordinate is taken to be a timelike vector. However, the rotation can be permitted in the limit of $M/r\ensuremath{\rightarrow}0$ (where $M$ is the black hole mass and $r$ is the radius). For a spacelike vector, the rotation can also be permitted for any value of the black hole mass.