Critical gravity in the Chern-Simons modified gravity

We perform the perturbation analysis of the Chern-Simons modified gravity around the AdS(4) spacetime (its curvature radius l) to obtain the critical gravity. In general, we could not obtain an explicit form of perturbed Einstein equation which shows a massive graviton propagation clearly, but for the Kerr-Schild perturbation and Chern-Simons coupling 0 = kx/y, we find the AdS wave as a single massive solution to the perturbed Einstein equation. Its mass squared is given by M-2 = [-9 + (2l(2)/k -1)(2)]/4l(2). At the critical point of M-2 = 0 (k = l(2)/2), the solution takes the log-form and the linearized excitation energies vanish.