Induced Einstein-Kalb-Ramond theory and the black hole

A four-dimensional induced Einstein-Kalb-Ramond theory with a conformally coupled Kalb-Ramond term is discussed. It is argued that the Kalb-Ramond field is difficult to interpret as an axion field from a dynamical point of view, and hence tends to be unstable. One can, however, apply Routh’s method to extract an effective action for the axion field. It is found that the Kalb-Ramond hair will turn the event horizon into a naked singularity. Such an exact solution is found and analyzed carefully.