An analysis of characteristics in nonlinear massive gravity

We study the Cauchy problem in a special case of nonlinear massive gravity. Despite being ghost free, it has recently been argued that the theory is inherently problematic due to the existence of superluminal shock waves. Furthermore, it is claimed that an acausal characteristic can arise for any choice of background. In order to further understand the causal structure of the theory, we carefully perform a detailed analysis of the characteristic equations and show that the theory does admit a well-posed Cauchy problem, i.e., there exists hypersurfaces that are not a characteristic hypersurface. Puzzles remain regarding the existence of a superluminal propagating mode in both the minimal ghost-free theory that we analyzed, as well as in the full nonlinear massive gravity. That is, our result should not be taken as any indication of the healthiness of the theory. We also give a detailed review of Cauchy-Kovalevskaya theorem and its application in the appendix, which should be useful for investigating causal structures of other theories of gravity.