An Analysis of Characteristics in Non-Linear Massive Gravity

We study the Cauchy problem in a special case of non-linear massive gravity: the two-tensor "f-g" theory. 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 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 exist hypersurfaces that are not characteristic hypersurface. Puzzles remain regarding the existence of a superluminal propagating mode in both the f-g theory, as well as in the full non-linear 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.