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Spherical gravitational waves and quasi-spherical waves scattered from black string in massive gravity

Spherical gravitational wave is strictly forbidden in vacuum space in frame of general relativity by the Birkhoff theorem. We prove that spherical gravitational waves do exist in non-linear massive gravity, and find the exact solution. Further more, we find exact gravitational wave solution with a singular string by meticulous studies of familiar equation, in which the horizon becomes non-compact. We analyze the properties of the congruence of graviton rays of these wave solution. We clarify subtle points of dispersion relation, velocity and mass of graviton in massive gravity with linear perturbations. We find that the graviton ray can be null in massive gravity by considering full back reaction of the massive gravitational waves to the metric. We demonstrate that massive gravity has deep and fundamental discrepancy from general relativity, for whatever a tiny mass of the graviton.