Vaidya solution and its generalization in de Rham-Gabadadze-Tolley massive gravity

We present a detailed study of the Vaidya solution and its generalization in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored with the St"{u}ckelberg fields ${\varphi }^a$ introduced, there is a new invariant $I^{ab}=g^{\mu \nu }{\mathrm{\partial }}_{\mu }{\varphi }^a{\mathrm{\partial }}_{\nu }{\varphi }^b$ in the massive gravity, which adds to the ones usually encountered in general relativity. There is no conventional Vaidya solution if we choose unitary gauge. In this paper, we obtain three types of self-consistent ansatz with some nonunitary gauge, and find accordingly the Vaidya, generalized Vaidya and furry Vaidya solution. As by-products, we obtain a series of furry black hole. The Vaidya solution and its generalization in dRGT massive gravity describe the black holes with a variable horizon.