Abstract
General Relativity(GR) has a great success to explain some astronomy problem such as universal expansion, Mercury precession, and gravitional lensing. The recent observation of grvitational wave from black hole converge also gives a strong evidence of GR. In the view of field theory, GR can be described as a massless spin-2 field. Physicists have been tried to construct a massive spin-2 gauge theory for a long time, however many of these attempts face the ghost problem, which involves a mode of infinite negative energy thus followed by inconsistence of the theory. A recent massive gravity theory called dRGT model and its extension model called bigravuty theory have been showed that they are ghost-free. The ghost may reappear when one concerns such theory coupling to matter fields, but recent study shows that a class of effective theory in bigravity model can be used to construct ghost-free interaction with matter. In this thesis we first take a brief review of dRGT model and bigravity, and then we solve a series of effective metrics by some known solution of bigravity theory. In the later part we consider a model in which a Dirac fermion couples to a effective metric and interacts with a U(1) gauge boson, which couples to another effective metric. Finally we give the modified Coulomb potential in this model.