On The Massive Gravity With And Without Lorentz Symmetry

We review massive gravity in Lorentz invariant type and Lorentz breaking type respectively in this paper. We extend the method to solve static sphere symmetry problem in Lorentz breaking massive gravity. The functional equations and commu-tative ring method give three types of black hole solution, which are Schwarzchild-(A)ds metric, furry black hole metric and a new type metric of black hole. Then, the stability of furry black hole and the physical phenomenology of the new black hole are discussed. We also find two branches black hole in Lorentz breaking massive gravity: the first branch is classical solutions (Schwarzchild metric or RN metric), the second branch is the furry type correspondingly. The no hairy theorem in dRGT is also been discussed. Well, we have to choose the nonunitary gauge to get all this solutions. Further more, the Birkhoff theorem guarantee that the black hole is a stationary metric if we choose the unitary gauge in dRGT. Only if we choose a special gauge, which we call the Birkhoff gauge, the black hole would take a static metric. The Birkhoff gauge would release the constraint in the spatial part of Stuckelberg field if some special parameters have been done in dRGT. Such model have more degrees of freedom than dRGT and is been called the minimal model of dRGT. The minimal model have no furry black hole branch and a new dynamics cosmology, which would explain both the late time expansion and the early time inflation at the same time. At last, the Misner-Sharp mass is been calculated.