Analytical expression for a class of spherically symmetric solutions in Lorentz-breaking massive gravity

We present a detailed study of the spherically symmetric solutions in Lorentzbreaking massive gravity. There is an undetermined function F(X, w(1), w(2), w(3)) in the action of stckelberg fields S phi = Lambda(4) integral d(4)x root-gF, which should be resolved through physical means. In general relativity, the spherically symmetric solution to the Einstein equation is a benchmark and its massive deformation also plays a crucial role in Lorentz-breaking massive gravity. F will satisfy the constraint equation T-0(1) = 0 from the spherically symmetric Einstein tensor G(0)(1) = 0 , if we maintain that any reasonable physical theory should possess the spherically symmetric solutions. The Stuckelberg field phi(i) is taken as a 'hedgehog' configuration phi(i) = phi(r) x(i)/r, whose stability is guaranteed by the topological one. Under this ansetz, T-0(1) = 0 is reduced to dF = 0. The functions. for dF = 0 form a commutative ring R-F. We obtain an expression of the solution to the functional differential equation with spherical symmetry if F is an element of R-F. If F is an element of R-F and partial derivative F/partial derivative X = 0, the functions. form a subring S-F is an element of R-F. We show that the metric is Schwarzschild, Schwarzschild-AdS or Schwarzschild-dS if F is an element of S-F. When F is an element of R-F. but F is not an element of S-F, we will obtain some new metric solutions, including the furry black hole and beyond. Using the general formula and the basic property of function ring R-F, we give some analytical examples and their phenomenological applications. Furthermore, we discuss the stability of the gravitational field by the analysis of the Komar integral and the results of quasinormal modes (QNMs).