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 Lorentz breaking massive gravity. There is an undetermined function $$\mathcal{F}$(X, w_1, w_2, w_3)$intheactionofSt\ddot{u}ckelbergfields$S_{\phi}=\Lambda^4\int{d^4x\sqrt{-g}$\mathcal{F}$}$, which should be resolved through physical means. In the general relativity, the spherically symmetric solution to the Einstein equation is a benchmark and its massive deformation also play a crucial role in Lorentz breaking massive gravity. $\mathcal{F}$ will satisfy the constraint equation $T_0^1=0$ from the spherically symmetric Einstein tensor $G_0^1=0$, if we maintain that any reasonable physical theory should possess the spherically symmetric solutions. The St"{u}ckelberg field ${\varphi }^i$ is taken as a ‘hedgehog’ configuration ${\varphi }^i=\varphi (r)x^i\mathrm{/}r$, whose stability is guaranteed by the topological one. Under this ans"{a}tz, $T_0^1=0$ is reduced to $d\mathcal{F}=0$. The functions $\mathcal{F}$ for $d\mathcal{F}=0$ form a commutative ring $R^{\mathcal{F}}$. We obtain a general expression of solution to the functional differential equation with spherically symmetry if $\mathcal{F}\in R^{\mathcal{F}}$. If $$\mathcal{F}$\in R^{$\mathcal{F}$}$and$\partial$\mathcal{F}$/\partial X=0$,thefunctions$$\mathcal{F}$$formasubring$S^{$\mathcal{F}$}\subset R^{$\mathcal{F}$}$. We show that the metric is Schwarzschild, AdS or dS if $$\mathcal{F}$\in S^{$\mathcal{F}$}$\mathrm{.}When$$\mathcal{F}$\in R^{$\mathcal{F}$}$but$$\mathcal{F}$\notin S^{$\mathcal{F}$}$, we will obtain some new metric solutions. Using the general formula and the basic property of function ring $R^{\mathcal{F}}$, we give some analytical examples and their phenomenological applications. Furthermore, we also discuss the stability of gravitational field by the analysis of Komar integral and the results of QNMs.