We present a detailed study of the spherically symmetric solutions in Lorentz breaking massive gravity. There is an undetermined function $mathcal{F}(X, w_1, w_2, w_3)$ in the action of St{u}ckelberg fields $S_{phi}=Lambda^4int{d^4xsqrt{-g}mathcal{F}
}$, 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 $phi^i$ is taken as a hedgehog configuration $phi^i=phi(r)x^i/r$, whose stability is guaranteed by the topological one. Under this ans{a}tz, $T_0^1=0$ is reduced to $dmathcal{F}=0$. The functions $mathcal{F}$ for $dmathcal{F}=0$ 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 $partialmathcal{F}/partial X=0$, the functions $mathcal{F}$ form a subring $S^{mathcal{F}}subset R^{mathcal{F}}$. We show that the metric is Schwarzschild, AdS or dS if $mathcal{F}in S^{mathcal{F}}$. When $mathcal{F}in R^{mathcal{F}}$ but $mathcal{F} otin S^{mathcal{F}}$, 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.
In a recent work, we present a new point of view to the relation of gravity and thermodynamics, in which we derive the sch~solution through thermodynamic laws by the aid of the Misner-Sharp mass in an adiabatic system. In this paper we continue to in
vestigate the relation between gravity and thermodynamics for obtaining solutions via thermodynamics. We generalize our studies on gravi-thermodynamics in Einstein gravity to modified gravity theories. By using the first law with the assumption that the Misner-Sharp mass is the mass for an adiabatic system, we reproduce the Boulware-Deser-Cai solution in Guass-Bonnet gravity. Using this gravi-thermodynamics thought, we obtain a NEW class of solution in $F(R)$ gravity in an $n$-dimensional (n$geq$3) spacetime which permits three-type $(n-2)$-dimensional maximally symmetric subspace, as an extension of our recent three-dimensional black hole solution, and four-dimensional Clifton-Barrow solution in $F(R)$ gravity.
We study the Misner-Sharp mass for the $f(R)$ gravity in an $n$-dimensional (n$geq$3) spacetime which permits three-type $(n-2)$-dimensional maximally symmetric subspace. We obtain the Misner-Sharp mass via two approaches. One is the inverse unified
first law method, and the other is the conserved charge method by using a generalized Kodama vector. In the first approach, we assume the unified first still holds in the $n$-dimensional $f(R)$ gravity, which requires a quasi-local mass form (We define it as the generalized Misner-Sharp mass). In the second approach, the conserved charge corresponding to the generalized local Kodama vector is the generalized Misner-Sharp mass. The two approaches are equivalent, which are bridged by a constraint. This constraint determines the existence of a well-defined Misner-Sharp mass. As an important special case, we present the explicit form for the static space, and we calculate the Misner-Sharp mass for Clifton-Barrow solution as an example.
