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

TL;DR

This paper derives analytical spherically symmetric solutions in Lorentz breaking massive gravity, classifying them via function rings, and explores their stability and phenomenological implications.

Contribution

It introduces a framework using function rings to systematically find and classify solutions in Lorentz breaking massive gravity, including new metric solutions.

Findings

Solutions include Schwarzschild, AdS, and dS metrics.

New metric solutions are obtained outside the subring $S^{\

},

Abstract

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 $S_{\phi}=\Lambda^4\int{d^4x\sqrt{-g}\mathcal{F}}$, 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 $\phi^i$ 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 $d\mathcal{F}=0$. 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 $\partial\mathcal{F}/\partial X=0$, 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 $\mathcal{F}\notin S^{\mathcal{F}}$, 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.