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

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 Stückelberg fields $S_{\varphi }={\Lambda }^4\int d^{4}x\sqrt{-g}\mathcal{F}$, 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 Stückelberg field ${\varphi }^i$ is taken as a 'hedgehog' configuration ${\varphi }^i=\varphi (r)x^i/r$, whose stability is guaranteed by the topological one. Under this ansätz, $T_0^1=0$ 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.

A typo is corrected. Some sentences are polished. 26pages, 1 figure. arXiv:1503.08952 [gr-qc]

References in corpus (12)

• The Confrontation between General Relativity and Experiment
• Massive Gravity
• Infrared-modified gravities and massive gravitons
• Degrees of Freedom in Massive Gravity
• The gravitational field of a global monopole
• Gravitating Global k-monopole
• Black hole solutions in de Rham-Gabadadze-Tolley massive gravity
• Stars and (Furry) Black Holes in Lorentz Breaking Massive Gravity
• Constraints on the massive graviton dark matter from pulsar timing and precision astrometry
• Global monopole surrounded by quintessence-like matter
• Quasinormal Modes and Late-Time Tails of Canonical Acoustic Black Holes
• Gravitating tensor monopole in a Lorentz-violating field theory

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