Vacuum solution within a metric-affine bumblebee gravity

Abstract

We consider a metric-affine extension to the gravitational sector of the Standard Model extension for the Lorentz-violating coefficients $u$ and ${s}^{\ensuremath{\mu}\ensuremath{\nu}}$. The general results, which are applied to a specific model called metric-affine bumblebee gravity, are obtained. A Schwarzschild-like solution, incorporating effects of the Lorentz symmetry breaking through the coefficient $X=\ensuremath{\xi}{b}^{2}$, is found. Furthermore, a complete study of the geodesic trajectories of particles is accomplished in this background, emphasizing the departure from general relativity. We also compute the advance of Mercury's perihelion and the deflection of light within the context of the weak-field approximation, and we verify that there exist two new contributions ascribed to the Lorentz symmetry breaking. As a phenomenological application, we compare our theoretical results with observational data in order to estimate the coefficient $X$.