Metric-affine bumblebee gravity: classical aspects

TL;DR

This work formulates the bumblebee model within metric-affine gravity, solving the independent connection to express the space-time metric in terms of an auxiliary metric and the bumblebee field, thereby generating a non-metricity sourced by $B_\mu$. The theory naturally couples the bumblebee to all matter through a non-minimal term with strength $\xi$, and can be recast in an Einstein-frame like form via an auxiliary metric, revealing nonlinear matter interactions. In the weak-field, post-Minkowskian regime, the effective metric induces Lorentz-violating corrections to scalar and Dirac fields, with explicit dispersion relations and potential instabilities depending on the sign of $\xi$ and the bumblebee VEV. The results provide a concrete, testable framework for Lorentz violation in metric-affine gravity and outline prospects for experimental constraints and further quantum and astrophysical applications.

Abstract

We consider the metric-affine formulation of bumblebee gravity, derive the field equations, and show that the connection can be written as Levi-Civita of a disformally related metric in which the bumblebee field determines the disformal part. As a consequence, the bumblebee field gets coupled to all the other matter fields present in the theory, potentially leading to nontrivial phenomenological effects. To explore this issue we compute the weak-field, post-Minkowskian limit and study the resulting effective theory. In this scenario, we couple scalar and spinorial matter to the effective metric, and then we explore the physical properties of the VEV of the bumblebee field, focusing mainly on the dispersion relations and the stability of the resulting effective theory.