[Paper Review] Vacuum solution within a metric-affine bumblebee gravity

A. A. Araújo Filho, J. R. Nascimento+2 more|arXiv (Cornell University)|Nov 21, 2022

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper presents the first exact vacuum solution in metric-affine bumblebee gravity, a Lorentz-symmetry-breaking extension of general relativity. By introducing a spacelike bumblebee vector field with a nontrivial vacuum expectation value, the authors derive a Schwarzschild-like metric with a nontrivial redshift factor induced by non-metricity, and constrain the Lorentz-violating coefficient $X = \xi b^2$ using Mercury's perihelion advance and light deflection data.

ABSTRACT

We consider a metric-affine extension to the gravitational sector of the Standard-Model Extension for the Lorentz-violating coefficients $u$ and $s^{μν}$. 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 coefficient $X=ξb^2$, is found. Furthermore, a complete study of the geodesics trajectories of particles has been 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 $X$.

Motivation & Objective

• To develop a metric-affine extension of the gravitational sector of the Standard-Model Extension (SME) with spontaneous Lorentz violation.
• To construct a consistent framework for Lorentz-violating gravity in a non-Riemannian spacetime with independent metric and connection.
• To derive exact solutions for a specific bumblebee gravity model in the vacuum, incorporating non-metricity and torsion.
• To compute observable corrections to Mercury's perihelion advance and light deflection due to Lorentz violation.
• To constrain the Lorentz-violating coefficient $X = \xi b^2$ using observational data from Solar system tests.

Proposed method

• Formulate a metric-affine gravity action with Lorentz-violating coefficients $u$ and $s^{\mu\nu}$, assuming $t^{\mu\nu\alpha\beta} = 0$ and $s^{\mu\nu}$ symmetric.
• Derive field equations for the metric and connection, showing that the connection equation yields Christoffel symbols of a disformally related metric.
• Obtain an Einstein-frame representation of the action, transforming the metric field equation into an Einstein-like vacuum equation for an auxiliary metric $h_{\mu\nu}$.
• Solve the metric field equation under vacuum conditions, yielding a Schwarzschild-like solution with a redshift factor dependent on $X = \xi b^2$.
• Compute geodesic equations for particles and weak-field approximations for perihelion advance and light deflection in the presence of $X$.
• Compare theoretical predictions with observational data to bound $X$ using Mercury’s perihelion shift and VLBI light-bending measurements.

Experimental results

Research questions

• RQ1 Can an exact vacuum solution be derived in metric-affine bumblebee gravity with spontaneous Lorentz violation?
• RQ2 How does the non-metricity induced by the bumblebee vector field modify the Schwarzschild geometry and geodesic motion?
• RQ3 What are the corrections to Mercury’s perihelion advance and light deflection due to the Lorentz-violating coefficient $X = \xi b^2$?
• RQ4 What observational constraints can be placed on $X$ using Solar system data?
• RQ5 How does the metric-affine formalism differ from the metric (Palatini) approach in bumblebee gravity?

Key findings

• An exact vacuum solution is found that generalizes the Schwarzschild metric, with a redshift factor modified by the Lorentz-violating coefficient $X = \xi b^2$ due to non-metricity.
• The solution exhibits non-geodesic behavior for particles, with deviations from general relativity increasing with $X$.
• The advance of Mercury’s perihelion receives a correction proportional to $X$, leading to an upper bound of $X \approx 10^{-12}$ from observational data.
• The deflection of light in the weak field regime receives an additional correction from $X$, and the most stringent constraint from VLBI data gives $X < 1.62 \times 10^{-10}$.
• The connection equation yields a disformal relation between the physical metric and an auxiliary metric, with the disformal factor determined by $s^{\mu\nu}$.
• The action admits an Einstein-frame formulation, simplifying the analysis of the metric field equation and confirming the consistency of the solution.