Dynamic Aspects of Bumblebee Gravity: Post-Newtonian Approach

TL;DR

This paper analyzes the dynamic properties of Bumblebee gravity using the parameterized post-Newtonian framework, revealing conditions for consistency, stability, and preferred-frame effects, with observational constraints from pulsar timing.

Contribution

It provides a detailed post-Newtonian analysis of Bumblebee gravity, identifying specific parameter conditions for consistency and stability, and deriving observational bounds.

Findings

PPN framework is consistent up to 1.5PN order under specific parameter relations.

The model exhibits preferred-frame effects due to Lorentz symmetry breaking.

Pulsar timing constrains the Lorentz-violating parameter to |1|  1.6  10^{-9}.

Abstract

In this work, we investigate the dynamic aspects of Bumblebee gravity via the parameterized post-Newtonian method. We find that the PPN framework is self-consistent up to 1.5PN order if and only if $\lambda = -\xi/2$, which corresponds to a direct coupling between the Bumblebee field $B_\mu$ and the Einstein tensor. The requirement of tachyonic stability restricts the Bumblebee potential to satisfy $V''(0)=0$. In the specific case where $\lambda = -\xi/2$, the resulting PPN metric yields non-vanishing values for the parameters $\alpha_1$ and $\alpha_2$, as well as a novel PPN potential $U_B$ that exhibits a logarithmic asymptotic growth. The vanishing of the potential $U_B$ necessitates the additional constraints $\xi = \kappa/2$ or $V^{(3)}(0) = 0$. These results signify the presence of preferred-frame effects, a direct consequence of the Lorentz symmetry breaking in the model. In the limit of small $\ell$, we obtain $\alpha_2 \simeq -\ell_2$, which yields a constraint of $|\ell| \lesssim 1.6\times 10^{-9}$ based on pulsar timing observations.