Dynamic Aspects of Bumblebee Gravity: Post-Newtonian Approach

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 $V^{\prime \prime }(0)=0$. 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.

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