Dynamic Aspects of Bumblebee Gravity: Post-Newtonian Approach

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.