Gravitational time advancement effect in Bumblebee gravity for Earth bound systems

This paper is a novel application of the new effect of gravitational time advancement or \textit{negative} time delay, first predicted for static black holes (spin $a=0$), that can be regarded as complementary to the well known effect of positive Shapiro time delay. We shall extend the Shapiro time delay formalism up to third PPN order using the recently proposed spinning ($a\neq 0$) black hole solution of the Lorentz symmetry breaking (LSB) Bumblebee gravity that is believed to reveal signatures of quantum gravity at low energies. Adopting two practical examples of signal propagation along Earth-Moon and Earth-Satellite configurations, we shall calculate the influence of the Bumblebee parameter $\ell$ on time advancement using terms up to the second PPN order $\varpropto aM$ and $M^{2}$ as the Bumblebee solution is valid only upto first order in $a$. It is shown that there is a critical radial distance $r_{c}$ above the Earth, where the Shapiro delay vanishes, and beyond $r_{c}$ the delay becomes negative, i.e., time advancement begins to set in, leading to the intriguing consequence that the measured LLR distance to Moon or any Satellite becomes \textit{less} than the zeroth order Euclidean distance. It is shown that the LSB correction arises from the conical geometry of the massless Bumblebee spacetime leading to upper bounds on the correction to the zeroth order Euclidean time interval as $\delta \tau _{\text{LSB}}^{\text{Eucl}}<0.8\times 10^{-4}$ (ns) and to time advancement as $\Delta \tau _{\text{LSB}}^{\text{adv}}<-4.5\times 10^{-13}$ (ns), both estimates based on the bound on $\ell$ corresponding to the Cassini spacecraft experiment. We shall also briefly touch upon the feasibility of direct experimental detection of the advancement effect.