Gravitational time advancement effect in Bumblebee gravity for Earth bound systems

Abstract

This paper is a novel application of the new effect of gravitational time advancement or 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 ($a\ne 0$) 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 up to 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 less than the zeroth order Euclidean distance. It is shown that the LSB correction arises from the conical geometry of the massless Bumblebee space-time 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.

Appendix 1: Bumblebee corrections

Our idea is to separate pure Bumblebee corrections from the mass effects, so we put $M=0$ in the metric (19), and for simplicity consider the static case ($a=0$), which yields a conical space-time

$$\begin{aligned} \textrm{d}\tau ^{2} = -\textrm{d}t^{2} + \left( 1 + \ell \right) \textrm{d}r^{2} + r^{2}\left( \textrm{d}\theta ^{2} + \sin ^{2}\theta \textrm{d}\phi ^{2}\right) , \end{aligned}$$

(A1)

where $0 \le \phi < 2\pi$. The Kretschmann scalar due to Bumblebee field is

$$\begin{aligned} R_{\mu \nu \alpha \beta } R^{\mu \nu \alpha \beta } = \frac{4\ell ^{2}}{r^{4}(\ell + 1)^{2}}. \end{aligned}$$

(A2)

Assuming the signal to travel on the equatorial plane $\theta = \pi /2$, consider the 2-surface S:(R, $\phi$) and using the method of Ford and Vilenkin [73], the metric (A1) can be rewritten as a “flat space-time” metric

$$\begin{aligned} \textrm{d}\tau ^{2}=-\textrm{d}t^{2}+\textrm{d}R^{2}+R^{2}\textrm{d}\phi ^{\prime 2}, \end{aligned}$$

(A3)

where $b = \frac{1}{\sqrt{1+\ell }}$, $R=r\sqrt{1+\ell }$ and $0\le \phi ^{\prime } < 2\pi b$. Therefore, the light ray moving tangentially along a straight line in the flat planar metric (A3) sweeps out an angle $\Delta \phi ^{\prime }= \pi$ or

$$\begin{aligned} \Delta \phi =\pi b^{-1}. \end{aligned}$$

(A4)

Hence, the light ray undergoes a deflection of

$$\begin{aligned} \delta \phi =\pi b^{-1}-\pi =\pi \left( b^{-1}-1\right) =\pi \left( \sqrt{1+\ell }-1\right) \simeq \frac{\pi \ell }{2}. \end{aligned}$$

(A5)

Note that, this is precisely the amount of correction to light bending by the LSB term, $\delta \phi _{{\textrm{LSB}}}$, obtained after elaborate calculations in [57].

Similarly, the time delay in the “flat space-time” (A3) is

$$\begin{aligned} \Delta t=\int \frac{R\textrm{d}R}{\sqrt{R^{2}-R_{0}^{2}}} = \sqrt{R^{2}-R_{0}^{2}}, \end{aligned}$$

(A6)

where $R_{0}$ is the closest approach distance of the light ray or radar signal from the origin. Using $R=r\sqrt{1+\ell }$ and $R_{0}= r_{0}\sqrt{1+\ell }$, we find the Euclidean part of time delay for two-way motion to be

$$\begin{aligned} \Delta \tau _{0}=2\Delta t=2\left( \sqrt{1+\ell }\right) \left( \sqrt{r^{2}-r_{0}^{2}}\right) =\delta \tau _{0}+\delta \tau _{{\textrm{LSB}}}^{{\textrm{Eucl}}}. \end{aligned}$$

(A7)

This is precisely Eq. (52), which is what we wanted to show. The correction to the Euclidean part of time delay for two-way motion in the original (t, r) coordinates, restoring c, then is

$$\begin{aligned} \delta \tau _{{\textrm{LSB}}}^{{\textrm{Euc}}}\simeq \frac{\ell }{c}\sqrt{r_{A}^{2}-r_{B}^{2}}, \end{aligned}$$

(A8)

confirming the LSB correction obtained after elaborate calculations in [57].