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.
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Notes
For instance, the distance to Moon as measured by the Lunar Laser Ranging (LLR) technique could appear to us on Earth to be shorter by 22.09 cm than the Euclidean distance of the zeroth order. It is a realizable physical prediction (see Sect. 6).
When \(\ell = 0\), the cubic Kerr terms in T(r) here differ from those of Wang and Lin [12]. However, their conclusion that the third-order mass effect is larger than the rotation effect for certain ranges of impact parameter is correct.
When \(\ell \ne 0\), the third-order term involving \(a^{2}\), i.e., \(a^{2}R_{\text {S}}\) is not to be considered at all since the Ding et al. [58] metric (13–16) is valid only up to first order in spin a [61]. However, the expressions for \(t(r_{0}\rightarrow r)\) are valid for all order in a only when \(\ell = 0\) (Kerr metric of general relativity).
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Acknowledgements
We thank an anonymous referee for many insightful comments that led to a considerable improvement of the paper. We are indebted to Gulnaz Kutlieva for her enlightening comments on different technical aspects of satellite communications relevant to the measurement of the time advancement effect. This research was funded by grant RB NOC-GMU-2022 (Prikaz No 2987 ot 29.11.2022).
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Appendix 1: Bumblebee corrections
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
where \(0 \le \phi < 2\pi \). The Kretschmann scalar due to Bumblebee field is
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
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
Hence, the light ray undergoes a deflection of
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
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
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
confirming the LSB correction obtained after elaborate calculations in [57].
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Tuleganova, G.Y., Karimov, R.K., Izmailov, R.N. et al. Gravitational time advancement effect in Bumblebee gravity for Earth bound systems. Eur. Phys. J. Plus 138, 94 (2023). https://doi.org/10.1140/epjp/s13360-023-03713-y
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DOI: https://doi.org/10.1140/epjp/s13360-023-03713-y