Possible altitudinal, latitudinal, and directional dependence of the relativistic Sagnac effect in Chern-Simons modified gravity

Toward a test of parity violation in a gravity theory, possible effects of Chern-Simons (CS) gravity on an interferometer have been recently discussed. Continuing work initiated in an earlier publication [H. Okawara, K. Yamada, and H. Asada, Phys. Rev. Lett. 109, 231101 (2012)], we study possible altitudinal and directional dependence of relativistic Sagnac effect in CS modified gravity. We compare the CS effects on Sagnac interferometers with the general relativistic Lense-Thirring (LT) effects. Numerical calculations show that the eastbound Sagnac interferometer might be preferred for testing CS separately, because LT effects on this interferometer cancel out. The size of the phase shift induced in the CS model might have an oscillatory dependence also on the altitude of the interferometer through the CS mass parameter $m_{\mathrm{CS}}$. Therefore, the international space station site as well as a ground-based experiment is also discussed.

Figure 1

Contour maps for the dependence of time shift on the interferometer direction angle $\alpha$ and the latitude $\varphi$, where the height corresponds to the angular part of $(c\mathrm{\Delta }t)$. Top: ${\mathrm{\Delta }}_{\mathrm{LT}}$ by LT effects in GR. Bottom: ${\mathrm{\Delta }}_{\mathrm{CS}}$ by CS effects.

Figure 2

The ratio of $(c\mathrm{\Delta }t{)}_{\mathrm{CS}}/(c\mathrm{\Delta }t{)}_{\mathrm{LT}}$ at the ground level and the ISS site at $h\sim 400\mathrm{km}$. For its simplicity, we assume the equatorial case as $\varphi =0^{\circ }$ and the northbound direction as $\alpha =0^{\circ }$. This figure suggests that the altitudinal effect might make a more complicated form of oscillating behavior in terms of $m_{\mathrm{CS}}$ compared with the ground level.

Figure 3

Time shift as a function of the orbital phase angle $\theta$ that is measured from the initial direction on the equator. For its simplicity, we assume the corotation of the ISS in a polar orbit around the Earth $(h\sim 400\mathrm{km})$. For the circular orbit, $\theta$ is proportional to time, where $\theta =0^{\circ }$, 90°, 180°, and 270° are corresponding to the passage time of the equatorial plane, a pole of Earth, the equatorial plane and the opposite pole, respectively. We consider three directions of the interferometer: The labels as x, y, and z denote the eastbound direction, northbound one and vertical one, respectively, at the initial time as $\theta =0^{\circ }$.