An Analytical Formula for Gravitational Faraday Rotation in the ADM Split of Spacetime

TL;DR

The paper provides an analytical, closed-form expression for the rate of gravitational Faraday rotation (GFR) as measured by Eulerian observers in the Kerr spacetime under the ADM (3+1) split. By aligning Fermi-Walker frames with the spatial wave vector and introducing the Shift Tetrad, the GFR rate is shown to equal the extrinsic-curvature component $K_{\grave{1}\grave{2}}$ projected in the Shift frame, yielding $\frac{d\chi}{d\lambda}=\omega_E K_{\grave{1}\grave{2}}$. The approach avoids ergosphere-related coordinate singularities and is validated against numerical results for both closed and ergosphere-crossing photon trajectories, with agreement at the $\sim 10^{-5}$ degree level. This Eulerian, curvature-based perspective complements traditional Lagrangian-frame analyses and provides a practical mechanism to compare polarization holonomy with analytic predictions for transiting light near a rotating black hole.

Abstract

An analytical expression is derived for the rate of gravitational Faraday rotation measured by Eulerian observers. The reference frame is a Fermi-Walker triad aligned with the spatial wave vector. Attention is restricted to the ADM split of Kerr spacetime and geometric optics. Our exact, closed-form GFR formula is implemented and verified to be consistent with numerical predictions. The approach offers a new perspective on Faraday rotation, and it allows a single Eulerian observer to compare experimentally measured polarization holonomy with analytical prediction. Sliced spacetime does not suffer from a mathematical singularity at the ergosphere associated with Boyer-Lindquist coordinates in the threading decomposition. These physically intuitive coordinates can therefore be used to analytically produce and study GFR predictions for transits of light that pierce the ergosphere.

Figures (4)

• Figure 1: Closed Trajectory Transit Outside the Ergosphere. (a) A closed photon trajectory is shown in yellow along with the common starting (green) and stopping (red) points. The ergosphere (white with black mesh) and outer event horizon (black with white mesh) make it clear that this trajectory stays outside the ergosphere while carrying out a non-symmetric, three-dimensional clover leaf path. (b) The radial component of the trajectory is plotted along with the associated ergosphere and event horizon radii at the same polar angle. This shows that the trajectory remains external to the ergosphere. $\psi = 0.766295, \eta = 17.9662, a = 0.99, r_i = r_f = 4$.
• Figure 2: Open Trajectory Transit Through the Ergosphere. (a) An open photon trajectory is shown in yellow along with distinct starting (green) and stopping (red) points. The ergosphere (white with black mesh) and outer event horizon (black with white mesh) make it clear that this trajectory starts and ends outside the ergosphere but transits through the ergosphere for much of the path shown. (b) The radial component of the trajectory is plotted along with the associated ergosphere and event horizon radii at the same polar angle. This shows that the trajectory pierces the ergosphere twice. $\psi = 2.05, \eta = 5, a = 0.99, r_i = r_f = 3$.
• Figure 3: GFR for Closed-Loop Trajectory of Fig. \ref{['Visualize_Closed_Traj']}. (a) Analytical prediction of GFR (green) along with numerical result (magenta), (b) Discrepancy $\Delta\chi(s)$, Eq. \ref{['Deltachi']}, is plotted on a log scale and is on the order of $10^{-5}$ °.
• Figure 4: GFR for Open Trajectory of Fig. \ref{['Visualize_Open_Traj']}. (a) Analytical prediction of GFR (green) along with numerical result (magenta), (b) Discrepancy $\Delta\chi(s)$, Eq. \ref{['Deltachi']}, is plotted on a log scale and is on the order of $10^{-5}$ °.