Kerr Polarization Transport: Accuracy and Performance in General Relativistic Light Propagation

TL;DR

This work addresses the challenge of accurately propagating polarized light in the Kerr spacetime, essential for interpreting strong-gravity polarization measurements. It introduces a compact Kerr polarization-transport framework that combines a Kerr-corrected light-bending mapping (via a fast Padé approximation) with full parallel transport of the polarization vector in a ZAMO frame, solved with a fourth-order Runge-Kutta scheme and cached Christoffel symbols. The approach delivers sub-degree EVPA accuracy (typically millidegree to a few millidegrees) and sub-percent PD accuracy while achieving ~5× speedups over strict reference integrators, satisfying current IXPE/NICER tolerances and approaching EHT requirements. This efficient, reproducible pipeline enables extensive parameter surveys and real-time polarization modeling for accreting black holes, offering a practical foundation for future GR-polarimetry and imaging studies across X-ray to submillimeter wavelengths.

Abstract

We present a compact and reproducible method for general relativistic polarization transport in the Kerr metric that achieves median electric vector position angle (EVPA) residuals of $\langle Δ\mathrm{PA} \rangle \approx 0.09^\circ$, a 95th percentile of $0.31^\circ$, and a worst case $Δ\mathrm{PA} \lesssim 0.32^\circ$ for spins up to $|a/M|=0.9$, while maintaining a fivefold or greater speedup relative to a strict reference integrator. Across the benchmark grid, typical residuals remain at the sub-tenth-degree level, with only modest degradation ($Δ\mathrm{PA} \lesssim 2^\circ$) near the Thorne spin limit. Photon four-momenta $k^μ$ and polarization four-vectors $f^μ$ are advanced using a fourth order Runge-Kutta scheme with cached Christoffel symbols, maintaining the constraints $u\cdot f=0$ and $n\cdot f=0$, where $u^μ$ is the ZAMO four-velocity and $n^μ$ is the disk normal, while keeping $k\cdot f \simeq 0$. A physically motivated gauge is enforced by projecting the polarization into the local zero-angular-momentum observer (ZAMO) screen at every substep, ensuring numerical stability of the orthogonality constraints. Accuracy and performance are benchmarked over a representative grid in spin, inclination, image-plane azimuth, and radius. The method comfortably meets IXPE and NICER polarization tolerances and approaches EHT requirements. The approach provides a practical foundation for future general relativistic polarimetry and simulation pipelines.

Figures (2)

• Figure 1: Residual accuracy validation for the accelerated Kerr polarization transport. Top row: EVPA residuals $\Delta{\rm PA}$ for inclinations $i=27^{\circ}$ (left) and $30^{\circ}$ (right) as functions of spin parameter $a/M$. Bottom row: corresponding polarization-degree residuals $\Delta{\rm PD}$. Dashed lines mark the target tolerances ($0.4^{\circ}$ and $0.05\%$). All values remain well below these limits, confirming sub-degree and sub-percent accuracy across the full spin range $|a/M|\!\le\!0.9$. Typical EVPA residuals are at the millidegree level ($\Delta{\rm PA}\!<\!0.005^{\circ}$), while the global worst case across the grid stays far below the $0.4^{\circ}$ adequacy threshold.
• Figure 2: Comparison of achieved residual $\Delta{\rm PA}\!\lesssim\!0.4^{\circ}$ (blue) with published observational tolerances (gray) for IXPE, NICER, and EHT. Error bars correspond to representative uncertainties quoted in Krawczynski2022, Steiner2024, and the Event Horizon Telescope Collaboration (2021, 2022).