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Achromatic, spin-odd Kerr EVPA as a null Frenet--Serret torsion integral on the photon ring

M. Baran Ökten

TL;DR

The paper addresses how strong-field Kerr geometry imprints an achromatic rotation of linear polarization on the photon ring. It develops a null Frenet--Serret framework that reduces EVPA transport to a single scalar evolution, yielding a Kerr-minus-Schwarzschild, parity-odd, ring-localized imprint described by the Geometric Polarization Invariant. A robust, two-parameter template in $a/M$ and $i$ is proposed, with a parity-odd estimator $oldsymbol{ extchi}_{ m odd}$ that is intrinsically achromatic after $oldsymbol{ extlambda^2}$ regression and protected from even-parity systematics. The method, validated by three independent computations and a second-order accurate numerical pipeline, provides a practical path to detecting frame-dragging–induced parallel transport in horizon-scale polarimetry of sources like M87$^*$ and Sgr A$^*$, or setting informative upper limits.

Abstract

We compute the achromatic gravitational imprint that Kerr spacetime leaves on linear polarization at the photon ring. Recasting parallel transport in a null Frenet--Serret frame yields a single scalar evolution law for the electric-vector position angle. On the observer's screen, the Kerr-minus-Schwarzschild pattern on the direct critical curve is nonzero, strictly odd under spin reversal after a half-turn azimuth relabelling, and tightly confined to a thin annulus. Using backward-shot, Carter-separated geodesics with midpoint RK2 transport, we achieve second-order convergence and degree-scale amplitudes that grow monotonically with spin and inclination (RMS $\simeq 0.5$--$2^\circ$ for $a/M\gtrsim 0.8$, $i\gtrsim 60^\circ$). Three independent constructions -- Frenet--Serret line integral, explicit Levi--Civita transport of the polarization vector, and the phase of the Walker--Penrose constant -- agree ray by ray. We then define a parity-odd ring estimator that is intrinsically achromatic after standard wavelength-squared regression, symmetry-protected against common even-parity systematics, and compressed into low azimuthal modes. This yields a minimal two-parameter template (spin and inclination) for mm/sub-mm polarimetry of horizon-scale rings in sources such as M87$^\ast$ and Sgr~A$^\ast$. The pipeline enables either a detection of the strong-field parallel-transport phase induced by frame dragging or informative upper limits.

Achromatic, spin-odd Kerr EVPA as a null Frenet--Serret torsion integral on the photon ring

TL;DR

The paper addresses how strong-field Kerr geometry imprints an achromatic rotation of linear polarization on the photon ring. It develops a null Frenet--Serret framework that reduces EVPA transport to a single scalar evolution, yielding a Kerr-minus-Schwarzschild, parity-odd, ring-localized imprint described by the Geometric Polarization Invariant. A robust, two-parameter template in and is proposed, with a parity-odd estimator that is intrinsically achromatic after regression and protected from even-parity systematics. The method, validated by three independent computations and a second-order accurate numerical pipeline, provides a practical path to detecting frame-dragging–induced parallel transport in horizon-scale polarimetry of sources like M87 and Sgr A, or setting informative upper limits.

Abstract

We compute the achromatic gravitational imprint that Kerr spacetime leaves on linear polarization at the photon ring. Recasting parallel transport in a null Frenet--Serret frame yields a single scalar evolution law for the electric-vector position angle. On the observer's screen, the Kerr-minus-Schwarzschild pattern on the direct critical curve is nonzero, strictly odd under spin reversal after a half-turn azimuth relabelling, and tightly confined to a thin annulus. Using backward-shot, Carter-separated geodesics with midpoint RK2 transport, we achieve second-order convergence and degree-scale amplitudes that grow monotonically with spin and inclination (RMS -- for , ). Three independent constructions -- Frenet--Serret line integral, explicit Levi--Civita transport of the polarization vector, and the phase of the Walker--Penrose constant -- agree ray by ray. We then define a parity-odd ring estimator that is intrinsically achromatic after standard wavelength-squared regression, symmetry-protected against common even-parity systematics, and compressed into low azimuthal modes. This yields a minimal two-parameter template (spin and inclination) for mm/sub-mm polarimetry of horizon-scale rings in sources such as M87 and Sgr~A. The pipeline enables either a detection of the strong-field parallel-transport phase induced by frame dragging or informative upper limits.

Paper Structure

This paper contains 9 sections, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theory: null geodesics, a null FS frame, and EVPA
  3. Kerr metric and Carter separation
  4. Screen mapping and the camera basis
  5. Null FS frame and polarization transport
  6. Spin oddness and ring geometry
  7. Computation
  8. Findings
  9. Conclusions

Figures (5)

  • Figure 1: Odd–parity EVPA $\chi_{\rm odd}(\phi)$ on the direct critical curve for a high‑spin, high‑inclination example ($a/M=0.9$, $i=80^\circ$). The curve is smooth and non‑sinusoidal, with degree‑scale amplitude and no aliasing artefacts. Odd differencing removes the gauge freedom and suppresses even–parity contributions.
  • Figure 2: Dependence on spin and inclination. Rows correspond to inclinations $i=20^\circ$, $50^\circ$ and $80^\circ$; columns display the Kerr–Schwarzschild difference $\Delta\chi_{\rm grav}(\phi)$, the odd channel $\chi_{\rm odd}(\phi)$ and the even channel $\chi_{\rm even}(\phi)$. Amplitudes rise with both $a/M$ and $i$, while the even channel remains near the noise floor. Vertical axes are in degrees; within each row the three panels share identical y-limits, while ranges differ by row solely for visibility.
  • Figure 3: Even‑channel detectability map. The colour scale shows the root‑mean‑square amplitude $A_{\rm even}$ (degrees) across spin $a/M$ and inclination $i$. The even channel is negligible (dark shading) across most of the parameter space, with modest values appearing only at low inclinations and large spins. These values remain small relative to the odd‑channel amplitude, consistent with the spin‑odd nature of the geometric signal.
  • Figure 4: Azimuthal derivative ${\rm d}(\Delta\chi_{\rm grav})/{\rm d}\phi$ along the ring at $i=20^\circ$ for $a/M=0$, $0.5$ and $0.9$. The derivative is flat except for a step at $\phi\simeq 180^\circ$, corresponding to the polar turning point. No random spikes or oscillations are present.
  • Figure 5: Signal–to–noise ratio as a function of the number of azimuthal samples $N$ for per–sample EVPA uncertainties $\sigma_\chi=0.3^\circ$, $0.5^\circ$, $1.0^\circ$ and $2.0^\circ$. Increasing $N$ boosts the SNR with diminishing returns; smaller $\sigma_\chi$ yields higher SNR.