Signatures of Black Hole Spin and Plasma Acceleration in Jet Polarimetry II: Off-Axis Jets

TL;DR

We address how jet polarimetry can reveal black hole spin and plasma acceleration for off-axis jets by combining an approximate GRMHD jet model with ray tracing and Lorentz-transformed polarization. The approach yields analytic and semi-analytic predictions for polarization swings at characteristic radii, tied to the light cylinder radius R_LC that scales as a^{-1}, and distinguishes face-on from off-axis viewing through spine and limb sub-patterns. The key contributions are (i) analytic swing locations tilde{R} for face-on and off-axis geometries, (ii) a spine-limb dichotomy in off-axis polarization with aberration and foreground/background effects, and (iii) observational strategies to constrain spin a, inclination i, and gamma_infty with current and upcoming VLBI facilities. The findings offer a practical, model-agnostic avenue to measure black hole spin in jets and to study jet acceleration, with implications for M87* and other AGN, while highlighting caveats such as Faraday rotation and optically thick regimes that motivate future extensions and Bayesian data fitting.

Abstract

We analyze the polarization of optically thin, stationary, axisymmetric black hole jets at scales of order the light cylinder radius. Our work generalizes the face-on results of Gelles et al. (2025) to arbitrary viewing inclination. Due to a combination of geometry and relativistic aberration, the polarization of the jet is not left-right symmetric, and the degree of asymmetry can shed light on both the viewing angle and the plasma bulk Lorentz factor. We show that there is always a radius in the jet at which the polarization transitions from azimuthal to radial; this radius is different along the spine and limb of the jet. We propose metrics that can be used to constrain the black hole spin, inclination angle, and plasma Lorentz factor from these polarimetric signatures, and we discuss the impact of limb-brightening on these measurements. We anticipate that these polarimetric signatures can be studied with current or forthcoming data in M87, NGC 315, NGC 4261, Centaurus A, Cygnus A, and other systems. Observations of the polarization of the base of the counter-jet in higher inclination sources would provide a particularly promising probe of black hole spin.

Figures (19)

• Figure 1: Three different viewing geometries $(i=0^\circ, i=30^\circ, \,i=90^\circ)$ for a sample jet, with coordinate axes plotted in units of $M$. For each inclination angle, the position of the observer is shown on top, with the corresponding projected image shown below. The filled in black circle corresponds to the black hole, and the black dashed line demarcates an impact parameter of $b=7M$. The solid blue line demarcates the origin of the observer's screen. In practice, the $i=90^\circ$ case is typically not observable due to lack of Doppler boosting, but it is conceptually useful.
• Figure 2: Polarization of a face-on jet with $a=0.5$ and $\gamma_\infty=5$, viewed along the image contour $\varphi=\pi/2$ (i.e. 12 o'clock). The jet and counter-jet each produce different polarization patterns, with the forward jet only becoming the dominant source of polarization outside the light cylinder. When a counter-jet is present, the net polarization therefore swings sharply at the light cylinder. The two crosses mark the lensed positions of the light cylinder in the forward jet and light cylinder in the counter-jet respectively. Note that this calculation includes GR lensing to match Figure 10 of Gelles_2025, while the Figures in other sections of this paper use a flat space approximation.
• Figure 3: Polarization of a face-on jet, again including fully general relativistic effects. On the left, the results are shown when a counter-jet is present (thus matching Figure \ref{['fig:faceonCJfig']}), and on the right, the results are shown when a counter-jet is not present. In each panel, polarization curves are plotted for two different spins ($a=0.5$ and $a=0.9$) and terminal Lorentz factors $(\gamma_\infty=5$ and force-free). When a counter-jet is present, the polarization curve always shows a sharp swing at the light cylinder (marked with black crosses), regardless of terminal Lorentz factor. But when the counter-jet is not present, the polarization swings more modestly at the true light cylinder, meaning that $\tilde{R}$ (shown as a star) can serve as a more easily identifiable metric.
• Figure 4: Effective EVPA swing location $(\tilde{R}_{\rm face-on})$ as a function of both spin (left panel) and terminal Lorentz factor (right panel) for a sample jet. Analytic solutions exist in the non-relativistic ($\gamma_\infty\sim 1$) regime and the force-free ($\gamma_\infty\gg 1$) regime, which are shown as blacked dashed lines in the left panel. These two analytic solutions can then be stitched together using Eq. \ref{['eq:seriesfaceon']}, which is overplotted as black dashed lines in the right panel. In all cases, $\tilde{R}_{\rm face-on}$ depends strongly on black hole spin and $\gamma_\infty$.
• Figure 5: Appearance of a radial $(\vec{f}\propto \hat{R})$ polarization pattern as viewed from $i=0^\circ$ (left) and $i=30^\circ$ (right). In the inclined image, the foreground emission (i.e. $Y<0$) is ignored. As the inclination angle is increased, the polarization angle "folds" up with the jet, as indicated by the black arrows. This is a purely geometric effect. The vertical axis of the image --- where polarization remains radial --- is referred to as the spine, while the regions to the sides are known as the limbs. The exact edge of the jet is the "true limb" and is shown as a black dashed line.