Polarization Signatures of Inspiraling Hotspots around Kerr Black Holes

Abstract

Polarimetric interferometry is a powerful tool for probing both black hole accretion physics and the background spacetime. Current models aimed at explaining the observed multiwavelength flares in Sgr A* often assume hotspots moving on geodesic, Keplerian orbits. In many scenarios, though, a hotspot may instead follow an inspiraling trajectory, potentially transitioning into a plunge toward the black hole. In this work, we present a general framework to simulate the polarized emission from generic equatorial inspiraling hotspots in Kerr spacetime using a parametric four-velocity profile. This parametrization defines a continuous family of flows, ranging from Cunningham's disk model (fixed radius orbits outside the innermost stable circular orbit and plunging motion within the innermost stable circular orbit) to purely radial motion, thereby extending the standard assumptions. Within this framework, we show that inspiral motion produces a distinctive observational signature: a precessing, unwinding evolution of the polarimetric Stokes Q-U looping pattern, in sharp contrast with the closed Q-U loops associated with stable orbits at a fixed radius. We then explore how the morphology of these signatures depends on black hole spin, observer inclination, and magnetic-field configuration. The presented model can be applied to current and near-future interferometric observations of linear polarization, offering a new avenue to probe the physics of matter spiraling inward and the relativistic velocities of plunging plasma.

Figures (9)

• Figure 1: This diagram summarizes the pipeline used to compute the Stokes $Q$ and $U$ parameters. It traces how the photon momentum ($p^\mu$), defined in the background geometry, is projected into the fluid frame, and how the synchrotron polarization is then transformed back to the background. The polarization is propagated to the observer’s screen via the Penrose--Walker constant (which is invariant along null geodesics) and then rescaled by a redshift factor to obtain the observed linear polarization in terms of the Stokes $Q$ and $U$ parameters. Dashed boxes indicate model inputs, while solid-line boxes denote quantities computed directly using the equations presented in Sec. \ref{['sec:theory']}. The calculation used to obtain the boosted set of orthonormal vectors and one-forms (highlighted in the figure with the symbol *) is detailed in Fig. \ref{['fig:transformation']}.
• Figure 2: This diagram shows the two orthonormal bases of vectors and one-forms used to transform quantities from the global geometry to either the ZAMO frame ($e_{\mu}$) or the fluid frame ($e'_{\mu}$). Boxes with dashed-line frames indicate model inputs, while boxes with solid-line frames denote quantities computed analytically.
• Figure 3: An inspiraling hotspot. The first column shows the hotspot’s trajectory around the source (top: $a=0$; bottom: $a=0.94$). The arrows indicate the equatorial velocity field vector, which represents the parametrization of the velocities in Eqs. \ref{['eq:iota,omega']}. The second column shows the corresponding linear polarization of the hotspot emission on the observer’s screen (located in the upper hemisphere of the coordinate system at $\phi=0$), represented in the $Q$--$U$ plane in geometrical units. In both examples, the local magnetic field is vertical, $\vec{B}=(0.0,1.0,0.0)$, and the observer is located at $\theta_{\rm{o}}=20^{\circ}$. The four-velocity parameters in Eq. \ref{['eq:four-velocity']} are $\xi = 0.95$, $\beta_r = 0.98$, and $\beta_\phi = 0.98$, and the inspiral is truncated after three complete angular revolutions.
• Figure 4: Polarimetric $Q$--$U$ loops for a hotspot inspiraling into a non-spinning black hole ($a= 0$). Columns vary the magnetic field configuration: purely vertical ($\vec{B}_1=(0.0,1.0,0.0)$, left) and predominantly vertical with non-axisymmetric components ($\vec{B}_2=(0.5,0.7,0.5)$, right). Rows vary the observer inclination ($\theta_o = 20^\circ, 45^\circ, 70^\circ$). The colored trajectory maps the instantaneous hotspot azimuth $\phi$; the black line is a stable circular orbit at $r=11\,M$ for reference. The key features that we can observe are: (1) the loops are circular at low inclination for $\vec{B}_1$ but become elongated as $\theta_{\rm{o}}$ increases, for $\vec{B}_2$, loops remain nearly circular at all inclinations; (2) a clear inner secondary loop is present at low $\theta_{\rm{o}}$ but vanishes at high inclination; and (3) the loop amplitude grows with $\theta_{\rm{o}}$; (4) at low inclination all orbital phases ($\phi = 0^\circ\text{-- }360^\circ$) are visible, and at high $\theta_{\rm{o}}$, emission from the receding phase ($\phi \sim 0^\circ\text{-- }180^\circ$) is suppressed by gravitational redshift; and (5) the primary effect of changing the field configuration is a rotation of the loop in the $Q$--$U$ plane, and considerable change in shape, suggesting that astrophysical effects dominate over gravitational effects during the polarization evolution. The parameters for the four-velocity (Eq. \ref{['eq:four-velocity']}) are $\xi = 0.95$, $\beta_r=0.98$ and $\beta_\phi=0.98$. The inspiral has been stopped after three complete angular revolutions in the geometry.
• Figure 5: Polarimetric $Q$--$U$ loops for a hotspot inspiraling into a rapidly spinning black hole ($a=0.94$). Columns vary the magnetic field configuration: purely vertical ($\vec{B}_1=(0.0,1.0,0.0)$, left) and predominantly vertical with non-axisymmetric components ($\vec{B}_2=(0.5,0.7,0.5)$, right). Rows vary the observer inclination ($\theta_{\rm{o}} = 20^\circ, 45^\circ, 70^\circ$). The colored trajectory maps the instantaneous hotspot azimuth $\phi$; the black line is a stable circular orbit at $r=11\,M$ for reference. The key features that we can observe are: (1) All loops for $\vec{B}_2$ remain nearly circular, with only slight elongation at the highest inclination, while the $\vec{B}_1$ case shows elongation for increasing $\theta_{\rm{o}}$; (2) the color map becomes increasingly biased to $\phi \sim 180^\circ\text{-- }360^\circ$ (approaching phase) as the inclination increases, demonstrating extreme Doppler boosting that renders the receding phase nearly undetectable; and (3) changing the field configuration results in a rotation of the loop in the $Q$--$U$ plane, evidenced in the low inclination case. The parameters for the four-velocity (Eq. \ref{['eq:four-velocity']}) are $\xi = 0.95$, $\beta_r=0.98$ and $\beta_\phi=0.98$. The inspiral has been stopped after three complete angular revolutions in the geometry.