Universal polarimetric signatures of the black hole photon ring

Black hole images present an annular region of enhanced brightness. In the absence of propagation effects, this “photon ring” has universal features that are completely governed by general relativity and independent of the details of the emission. Here, we show that the polarimetric image of a black hole also displays universal properties. In particular, the photon ring exhibits a self-similar pattern of polarization that encodes the black hole spin. We explore the corresponding universal polarimetric signatures of the photon ring on long interferometric baselines, and propose a method for measuring the black hole spin using a sparse interferometric array. These signatures could enable spin measurements of the supermassive black hole in M87, as well as precision tests of general relativity in the strong field regime, via a future extension of the Event Horizon Telescope to space.

Figure 1

Ray-tracing into a geometrically thick, optically thin bulk matter distribution. Light rays near a point $(\tilde{\alpha }(\tilde{r}),\tilde{\beta }(\tilde{r}))$ on the critical curve are nearly bound at the photon shell radius $r=\tilde{r}$ and can complete multiple orbits through the emission region. Even/odd half-orbits travel through the same matter distribution at the same inclination, and hence are loaded with photons making identical contributions to the Stokes parameters of the ray. Light rays executing $n\in [n_{-},n_{+}]$ half-orbits appear within a narrow window $\mathrm{\Delta }d\sim e^{-\gamma (n_{+}-n_{-})}$ near the critical curve on the screen.

Figure 2

The photon ring and its universal substructure: a self-similar sequence of subrings labeled by half-orbit number $n$ and exponentially converging to the critical curve (black). The subrings are exponentially demagnified images of the main emission appearing at perpendicular distance from the critical curve $d_{n+1}\sim e^{-\gamma }{d}_n$. Hence, on a logarithmic scale, the subrings have equal width and equal spacing. Even/odd subrings display the same polarization $\overrightarrow{\mathcal{E}}$, depicted with green/purple ticks and corresponding to the contributions collected on the green/purple passes through the matter region depicted in Fig. 1. Importantly, the polarization alternates between even and odd subrings in a manner that reflects the black hole spin.

Figure 3

Schematic showing polarized visibility amplitude as a function of baseline length for a photon ring with $d=40\mu \text{as}$. Visibility amplitudes are shown for the complete image (black) as well as for individual subrings ($n=0$: blue, 1: green, 2: red). Progressively longer baselines are dominated by visibilities for a subring with correspondingly larger index $n$. The visibility amplitudes in each range are determined by a pair of complex coefficients, $P_{\pm }^{\mathcal{O}}$ or $P_{\pm }^{\mathcal{E}}$ depending upon whether $n$ is odd or even. The periodicity of the visibility amplitudes gives the subring diameter in the baseline direction, while the coefficients $P_{\pm }^{\mathcal{O}}$ and $P_{\pm }^{\mathcal{E}}$ carry information about the black hole spin and azimuthal brightness asymmetry.

Figure 4

The offset function $\mathrm{\Lambda }(a,{\theta }_{\mathrm{o}})$ is a bijective function of the black hole spin parameter $a$. We set $M=1$ in this plot.