Spacetime Measurements with the Photon Ring

TL;DR

The paper investigates how higher-order photon-ring images around black holes encode near-horizon spacetime geometry beyond Kerr by introducing three geometry-driven critical parameters $\gamma_p$, $\tau_p$, and $\delta_p$ that govern demagnification, time delay, and rotation of photon subrings. Using the Johannsen metric and a set of non-Kerr spacetimes (Kerr-Newman, Kerr-Sen, Kerr-Bardeen, Kerr-Hayward), it derives how these parameters map onto the image plane via $\xi_p$ and $\mathscr{I}_p$, and introduces image-plane averages $\langle \gamma \rangle_\psi$, $\langle \tau \rangle_\psi$, $\langle \delta \rangle_\psi$ relative to Schwarzschild baselines. The results show that $\langle \bar{\gamma} \rangle_\psi \le 0$ and $\langle \bar{\tau} \rangle_\psi \le 0$ while $\langle \bar{\delta} \rangle_\psi \ge 0$, with Kerr-wide variations of roughly $20\%$, $10\%$, and $60\%$ respectively, and a shadow-size variation of $\lesssim 8\%$, highlighting that $\tau$ is sensitive to inclination and near-extremality, $\gamma$ to charge and spin, and $\delta$ to spin. A joint measurement of these parameters with the shadow radius can break degeneracies between spin and non-Kerr deviations, enabling precise determinations of $a$, $\mathscr{i}$, and $Q$, with practical prospects for ngEHT/BHEX observations of $n=1$ photon rings.

Abstract

We explore the universal symmetries of the black hole photon ring in a wide range of non-Kerr spacetimes, including the Kerr-Newman, Kerr-Sen, Kerr-Bardeen, and Kerr-Hayward metrics. The demagnification exponent ($γ$) controls the size and flux scaling of higher-order images, which appear in the photon ring, the time delay ($τ$) determines the timing of their appearance, and the rotation parameter ($δ$) relates their relative orientations on the image plane. Our investigation reveals that these critical parameters respond distinctly to variations in black hole spin, generalized charge, and observer inclination, establishing them as complementary probes of spacetime geometry: $γ$ is predominantly influenced by charge and spin, $τ$ is strongly affected by inclination, especially for near-extremal black holes, and $δ$ is highly sensitive to spin. Notably, we find that the time delay provides an independent constraint on shadow size for polar observers, while the rotation parameter facilitates metric-independent spin measurements. Specifically, for Kerr black holes, the total variation in $γ$, $τ$, and $δ$ across all possible inclinations and spins is $\lesssim 20\%$, $\lesssim 10\%$, and $\lesssim 60\%$, respectively. By contrast, the Kerr shadow radius varies by only $\lesssim 8\%$. A future joint measurement of these critical parameters -- along with the black hole shadow size -- will enable precise spacetime characterization, including measurements of the spin, inclination, and generalized charge.

Figures (8)

• Figure 1: Fractional deviations of the mean demagnification exponent (left panel), time delay (center panel), and rotation parameter (right panel) of the Kerr, Kerr-Newman, Kerr-Sen, Kerr-Bardeen and Kerr-Hayward BHs from their corresponding values for Schwarzschild BH, for varying observer inclinations. The spin, $a$, of each BH is fixed to $0.5 M$ and the "generalized charge," $Q$, of the non-Kerr BHs is set to $0.4 M$. We find modest variations ($\lesssim 5\%$) in all three critical parameters across the different spacetimes.
• Figure 2: Fractional deviation of the (mean) critical parameters of the Kerr (top row), Kerr-Newman (middle row) and Kerr-Bardeen (bottom row) BHs from their corresponding values for the Schwarzschild BH, as a function of BH spin and observer inclination. The charge of these non-Kerr BHs is set to a middling value of $0.4 M$, as in Fig. \ref{['fig:half spin and extra charge']}. Isocontours of critical parameters with constant values smaller (or larger) than the corresponding values for the Schwarzschild BH are shown by dashed (or solid) lines. The demagnification exponent (left column) is always smaller than its Schwarzschild value; thus, secondary images are larger in non-Schwarzschild spacetimes. For most values of ($a, \mathscr{i}$), the time delay (middle column) is smaller than the Schwarzschild BH value but increases sharply for near-extremal BHs viewed by an equatorial observer, due to the influence of photons orbiting extremely close to the event horizon. The rotation parameter (right column) always exceeds the Schwarzschild BH value and serves as an excellent measure of BH spin, as it remains largely unaffected by the observer's inclination or the specifics of the BH spacetime geometry. For moderate spin values ($a\lesssim 0.5M$), the rotation parameter values remain consistent across all spacetimes, with constant $\langle \bar{\delta} \rangle_{\psi}$ lines appearing nearly vertical and overlapping across different spacetimes. This consistency makes $\delta$ a sensitive and reliable indicator of BH spin. The red vertical line in each figure identify BHs with $a=0.5M$ and $Q=0.4M$, same as considered in Fig. \ref{['fig:half spin and extra charge']}.
• Figure 3: Variation of the fractional deviation in mean critical parameters---demagnification exponent (left column), time delay (center column), and rotation parameter (right column)---for Kerr-Newman BHs with charge and spin, as observed by an observer at a low inclination of $17^\circ$ (top row) and a high inclination of $89^\circ$(bottom row).
• Figure 4: Combining measurements of various critical parameters could pin down physical properties of astrophysical BHs. Shown here are the deviations in the shadow size (blue lines and background color), the demagnification exponent (orange), the time delay (pink), and the rotation parameter (purple) of a Kerr-Newman BH from their Schwarzschild values. The isocontours display 5% (solid), 10% (dashed), 20% (dot-dashed) and 30% (dotted) deviations. The white-shaded region is excluded by the 2017 EHT shadow size measurement of M87$^*$. The isocontours of shadow size and time delay are virtually indistinguishable.
• Figure 5: Similar plot to Fig. \ref{['fig:m87_charge_spin_contour_plot']} but for an observer present close to the equatorial plane. Notably, the time delay and the shadow size display distinct behaviors for Kerr-Newman BHs of different spin and charge for general observer inclinations. Nevertheless, the joint measurement of the shadow size and either of these critical parameters can precisely determine the BH spin and charge.