Overlapping of photon rings in black hole imaging

TL;DR

This work introduces the matrix of merging, a universal, metric-dependent framework for predicting overlaps among photon rings around black holes in static, spherically symmetric spacetimes. By modeling a geometrically thin equatorial disk with a single inner radius, the authors define radii of merging $r_{nn'}$ and organize them into an upper-right triangular matrix that serves as a spacetime signature. They derive general properties of the matrix, show that certain overlap patterns are universally forbidden, and demonstrate the approach with Schwarzschild and Reissner–Nordström examples, including strong deflection-limit analytics for higher-order rings. The methodology enables constraining either the spacetime metric or the accretion model from observed photon-ring overlaps, providing a principled tool for interpreting future high-resolution black hole images. The results have direct implications for testing gravity in the strong field and for informing models of accretion in ultracompact objects.

Abstract

In this paper, we investigate the overlapping of photon rings - higher-order images of a black hole's luminous environment, concentrated near the shadow boundary and expected to be resolved in future observations. We consider a broad class of static spherically symmetric spacetimes and geometrically thin equatorial accretion disk with a prescribed inner radius and infinite outer extent, viewed by a polar observer. Depending on the inner radius of the disk, the thickness of each photon ring varies, and the rings may or may not overlap. To characterize the overlapping, we introduce the radius of merging - the value of the disk's inner radius at which two photon rings of given orders begin to overlap. Since each radius of merging is labeled by two indices corresponding to the image orders, it becomes possible to arrange these radii in the form of an infinite-dimensional matrix where only the upper right-hand corner is filled. This matrix, which we call the "matrix of merging", is a signature of spacetime only, and, once known, it provides a qualitative understanding of the overlapping pattern for any chosen value of the inner radius of the disk. Remarkably, the matrix of merging exhibits several universal properties that hold for all spherically symmetric metrics and can be established even without explicit calculation of light trajectories. Based on these properties, we demonstrate that certain overlapping patterns are universally forbidden across all such spacetimes and for any inner radius of the disk. Examples for the Schwarzschild and Reissner-Nordström black holes are provided. The main application of our study is constraining the spacetime metric and the accretion model using observed photon ring overlaps.

Figures (9)

• Figure 1: Definition of the radius of merging (Subsec. \ref{['subsec:radius-merging-def']}), illustrated through the example of merging primary and secondary images. The left panels show an equatorial accretion disk and a polar observer, along with the light rays that form the borders of primary and secondary images ($n = 0$ and $n = 1$ photon rings, respectively). The corresponding images, as they appear on the observer’s sky, are shown in the right panels. In our simplified disk model, the angular thickness of each image, and therefore the overlap of images, depends only on the value of the inner radius of the accretion disk, $r_S^{\mathrm{in}}$. In the upper panels, the inner radius $r_S^{\mathrm{in}}$ is large enough, such as the primary and secondary images are relatively thin and remain separated. If $r_S^{\mathrm{in}}$ is smaller, all images become thicker, since their inner boundaries shift closer to the center, while their outer boundaries remain fixed. In particular, the shrinking inner boundary of the primary image approaches the fixed outer boundary of the secondary image. The radius of merging, denoted $r_{01}$, is defined as the value of $r_S^{\mathrm{in}}$ at which the inner edge of the primary ($n=0$) image just touches the outer edge of the secondary ($n=1$) image. This configuration is shown in the lower panels. Note that, for visualization purposes, the photon rings are not drawn to scale. In particular, throughout the paper, the outer radius of the disk is assumed to be infinite. The lower left panel demonstrates that the conditions (\ref{['eq:mergingcondition-theta']}) and (\ref{['eq:mergingcondition']}) cause the final segment of the ray forming the outer boundary of the secondary image coincide with the trajectory of the ray forming the inner boundary of the primary image (see the text for more details).
• Figure 2: Graphical determination of the relationship between the elements of one column in the matrix of merging \ref{['eq:rmatrix-general']}. Each panel shows a light ray emitted from the outer edge of the accretion disk and forming the outer boundary of an image of order $n'$: $n'=1$ (upper left), $n'=2$ (upper middle), $n'=3$ (upper right), and $n'=4$ (lower panels, where the right panel is a zoom-in of the left). The radial coordinates of the points where the ray intersects the equatorial plane (marked with open circles) correspond to the radii of merging in the $n'$-th column of the matrix (see text for a detailed explanation). All trajectories are found numerically in the Schwarzschild metric. For visualization purposes, the outer radius of the accretion disk is set to $15m$ (instead of being infinite, as assumed in our model), and the observer is also placed at $15m$, ensuring symmetry of the ray paths. Due to the symmetry of each ray with respect to the diagonal line, the relationships between the matrix entries in a given column can be easily found based on their angular "separation" from the point of closest approach, shown by the filled red dot. In particular, regardless of the column number $n'$, the value of $r_{0n'}$ is the largest in that column.
• Figure 3: Overlapping patterns of the first three images for different inner radii of the accretion disk, including forbidden cases. Images are shown separately for clarity. Vertical lines indicate the inner boundaries of the $n=0$ and $n=1$ images, making it easier to identify overlaps visually. The upper row of panels shows overlapping patterns where $n=1$ and $n=2$ images are separated: (left) all images separated; (middle) $n=0$ image overlaps with $n=1$; (right) $n=0$ overlaps with both $n=1$ and $n=2$. The lower row of panels shows patterns where $n=1$ and $n=2$ overlap: (left) $n=0$ overlaps with both; (middle) $n=1$ and $n=2$ overlap, but $n=0$ does not overlap with $n=1$ and, consequently, not with $n=2$ --- forbidden; (right) $n=0$ overlaps with $n=1$ but not with $n=2$ --- forbidden.
• Figure 4: $\Delta \tilde{\phi}$ as a function of the impact parameter $b$ for the metric (\ref{['eq:Example']}). The source is at $r_S = 6m$, $\vartheta _S = \pi/2$ and the observer is at $r_O = \infty$, $\vartheta _O = \pi$. Then there are three images of order 1 whose impact parameters $b_1$, $b'_1$ and $b"_1$ are marked in the picture.
• Figure 5: $\Delta \tilde{\phi}$ as a function of $b$ for the Schwarzschild spacetime with $r_S=8m$ and $r_O = \infty$.