Signatures of the Israel Junction II: Double Photon Rings in Slowly Rotating Kerr Spacetime with Thin Shell

TL;DR

Addresses how a thin shell between two slowly rotating Kerr spacetimes affects photon propagation and black hole imaging. By applying the Israel junction conditions in the slow-spin limit, the authors show that while $L$ and $C$ are conserved across the shell, the photon energy $E$ jumps, producing a refraction-like transformation of reduced impact parameters $\eta=L/E$ and $\xi=C/E^2$. Backward ray tracing of a thin equatorial disk reveals novel image features including double photon rings, step-like brightness, and truncated photon regions, with shadow boundaries not in one-to-one correspondence with photon rings. The work offers a framework to test the viability of the Israel junction condition and thin-shell models in astrophysical settings and points to future extensions beyond slow rotation and to other spacetimes such as Reissner–Nordström.

Abstract

Applying the junction conditions to the slowly rotating Kerr spacetime with a thin shell, we find that while the angular momentum $L$ and Carter constant $C$ of the ray remain unchanged upon crossing the shell, its energy $E$ does not. Consequently, the impact parameters $η=L/E$ and $ξ=C/E^2$ of the ray are discontinued at the shell. Utilizing this transformation, we study the shadow of this spacetime and the corresponding images from an equatorial thin accretion disk. The presence of the shell gives rise to distinctive features in the observed images. Notably, we observe distinct double photon rings in the images, which can gradually merge into a single ring. Moreover, the shadow boundaries and the photon rings do not exhibit a one-to-one correspondence. The abrupt changes in redshift factor and the truncated photon regions profoundly influence the image, producing distinctive features such as the step-like structures. These features in shell-equipped spacetimes can help evaluate, through future astronomical observations, the applicability of the Israel junction condition and the shell model in real astrophysical systems.

Figures (8)

• Figure 1: The shadow (a), images (b) and enlarged view of the image (c) and (d) of slowly rotating Kerr black hole with $m_-=0.6,a_-=0.3,m_+=1.13,a_+=0.01,R=2.7$. In (a), the orange curve represents the shadow boundary corresponds to the outer photon region and the blue one corresponds to the inner photon region. The brown dashed curve represents the shadow boundary corresponds to the inner photon region when no thin shell is present. In (b), the dashed orange and blue curve represents the outer and inner photon ring, respectively. The green dashed curves indicates the "step" feature resulting from a discontinuity in the redshift factor.
• Figure 2: (a) The observed intensity of $y=1$. The black, orange and red curves correspond to the intensity contributions from rays intersecting the accretion disk for the first, second and third time, respectively. (b) The trajectory of rays that intersect the accretion disk for the second time. The light blue sphere represents the shell. The orange plane denotes the accretion disk inside the shell, while the gray plane represents the accretion disk outside the shell. Blue rays represent intersections occurring outside the spherical shell, and red and green rays represent intersections inside the shell.
• Figure 3: The transfer function at $y=1$. The black, orange and red curves represent the transfer functions for rays intersecting the accretion disk for the first, second, and third times, respectively.
• Figure 4: The shadow (a), images (b) and enlarged view of the image (c) and (d) of slowly rotating Kerr black hole with $m_-=1,a_-=0.4,m_+=1.2,a_+=0.3,R=3.3$. In (a), the orange curve represents the shadow boundary corresponds to the outer photon region and the blue one corresponds to the inner photon region. The dashed curve represents the hypothetical shadow that would exist if it were not truncated by the shell — a shadow that does not actually form due to the interruption caused by the shell. The brown dashed curve represents the shadow boundary corresponds to the inner photon region when no thin shell is present.
• Figure 5: The shadow (a), images (b) and enlarged view of the image (c) and (d) of slowly rotating Kerr black hole with $m_-=0.5,a_-=0.01,m_+=1.4,a_+=0.2,R=4.3$. In (a), the orange curve represents the shadow boundary corresponds to the outer photon region and the blue one corresponds to the inner photon region. The orange dashed curve represents the hypothetical shadow that would exist if it were not truncated by the shell. The brown dashed curve represents the shadow boundary corresponds to the inner photon region when no thin shell is present. In (b), the dashed orange and blue curve represents the outer and inner photon ring, respectively. The green dashed curves indicates the "step" feature resulting from a discontinuity in the redshift factor.