Imaging signatures of the Israel junction: Photon ring evolution in dynamical thin shell Schwarzschild spacetimes

We study the images of black holes by gluing two Schwarzschild spacetimes with a thin shell where the Israel junction conditions are satisfied. By studying the refraction law for null geodesics at the spherical shell, and taking account of the light travel time delay, the images are obtained by ray tracing a geometrically and optically thin accretion disk. For a static shell, we identify three signatures: a redshift cusp at the shell, a V-shaped profile of the transfer function $r(b)$, and a loss of the one-to-one correspondence between photon spheres and photon rings on the observer's screen. During the collapse of the shell, the spacetime evolves from a stage with a single photon sphere inside the shell, through an intermediate stage with double photon spheres, and finally to a spacetime with a single photon sphere outside the shell. However, when the shell is released from a large distance, the corresponding images never show two separate photon rings, even in the stage with two photon spheres. In addition, the motion of the shell leads to a discontinuity in the redshift factor. These signatures provide a practical basis for testing the Israel junction in black hole spacetimes.