Imaging Signatures of the Israel Junction: Photon Ring Evolution in Dynamical Thin Shell Schwarzschild Spacetimes

TL;DR

The study investigates imaging in spacetimes formed by gluing two Schwarzschild regions across a spherically symmetric thin shell via the Israel junction conditions. It derives the shell dynamics, the null-geodesic transmission (refraction) laws, and the redshift accumulation across multiple shell crossings, then performs ray tracing of a geometrically thin accretion disk for both static and collapsing shells. The static case reveals redshift cusps at the shell, a refraction-induced V-shaped transfer function, and a nontrivial mapping between photon spheres and photon rings, with photon-ring features not strictly determined by the presence of external photon spheres. In the dynamical case, light-travel delays and shell motion produce step-like intensity features and generally suppress a robust double photon-ring appearance, except in finely tuned configurations where a genuine double ring can briefly occur; these findings offer observational handles to test Israel junctions in strong gravity.

Abstract

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.

Figures (14)

• Figure 1: Schematic effective potential $V(R)$ for the thin spherical shell. The solid black curve shows $V(R)$; the horizontal dashed lines in red, blue, and green mark $e^{2}$ corresponding to the cases $e^{2}\!<\!1$, $e^{2}\!=\!1$, and $e^{2}\!>\!1$, respectively. For $e^{2}<1$, the shell undergoes bound motion: it can reach at most the turning point $R_{\max}$, i.e., the intersection of the red dashed line with the black curve $V(R)$.
• Figure 2: Photon sphere structure as a function of the inner Schwarzschild mass $m_{-}$ and the shell radius $R$. We choose $m_{+}=1$. Region I contains only the outer photon sphere at $r_{\rm ph}^{+}=3m_{+}$; Region II contains two photon spheres at $r_{\rm ph}^{+}=3m_{+}$ and $r_{\rm ph}^{-}=3m_{-}$; Region III contains only the inner photon sphere at $r_{\rm ph}^{-}=3m_{-}$.
• Figure 3: For parameters $m_-=0.1$ and $R_{\rm sh}=3.1$. (a) Observed intensity $I_{\rm obs}$ versus impact parameter $b$. (b) Transfer function $r(b)$ for rays with different numbers of disk crossings. (c) Corresponding redshift factors $g_{n}(b)$. In panels (b) and (c), black, orange, and red curves denote rays that intersect the disk once, twice, and three times, respectively.
• Figure 4: Left: ray trajectories for $m_-=0.1$ and $R_{\rm sh}=3.1$ that intersect the accretion disk twice. The orange trajectory does not cross the shell, while the blue trajectory does. The blue dashed circle marks the shell location, the black circle marks the inner Schwarzschild horizon, and the vertical black dashed line marks the accretion disk. Refraction at the shell is clear along the blue ray: rays with smaller impact parameter $b$ reach a larger radius $r$ at their second intersection with the disk. Right: black hole image corresponding to the intensity in Fig. \ref{['mminus0.1,mplus1,ms3.1Refractioneffect']}(a). The bright central ring arises from the inner Schwarzschild photon sphere. A broader and fainter outer ring corresponds to the peak at large $b$ in Fig. \ref{['mminus0.1,mplus1,ms3.1Refractioneffect']}(a). It is not associated with an external photon sphere.
• Figure 5: intensity profiles and transfer function $r(b)$ for several shell radii $R_{\rm sh}$ at fixed $m_-=0.1$.