Images and photon regions of continuous photon sphere spacetime

TL;DR

We address imaging of spacetimes with continuous photon spheres (CPS), focusing first on a self-gravitating, isotropic, spherically symmetric CPS core and then on a rotating CPS geometry. For static CPS cores surrounded by thin or spherical accretion, we show that thin-disk images are practically indistinguishable from Schwarzschild images, and, in the static case, the observed image profiles factorize into a universal $b_c$-dependent form that is independent of the emissivity profile or gravity theory, differing only by an overall normalization. When accretion is infalling rather than static, this universality breaks as Doppler shifts introduce $b$-dependent factors tied to the emission model. For rotating CPS spacetimes, the photon region in a constant-$\phi$ slice consists of one or two angular sectors with all unstable photon orbits sharing the same $b$ and reduced Carter constant $K_E$, offering observational signatures that distinguish CPS from Kerr.

Abstract

We study images of spacetimes containing continuous photon spheres (CPS). For a self-gravitating, isotropic, spherically symmetric spacetime with CPS, we find that a thin accretion disk produces images that closely resemble those of a Schwarzschild black hole, despite significant differences in photon dynamics. More generally, for any static, pherically symmetric spacetime with a luminous CPS core, the image profile is universal: members of this class produce identical image shapes, differing only by an overall normalization factor. This universality is, however, sensitive to the nature of the accretion flow and breaks down for spherically symmetric infalling accretion, where Doppler shifts and non-static emission introduce image features that depend on the flow dynamics and the metric. Finally, we investigate photon regions in a rotating CPS spacetime and find that unlike in Kerr spacetime, the photon region appears as one or two angular sectors in a constant-$φ$ cross section. These distinctive photon region properties could produce observable signatures that distinguish rotating CPS spacetimes from the Kerr one.

Figures (10)

• Figure 1: The ray trajectory with $r_+=3M$. The dashed circle is $r=3M$. The green curve and blue curve represent rays with $b>b_c$. The orange curve represents the ray with $b=b_c$. The red curve represents the ray with $b<b_c$.
• Figure 2: The observed intensities (left column) and images (right column) of Schwarzschild black hole (top) and the SSSGI naked singularity with a CPS core for $y=3$ (bottom). $I_0$ is the maximum emitted intensity.
• Figure 3: The observed intensity (left) and image (right) of the SSSGI with a CPS core for $y=3.5$.
• Figure 4: The ray trajectory with $r_+=2.5M$. The dashed circle is $r=3M$ corresponding to the photon sphere with $b = b_{c, Sch} = 5.20M$. It is an isolated photon sphere. The gray region is the CPS ($r\leqslant2.5M$). Every circle in this region, such as the magenta curve, is a photon sphere. The red curve represents the ray with $b>5.20M$. The green curve represents the ray with $5.20M< b<5.59M$. The blue curve represents the ray with $b<b_c$.
• Figure 5: The observed intensity (left) and image (right) of the SSSGI naked singularity with a CPS core for $y=2.5$.