On the Cuspy Structure of Rotating Wormhole Shadows

Abstract

We investigate the shadow cast by a rotating traversable wormhole in the Teo class endowed with a general redshift function, with particular emphasis on the emergence of cuspy structures. The shadow boundary is the common envelope of two critical orbit families: unstable circular orbits outside the throat and orbits at the throat itself. The formation of cusps, marking the transition between smooth and cuspy shadow boundaries, only becomes possible when the redshift parameter $λ$ is allowed to vary. Moreover, we uncover a universal critical value $λ_c$ that signals the onset of the cusp. A phase diagram characterized by the spin and redshift parameters reveals four distinct morphologies: smooth, cuspy, ears touching, and throat drowning. The morphology of the wormhole shadow may provide observational diagnostics for the different compact objects in future high-resolution imaging observations.

Figures (8)

• Figure 1: The shadow of a compact object in the observer's sky. Bardeen's impact parameters can be related to the observer's local frame $(\hat{r},\hat{\theta},\hat{\phi})$.
• Figure 2: The shadow of a rotating traversable wormhole with $a=0.07$. The shadow boundary is composed of two sets of orbits: the throat orbits (red curves) and the outer unstable circular orbits (blue curves). The real shadow is the shaded region enveloped by the two sets of curves.
• Figure 3: The cusp and swallowtail for the unstable circular orbits, with $a=0.07$ and $\lambda=0.8$. The outer unstable circular orbit (blue) and throat orbit (red) meet at the purple dot, with the same slope $\mathcal{F}_{\text{out}}=\mathcal{F}_{\text{throat}}$. The cusp is illustrated by the red dot.
• Figure 4: The rotating traversable shadow with $\lambda$ (with $a=0.07$). For $\lambda\geq \lambda_{\text{drown}}$, with increasing $\lambda$, the wormhole throat orbit first detaches from the shadow, and then becomes tangent with the shadow at $\lambda=5.41$. For even larger $\lambda$, the shadow boundary comprises the throat orbits and the outer unstable circular orbits.
• Figure 5: Phases of the shadow boundary illustrated with different $a$ and $\lambda$. For $\lambda<\lambda_c$, the shadow boundary is smooth. For larger $\lambda$, there are three different phases: cuspy shadow, ears touching, and throat drowning. The blue dotted line represents $\lambda=2$, and the red dotted line is $a=0.07$.