Shadow structure of generalized $k-n$ black-bounce metrics

TL;DR

We analyze shadows in generalized black-bounce spacetimes with metric $ds^{2}=f(r)dt^{2}-f(r)^{-1}dr^{2}-\Sigma(r)^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$, where $f(r)=1-2M(r)/\Sigma(r)$, $\Sigma(r)=\sqrt{r^{2}+a^{2}}$, and $M(r)=m\Sigma(r)r^{k}/(r^{2n}+a^{2n})^{(k+1)/(2n)}$. We combine semi-analytic photon-sphere analysis to obtain the critical impact parameter $b_{crit}$ and shadow radius, with full ray-tracing using GYOTO and Page-Thorne disk emission to generate realistic images. The results reveal a double-ring shadow structure and radius deformations that depend on $(a,k,n)$, including horizon thresholds $a_{hor}$ and optical thresholds $a_{*}$ that separate BH-like shadows, horizonless two-ring regimes, and no-shadow configurations. The work provides concrete observational discriminants for generalized black-bounce geometries and demonstrates how parameter interplays can constrain modified gravity with current or future high-resolution observations.

Abstract

The existence of black hole shadows is one of the most interesting effects of the strong field regime of general relativity (GR). Recent observations by the Event Horizon Telescope (EHT) have provided high-resolution images of the vicinity of supermassive black holes, ushering in a new era for testing gravitation on astrophysical scales. In this work, we continue the investigation initiated by \cite{furtado2025gravitational}, focusing on shadows associated with generalized $k-n$ \emph{black-bounce} type spacetimes, which smoothly interpolate between regular black holes and wormholes. We consider a generalization of the metric with free parameters $(a,k,n)$ that modify the mass function and enrich the possible phenomenology. We develop a semi-analytical study of photon orbits, obtaining the critical impact parameter and the shadow radius for different parameter combinations. Subsequently, we perform numerical ray-tracing simulations using the \textsc{GYOTO} code, incorporating optically thick accretion disks and varying the observation angle. Our results reveal characteristic signatures, including the formation of double-ring structures and deformations of the shadow radius, which can serve as observational discriminators between classical black holes and \emph{black-bounce} solutions.

Figures (12)

• Figure 1: In the figure, we observe the difference in the shadow radius according to different values of $a$.
• Figure 2: Comparison of the shadow and intensity profile for different values of the bounce parameter $a$, with $k=n=1$ and $m=1$. The top panels show the radial intensity profile, while the bottom panels display the corresponding shadow. (a) The Schwarzschild case ($a=0$). (b) A black-bounce with $a=0.3 \ m$. (c) A black-bounce with $a=0.6 \ m$. (d) A black-bounce with $a=1.0 \ m$. The blue dashed lines indicate the photon ring ($b_{crit}$) and the orange solid lines indicate the peak of the lensing ring.
• Figure 3: Comparison of the shadow and intensity profile for different values of parameter $k$, with $a=0.3 \ m$ and $n=1$ fixed. Increasing the value of $k$ results in a progressive decrease of the shadow radius, $b_{crit}$, and, consequently, of the lensing ring radius, $b_{peak}$.
• Figure 4: Simulation of the shadow and intensity profile for the case where an observable shadow is formed, with $n=4.0$ (and $a=0.3 \ m$, $k=4.0$ fixed). The image displays the characteristic double-ring structure of the black-bounce.
• Figure 5: Analysis of the shadow for different combinations of the parameters $a$, $n$, and $k$. Each panel demonstrates how the interaction between the parameters affects both the shadow size and the separation of the light rings, revealing the complex phenomenology of the spacetime.