Periodic equatorial orbits in a black bounce scenario

TL;DR

The paper extends the Levin-Perez-Giz topological classification of equatorial periodic orbits to the Simpson-Visser black bounce family, including its rotating version, by associating each closed orbit with a triplet $(z,w,v)$ and a rational label $q = w + \frac{v}{z}$. It shows that the accumulated azimuth $\Delta \varphi_r$ and the frequency ratio $\frac{\omega_\varphi}{\omega_r}$ map directly to $q$, allowing a complete description of equatorial dynamics across Schwarzschild, Kerr, SV regular black holes, and wormholes. The analysis highlights how the SV parameter $r_{min}$ shifts the effective potential and raises $q$, potentially causing large orbital reconfigurations with small $r_{min}$ changes, and demonstrates that the same classification scheme remains valid for traversing wormholes with an observationally adjusted $q_{obs}$. These results provide a framework for modeling EMRIs and probing the internal structure of compact objects via equatorial geodesics, potentially aiding observational discrimination between black holes and black-bounce alternatives.

Abstract

We study equatorial closed orbits in a popular black bounce model to see if the internal structure of these objects could lead to peculiar observable features. Paralleling the analysis of the Schwarzschild and Kerr metrics, we show that in the black bounce case each orbit can also be associated with a triplet of integers which can then be used to construct a rational number characterizing each periodic orbit. When the black bounce solution represents a traversable wormhole, we show that the previous classification scheme is still applicable with minor adaptations. We confirm in this way that this established framework enables a complete description of the equatorial dynamics across a spectrum of cases, from regular black holes to wormholes. Varying the black bounce parameter $r_{min}$, we compare the trajectories in the Simpson-Visser model with those in the Schwarzschild metric (and the rotating case with Kerr). We find that in some cases even small increments in $r_{min}$ can lead to significant changes in the orbits.

Figures (21)

• Figure 1: Periodic orbit in Schwarzschild black hole (the black circle represents the event horizon) with the triplet $(z = 2, w = 0, v = 1)$. Specifically, this orbit has $L = 4$ and $E = 0.975624$.
• Figure 2: Periodic orbit in Schwarzschild black hole with the triplet $(z = 2, w = 1, v = 1)$. Specifically, this orbit has $L = 3.8$ and $E = 0.975685$.
• Figure 3: Periodic orbits in Schwarzschild black hole. Left: orbit with the triplet $(z = 3, w = 1, v = 1)$, with parameters $L = 3.8$ and $E = 0.975342$. Right: orbit with the triplet $(z = 3, w = 1, v = 2)$, with parameters $L = 3.8$ and $E = 0.975858$.
• Figure 4: The effective radial potential $V_{\rm eff}(r)$ (plotted in a $\log{r}$ scale) for different values of angular momentum $L$. The horizontal line at $V_{\rm eff} = 0.5$ corresponds to the bound energy of $E = 1$. Note that $L_{IBCO} = 4$ and the lower curve have angular momentum $L_{ISCO} = \sqrt{12}$.
• Figure 5: Lower bound of the associated rational $q$ versus the radius of the (co-rotating) stable circular orbit in the Kerr metric, shown for various values of the spin parameter $a$. Above the horizontal line at $y = 1$, all orbits must exhibit whirls, a feature exclusive to strong-field gravity near the black hole. As the ISCO is approached, the minimum diverges due to the merger of the unstable and stable circular orbits. As $a$ increases, the ISCO radius decreases for co-rotating orbits (and increases for counter-rotating orbits).